Policy Research Working Paper 10782 Production Networks and Firm-level Elasticities of Substitution Brian C. Fujiy Devaki Ghose Gaurav Khanna Development Economics Development Research Group May 2024 Policy Research Working Paper 10782 Abstract This paper provides one of the first estimates of elasticities quality of institutions, input specificity, inventories, and of substitution across suppliers within the same product. time horizons explain the low elasticity. These firm-level This paper estimates these elasticities using new real-time complementarities amplify the propagation of negative administrative tax data on firm-to-firm transactions, with shocks through production networks, and make con- product-level prices and quantities, leveraging geographic nected firms important for shock propagation. In policy and temporal variation from India’s Covid-19 lockdowns counterfactuals, the paper shows that given these comple- to derive causal estimates of these elasticities. Suppliers are mentarities, allowing more connected firms to operate in highly complementary even at this granular level, with an the face of shocks mitigates output declines non-linearly estimated elasticity of $0.55$. The paper shows that the with the size of the productivity shock. This paper is a product of the Development Research Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at dghose@worldbank.org. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team Production Networks and Firm-level Elasticities of Substitution* Brian C. Fujiy† Devaki Ghose‡ Gaurav Khanna§ Keywords: production networks, elasticities of substitution, shock propagation, resilience JEL Codes: D57, E32, E61, F10, L14 * We thank participants at the Danish International Economics Workshop, AEA 2024, World Bank, CEPR’s Glob- alisation, Strategic Autonomy, and Economic Resilience in Time of Crisis, Annual Mid-Atlantic International Trade Workshop, Midwest International Economic Development Conference, STEG’s Annual Conference, IMF’s Jacques Polak Annual Research Conference, Bank of Italy’s Global Value Chains, George Washington University, and Lund University for their comments and suggestions. We thank David Baqaee, Christoph Boehm, Ana Cecilia Fieler, Banu Demir, Yuhei Miyauchi, Pablo Garriga, Andrei Levchenko, Sebnem Kalemli-Ozcan, Scott Orr, Ludovic Panon, Car- olina Sanchez, Ritam Chaurey, and Sebastian Sotelo for insightful discussions. We thank Ahmed Magdy and Eli Mogel for their excellent research assistance and Sushil Khanna for guidance on the data and details on the analysis. The authors gratefully acknowledge support by the Umbrella Facility for Trade, which is funded by the governments of The Netherlands, Norway, Switzerland, Sweden and the United Kingdom, and the Research Support Budget of the World Bank. The views expressed in this paper do not represent the views of the World Bank or its partner organizations and solely represent the authors’ personal views. † U.S. Census Bureau, brian.j.cevallos@census.gov. ‡ The World Bank, dghose@worldbank.org. § University of California, San Diego, gakhanna@ucsd.edu. 1 I NTRODUCTION The ability of firms to substitute inputs across suppliers is critical for the resilience of supply chains and the transmission of supply shocks. If firms are unlikely to substitute across suppliers, shocks could amplify by transmitting further downstream and upstream through the supply chain. The importance of this mechanism was reflected during the Covid-19 pandemic, where supply chain disruptions drove dramatic reductions in GDP worldwide. For instance, India reported a −7.3% growth rate for the 2020/21 financial year, one of the most significant contractions worldwide and the largest decline in GDP since India’s independence.1 In this paper, we quantify the importance of firm-level elasticities of substitution across sup- pliers (firms) of the same product category to explain large fluctuations in GDP. We provide new estimation strategies and estimates for these elasticities by leveraging regional variation in supply- side shocks induced by the Indian government’s massive lockdown policy. We explore explana- tions behind the magnitude of our estimated elasticity, showing that contract specificity of inputs, the level of inventories, and response time horizons matter. We show that this elasticity is key to partly explaining the large income declines during the Covid-19 pandemic. Using new big data computational techniques, we quantify this decline in GDP by directly leveraging information on the economy-wide firm-to-firm network, and highlight how protecting more connected firms miti- gates output declines. We pose two main research questions. First, are suppliers of intermediate inputs within a product category complements or substitutes?2 The degree of complementarity or substitutabil- ity determines how shocks propagate through supply chains. We expect shocks to propagate less across firm networks if input suppliers are substitutable. However, if input suppliers are com- plements, the effects of adverse shocks can easily propagate through buyer-supplier networks. Second, we ask how this newly estimated elasticity affects firm-level sales, and ultimately GDP, by amplifying negative shocks through firm-level input-output linkages, and how the effect of the shock depends on the connectivity of affected firms. Two unique features of our setting allow us to answer these questions credibly. First, India had a distinct mosaic of lockdown policies, whereby the roughly 600 districts were classified into three different zones with varying degrees of restrictions. This allows us to isolate variation in the 1 https://www.economicsobservatory.com/how-has-Covid-19-affected-indias-economy. More broadly, during the 2020/21 financial year, GDP fell by −3.3% in emerging market economies, and by −2.2% in developing countries. 2 Complementarity in this setting means that a buyer’s expenditure share for a given supplier rises with the input price of that same supplier in the short run. That is, the buyer is unable to replace a sufficient amount of the input from the high-cost supplier with one from the lower-cost substitute to lower the expenditure share attributed to the high-cost supplier. 1 ability to trade and transport goods over this period, thereby inducing variation in input prices at the supplier level. Second, we obtain new granular and high-frequency administrative data on the universe of firm-to-firm transactions for a state in India, with unique information on unit values and HS-product classifications. This previously unused data on prices and purchase quantities at the buyer-supplier level, combined with exogenous variation in prices induced by the lockdown, allows us to estimate new elasticities at the firm level and across different suppliers of a product. We first document whether the lockdown indeed induced variation in prices we need to es- timate these elasticities. We begin by presenting reduced-form evidence that leverages the nation- wide, sudden, and unprecedented lockdown imposed by the Indian government in March 2020. Importantly, these lockdowns induced geographic variation: districts were categorized into Green (mild lockdown), Orange (medium lockdown), and Red (severe lockdown). Since the lockdowns were sudden and unexpected, they were likely implemented independent of economic fundamen- tals and induced strong variation in transactions between firms across India.3 We show, conditional on high-dimensional fixed effects, that these adverse supply shocks led to a sharp increase in unit values (prices), and dramatic fall in transactions of intermediate inputs, if either buyers or sellers were located in high lockdown zones. We estimate the elasticity of substitution across suppliers leveraging the variation in prices we document in our reduced form. To do this, we construct a standard multi-sector firm-level model of input-output linkages with a production function augmented with substitution across sup- pliers within the same product category. Our model is intentionally parsimonious to fix ideas, but importantly, unlike standard trade models with Constant Elasticity of Substitution across products, it does not assume that suppliers of the same product are perfect substitutes. We derive analyti- cal expressions that connect the relative values of quantities purchased of the same product from different suppliers to the equilibrium relative prices. A combination of this analytical framework, and the lockdown-induced variation in sellers’ marginal costs and transportation costs allows us to estimate how substitutable the different suppliers are, within each product category. Yet, Covid-19 was not just a supply shock. The pandemic outbreak was a combination of exogenous shocks to the quantities of factors supplied, the productivity of producers, and the composition of final demand by consumers across industries (Baqaee and Farhi, 2020). To estimate the elasticity of substitution across suppliers of inputs, we leverage variation in input prices driven by the sudden restrictions in economic activity in lockdown districts where these suppliers were located. In addition, we leverage variation in trade costs arising from transportation restrictions in districts through which the goods need to pass from the seller to the buyer. While our instruments help derive the necessary variation, to further isolate supply shocks from other shocks, we control 3 See the BBC (link), and The Wire (link) 2 for an entire array of high-dimensional fixed effects, such as buyer-by-product-by-time level fixed effects to account for demand-side shocks, and product-specific shocks. Given the richness of our data, we can also include seller-by-product fixed effects, and control for other factors, such as firms’ exposure to foreign shocks transmitted through trade (Hummels et al., 2014), and the caseload and severity of Covid-19 cases. We find that suppliers within the same HS-4 product category are highly complementary. Our estimated elasticity of substitution across suppliers of the same product is 0.55. In various speci- fication tests employing different combinations of fixed effects and different sources of variation, we find that the estimated elasticities lie within a range of 0.50 to 0.66. Our new elasticities show that inputs across firms are highly complementary even within the same HS-4 product category. The elasticities are similar at the HS-6 and HS-8 product level.4 As a result, second-order effects amplify adverse firm-level shocks by propagating through the network, leading to an additional 25 USD per capita per quarter GDP loss compared to if firms were substitutes. Additionally, we also estimate a more aggregate elasticity of substitution across different industries of 0.69, which implies complementarity across industries as in Atalay (2017) and Boehm et al. (2019). A challenge in the estimation of this elasticity arises if sellers and buyers decide to trade due to unobservable reasons. To account for this extensive margin, we leverage variation from coun- terfactual prices based on transactions that did not happen in a particular period. Since variation in prices that arise outside of a seller-by-buyer relationship are arguably orthogonal to seller-by-buyer unobservables, we are able to account for this concern. Given we uniquely see prices that sellers and buyers trade with other sellers and buyers, we can construct counterfactual prices. We expand our estimating sample to include all counterfactual suppliers of a buyer, using counterfactual prices charged by the supplier to other buyers in the same time period.5 We continue to find that across all specifications, there is a high level of complementarity. But why, in the short run, do buyers not substitute much away from high-cost suppliers? We investigate several candidate mechanisms. First, we find that sellers that offer differentiated intermediate inputs exhibit higher complementarity. This lends support to the hypothesis that buy- ers may not be able to substitute because inputs are often sourced from specific suppliers and customized, rather than bought “off-the-shelf" (Elliott and Golub, 2022). Second, we find that in- termediate inputs purchased from buyers in locations with better institutional quality display higher complementarity. This reflects the importance of contract enforcement for the trade of more differ- 4 We use the term “product" and “product category" interchangeably and define whether we refer to HS-4, HS-6, or HS-8 codes when relevant. We use the term “industry" to refer to the broad HS Sections. 5 For the construction of these counterfactual prices to be relevant, the prices charged by a seller in a particular time should be approximately the same across all buyers, that is, the counterfactual price will be difficult to measure if the seller charges differential prices across buyers within the same year-month. We show evidence for this in section 5.1 and in Figure A3 in the appendix. 3 entiated intermediate inputs. Third, in line with the previous literature, we find that the estimation of this elasticity is necessarily tied to the time horizon (Ruhl et al., 2008; Peter and Ruane, 2022). Allowing buyers a longer time to substitute—e.g., a quarter rather than a month—increases the value of the elasticity of substitution. Given the expectations that many large shocks may be short- lived, our estimates are relevant for short-run shocks rather than longer-term structural changes. Finally, we find that the level of inventory in the buyer’s industry helps to substitute inputs to some degree. When buyers are in industries with high levels of inventory, inputs across suppliers of the same product are less complementary. This is because the buyer can temporarily substitute a costly input from a seller with its input inventory holdings. As our model framework is intentionally parsimonious, we recognize that our elasticity esti- mates should be thought of as an empirical average over industries, locations, and types of firms. That is, we do not think that the elasticity is a fixed exogenous parameter that applies to every buyer. Rather, it likely depends on various factors, such as the contracting structure and the en- forcement of contracts, the level of inventories by industry, and the types of products produced. This is why our heterogeneity analysis adds much nuance to our understanding of why (on average) we find low substitutability. In the final part of our paper, we embed the estimated elasticities in our model and quantify how input complementarities at the firm level affect aggregate economic outcomes. Considering a firm’s full network is a highly computationally demanding task. In our case, it involves inverting a 94,555 by 94,555 input-output matrix.6 We apply state-of-the-art techniques from computer science to our input-output matrix to derive the full connectivity of each firm. Compared to our baseline case ( = 0.55), we find that the quarterly fall in GDP induced by a negative 25% shock to firms in Red zones would be 2.68 pp less in a model where firms in the same HS-4 product category are substitutes ( = 2) and 0.99 pp more when they are almost Leontief ( = 0.001).7 The additional losses due to firm-level complementarities translate into a GDP loss of 870 million USD, which is about 25 USD per capita per quarter. Additionally, under our estimated elasticity, the shock to Red zone firms spreads widely through the network to other parts of the economy. In our policy counterfactuals, we quantify the importance of firm connectivity separately from firm size and the effects of allowing the most connected firms (measured by all direct and indirect connections) to operate during lockdowns as opposed to allowing the most directly con- nected firms to operate. In policy and academic circles, much importance has been paid to large firms, as Hulten (1978) emphasized the importance of firm sizes in the propagation of shocks through production networks. We show that controlling for size, the fall in GDP is much larger if 6 As a reference, the typical sector-level input-output matrix from the BEA 2012 is 405 by 405. 7 We find that a 25% productivity shock to firms in Red zones reduces GDP by 10.95%. As an empirical benchmark, the state’s annual GDP fell by 11.3% in 2020/21. 4 the most connected firms are affected compared to the least connected or a random set of firms. The importance of the most connected firms increases non-linearly with the size of the negative productivity shock and decreases as firms become more substitutable. Finally, we quantify how important it is to also consider a firm’s indirect connectivity and find that under our elasticity and a negative productivity shock of 25%, the fall in GDP would be 2.56 pp less if the most connected firms were allowed to operate during the lockdowns compared to only allowing the directly con- nected firms (counting only the number of direct buyers) to operate.8 We see that as the shock gets larger, the difference in aggregate GDP between these two experiments rises, emphasizing the importance of measuring a firm’s indirect connections as well. Related Work. Our paper contributes to two strands of the literature. First, we contribute to the literature on shock propagation and amplification through supply chains and production net- works (Barrot and Sauvagnat, 2018; Carvalho et al., 2021; Peter and Ruane, 2022; Boehm et al., 2019; Korovkin and Makarin, 2020; Ferrari, 2022; Dew-Becker, 2023; Huneeus, 2018; Arkolakis et al., 2023).9 A crucial parameter determining the degree of shock propagation through supply chains is the firm-level elasticity of substitution across suppliers within the same product category. Existing work has estimated firm-level elasticities of substitution between product categories or between domestic and foreign industries (Peter and Ruane, 2022; Carvalho et al., 2021; Boehm et al., 2019; Atalay, 2017). The challenge highlighted by this literature in estimating the elasticity across suppliers within a product category lies in the fact that most firm-to-firm datasets do not contain product-level (unit) prices from each supplying firm to a buying firm, or lack the required variation in such prices to estimate these elasticities.10 The lack of firm-level elasticities across suppliers has so far constrained our assessment of the importance of nodal firms, such as the most connected firms, in the propagation of shocks through production networks. We contribute to the literature in each of these dimensions. First, we measure unit val- ues (prices) and quantities at the transaction (seller-buyer-product-time) level. We derive price changes from supply and transportation disruptions in lockdown-affected districts and estimate the firm-level elasticity of substitution between suppliers within a product category. We then quantify how this elasticity amplifies firm-specific supply shocks through a roundabout production network 8 A firm’s indirect connections measure not only the number of direct buyers of a supplier but also the buyers’ buyers and their buyers, and so on. 9 See Bernard and Moxnes (2018) and Carvalho and Tahbaz-Salehi (2019) for a comprehensive literature review. 10 Carvalho et al. (2021) observe a binary measure of whether firms were connected via buyer-supplier relationships rather than quantities and unit values associated with such transactions. They use a proportionality assumption, which precludes estimating the elasticity of substitution across suppliers within a product category, as a buyer sourcing from two suppliers in the same industry will source the same amount given the assumption. Although lacking firm-to-firm price data, Dhyne et al. (2022) structurally estimate a similar elasticity in the context of imperfect competition models where they restrict the elasticity to be larger than 1 for mark-ups to be relevant. 5 (Baqaee and Farhi, 2019).11 Finally, given the presence of complementarity across suppliers of the same product, we address previously unanswered questions on the importance of firms’ overall connectivity within a production network for the amplification of shocks. We leverage computa- tional innovations in big data to compute the second-order effects of productivity shocks using the entire matrix of production linkages between firms and industries. This innovation helps quan- tify the non-linear effects of productivity shocks directly using the network without relying on approximations using final sales.12 Our paper is also related to research on trade collapses during adverse shocks (Behrens et al., 2013; Giovanni and Levchenko, 2009; Bricongne et al., 2012), and shock transmission through GVCs during the Covid-19 pandemic via disruptions to imports, exports, or aggregate production (Bonadio et al., 2021; Baqaee and Farhi, 2020; Cakmakli et al., 2021; Demir and Javorcik, 2020; Gerschel et al., 2020; Heise, 2020; Lafrogne-Joussier et al., 2022; Bas et al., 2023; Chakrabati et al., 2021; Khanna et al., 2022; Miranda-Pinto et al., 2022). In contrast, we analyze how domestic transactions were affected during Covid-19 lockdowns in a large developing country. Our policy motivation stems from the observation that policymakers worldwide are interested in quantifying the trade-off between strict lockdowns that affect GDP through complex buyer-seller networks and more lenient measures that increase production and trade. More importantly, even beyond the Covid-19 crisis, our estimates of how substitutable suppliers are within a product category will help policymakers quantify the economy-wide effects of any disruptive events (e.g., natural disasters or sanctions) on trade and production that are expected to be reasonably short lived.13 2 DATA AND C ONTEXT Firm-to-firm trade. Our primary data source is daily establishment-level transactions with dis- tinct information on establishment locations.14 This data is provided by the tax authority of a large Indian state with a diversified production structure, around 50% urbanization rates, and high lev- els of population density. To benchmark the size of this Indian state to other firm-to-firm trade datasets, the population of this state is roughly three times the population of Belgium, seven times the population of Costa Rica, and double the population of Chile. The data contains daily transactions between all registered establishments in this state and 11 Prior work has highlighted the aggregate implications of supplier churn (Baqaee et al., 2023; Khanna et al., 2022). We focus on the substitution among suppliers given a fixed set of suppliers as in Baqaee and Farhi (2019). 12 As firm-to-firm data become common (Panigrahi, 2021; Demir et al., 2024; Dhyne et al., 2021; Alfaro-Urena et al., 2022; Chacha et al., 2022), our methods can be used to quantify shock propagation through complex networks. 13 We may hesitate to use these elasticities for exercises on long-term structural transformations. 14 While we use the term “firm" throughout the paper, our data is actually at the more granular establishment level, and we can identify the parent firms for each establishment as well. 6 all registered establishments in India and abroad from April 2018 to October 2020. This data is collected by the tax authority’s E-way Bill system to increase compliance for tax purposes. This is an advantage over standard VAT firm-to-firm trade datasets in developing countries, which suffer from severe under-reporting. By law, anyone dealing with the supply of goods and services whose transaction value exceeds Rs 50,000 (700 USD) must generate E-way bills. Transactions with values lower than 700 USD can also be registered, but it is not mandatory. The E-way bill is generated before transport (usually via truck, rail, air, or ship), and the vehicle driver must carry the bill with them, or the entire extent of goods can be confiscated. Our data is generated from these bills. This implies that our network is representative of relatively larger firms, but the threshold is sufficiently low that we are likely capturing small firms as well. Each transaction reports a unique tax code identifier for both the selling and buying estab- lishments, all the items contained within the transaction, the value of the whole transaction, the value of the items being traded up to 8-digit HS codes,15 quantity of each item, unit, and mode of transportation. Each transaction also reports the ZIP code of both the selling and buying firms, which we use to merge with other district-level datasets. Since the data report both value and quantity of traded items, we construct unit values for each transaction. We also calculate average unit values at the product-by-time-by-seller-by-buyer level, the number of transactions, and the total value of the goods transacted. This is the foundation of our firm-to-firm dataset used in the analysis. Lockdowns. On March 25, 2020, India unexpectedly imposed strict lockdown policies nation- wide. Districts were classified into Red , Orange, and Green zones according to each district’s severity of Covid-19 cases. Each color corresponds to different lockdown degrees, where Red was severe lockdown, Orange was medium lockdown, and Green was mild. Yet, at that time, there were barely any Covid-19 cases in India, as the entire country averaged about 50 cases a day (as opposed to about 400,000 cases a day the following year). Districts in the Red zone experienced the strictest lockdowns, where rickshaws, taxis and cabs, public transport, barbers, spas, and salons remained shut. E-commerce was allowed for es- sential services. Orange and Green zone districts experienced fewer restrictions. Orange zones allowed the operation of taxis and cab aggregators, as well as the inter-district movement of indi- viduals and vehicles for permitted activities. In addition to the activities allowed in Orange zones, buses were allowed to operate with up to 50% seating capacity and bus depots with 50% capacity 15 The data partially reports items up to 8-digit HS codes. Until April 2021, in India, it was only mandatory to report 4-digit HS codes of goods traded (See the Economic Times). 97% of transactions report 4-digit HS codes, and 40% report 8-digit HS codes. Given this, our main specifications are based on 4-digit HS codes, and we show robustness using HS-8 codes. 7 in Green zones.16 Throughout the paper, we use this color scheme as the treatment across Indian districts. In particular, each treated firm is located in a Red or Orange district, while each control firm is located in a Green district. Physical and cultural distance. We use different measures of distance, which we include as controls in our empirical results. The measures of geographic distance between districts calculate the length of the shortest distance between district centroids. The measure of linguistic distance between Indian districts is from Kone et al. (2018) using the commonly used ethnolinguistic frac- tionalization (EFL) index (Mira, 1964). This index measures the probability of two randomly chosen individuals from different districts speaking the same language. Other controls. We control for different firm and district-level time-varying variables such as data on the monthly number of cases, deaths, and recoveries from Covid-19 for all of India at the district level from Covid India. For each firm, we construct two variables that measure the firm’s exposure to global demand and supply shocks that vary at the product and country level, following Hummels et al. (2014). The construction of these exposure variables is described in detail in online data Appendix C. Summary statistics. We present some key summary statistics from our firm-to-firm trade dataset in Table A1 and Figure A1. They report the unique numbers of sellers and buyers, total sales (in million rupees), and total number of transactions during the months of January-March, April-June, and July-September for both 2019 and 2020. The most noticeable pattern is the large drop in all variables in 2020 compared to 2019, particularly during the April-June period, which coincided with the lockdown policies. The total value of sales and the number of transactions fell by almost 60% during April-June of 2020 compared to 2019. For reference, the fall in the value of sales was only 25% after the strict centralized lockdown was over (July-September) and only 15.6% before the lockdown (January-March) compared to the corresponding months in 2019. The characteristics of the network, including measures of network in-degree (i.e., number of sellers per buyer) and outdegree (i.e., number of buyers per seller), are in Tables A2 and A3. Similar to other production network datasets, our network is sparse. In terms of network indegree, for 4-digit HS products, the median buyer has 2 suppliers per product but 6 suppliers on average with a standard deviation of 17. In terms of network outdegree, the median seller has 5 buyers 16 See the Economic Times. On April 30, one Red zone district was reclassified to the Green zone, but we maintain the initial classification as it is likely to be more exogenous. 8 but 64 buyers on average, with a standard deviation of 532. Finally, the sparsity of the network increases with the narrowness when defining a product and the frequency of the data. To further understand the composition of economic activity of the Indian state of our analysis, in Table A4, we show the types of goods firms sell and buy and what fraction crosses state and country borders. In our state, firms are mostly in the business of selling vegetables, plastics, and minerals and buying machinery, metals, and vegetables. In terms of the type of trade, firms in our state are more likely to sell to other firms in the state. This contrasts with how firms in our state buy goods, where the share of purchases that come from within the state is almost the same as from other Indian states. Finally, international exports and imports represent a non-negligible but small share of sales and purchases. Before using the lockdown variation to understand how firm-to-firm transactions are affected, we verify the stringency of these lockdowns in Figure A2 using Google mobility data. The data shows how the number of visitors to (or the time spent in) categorized places changes in compari- son to baseline days. The baseline day is the median value from the 5-week period Jan 3 – Feb 6, 2020.17 Until the beginning of March 2020, there were essentially no differences in mobility trends across Red , Orange, or Green zones. But starting at the end of March 2020, we see that there is a substantial reduction in different types of activities (time spent in retail and recreation, grocery and pharmacy, parks, commuting, and workplaces) in Red zones compared to Green zones; with Orange zones in between. People in Red zones also spent more time at home than people in either Orange or Green zones. We notice that starting in August 2020, a few months after the central- ized lockdown was over, these differences reduced, and by December 2020, these differences, especially in workplace mobility, become negligible. 3 R EDUCED - FORM EVIDENCE In this section, we outline a simple empirical specification to provide evidence showing the role of lockdown policies on key outcome variables for firm-to-firm trade. We show that the sudden Covid-19 lockdown policies between March and May 2020 led to a rise in unit values and a fall in the monthly number of transactions between firms.18 In the next section, we exploit this variation to estimate firm-level elasticities of substitution across intermediate suppliers of the same product. 17 See the Google Covid-19 Community Mobility Reports. 18 To see a similar application of this empirical strategy for domestic violence and economic activity in India, see Ravindran and Shah (2020) and Beyer et al. (2021). 9 3.1 Empirical specifications Our reduced-form specifications implement a difference-in-differences approach where we com- pare the unit values and the number of transactions both at the seller and seller-buyer level across Red , Orange, and Green districts, before and after the lockdown. In our analysis at the seller level, the omitted (control) group are sellers located in Green districts and the base month is February 2020, the month before the lockdown enforcement. At the seller-buyer level, the omitted groups are sellers and buyers located in Green zones, and the base month is February 2020. Seller-level regressions. We estimate the following specification: Ys,i,t = ωo(s) ,i + ωi,t + βt Redo(s) + γt Orangeo(s) + Xδ + s,i,t , (1) t =−1 t =−1 where Ys,i,t are either unit values or the log number of transactions for seller s of product i in month t , ωi,t are product-by-time fixed effects, ωo(s) ,i are product-by-district fixed effects (i.e. fixed effects based on the district o where seller s resides). Product-by-time fixed effects control for any unobserved demand shocks at the product level. X are controls that include the number of Covid- 19 cases, deaths, recoveries, and exposure to international demand and supply shocks as discussed in Appendix C. We control for the Covid-19 cases and deaths since these are the variables on which the government based its lockdown decisions (Ravindran and Shah, 2020). The covariates of interest are Redo(s) and Orangeo(s) . The first is an indicator variable that equals 1 if seller s located in district o(s) experienced a severe lockdown, 0 otherwise. The second equals 1 if seller s located in district o(s) experienced a mid-level lockdown, 0 otherwise. The excluded category is Greeno districts, where mild lockdown was imposed. The estimates of interest are βt and γt . Our base time category is February 2020, just before lockdowns began. Standard errors are clustered at the seller’s origin district level. Seller-buyer level regressions. At the seller-buyer level, we estimate the specification: Ys,i,b,t = v βtvz γo(s) z × γd (b) + δo(s) + δd(b) + δi,t + β1 log disto(s) ,d(b) + Xδ + si,b,t , (2) (v,z)∈Ω t =−1 where Ys,i,b,t are unit values or the number of transactions in logs between seller s of product i and a buyer b in time t . δo(s) , δd(b) , and δi,t are origin, destination, product-by-time fixed effects. disto(s) ,d(b) is a vector of cultural and geographic distance variables, and X are controls that include the number of Covid-19 cases, deaths, recoveries, and exposures to international demand and supply shocks. 10 The first term of the right-hand side contains our estimates of interest. (v, z) ∈ Ω is a duple that contains the color y of seller’s district, and the color z of buyer’s district. Ω is the set that includes all pairs except (Green, Green), such that this is the excluded category when estimating Equation (2). v z γo (s) and γd (b) are thus indicator variables that equal 1 when seller s is located in district o located in lockdown zone v, and when buyer b is located in district d located in lockdown zone z, respectively. The estimates of interest are βtvz . Our base time category is February 2020, just before lockdowns began. Standard errors are two-way clustered at the origin and destination district levels. 3.2 Reduced-Form Facts Fact 1: Sellers’ unit values disproportionately rose, and trade fell in more severe lockdown zones. The first two panels of Figure 1 plot the coefficients βt and γt from Equation (1), repre- senting changes in log unit values and log number of transactions with respect to Green districts in February 2020 (the base category). In May 2020, sellers’ unit values in Red districts rose by 25 pp, and in Orange districts rose by around 10 pp with respect to the base category. 11 F IGURE 1: Seller-level reduced-form event studies (a) Unit value, 4-digit HS (b) # Transactions, 4-digit HS (c) Unit value, 8-digit HS (d) # Transactions, 8-digit HS (e) Unit value, 8-digit HS, strong FEs (f) # Transactions, 8-digit HS, strong FEs Notes: This figure consists of 6 plots. Each plot shows estimates for βt and γt from Equation (1). The estimated values are all in comparison to sellers in Green districts in February 2020. The dependent variable for the plots on the left-hand side is log unit values. The dependent variable for the plots on the right-hand side is the log number of transactions. Each row varies by the definition of a product-group and the fixed effects included in the regression. In the first row, a product is a 4-digit HS code, and the fixed effects are product-by-time and district. In the second row, a product is an 8-digit HS code, and the fixed effects are product-by-time and district. In the third row, a product is an 8-digit HS code, and the fixed effects are product-by-time12 and district-by-product. Standard errors are clustered at the district level. All controls mentioned in the paper are included. The shaded areas are 95% confidence intervals. At the same time, sellers’ number of transactions in Red districts declined by around 20 pp, and in Orange districts declined by around 3 pp with respect to the base category. Additionally, as expected by the severity of the lockdown policies by color, the rise in unit values and fall in the number of transactions was larger for sellers in Red districts than for Orange ones. In both figures, we find no evidence of pre-trends, implying that there were likely no differences in the trends of unit values or number of transactions between Red , Orange, and Green districts before the lockdown. The middle two panels of Figure 1 repeat the same exercise with a finer product definition, using 8-digit HS codes. Results remain virtually the same. In the last row of Figure 1, we include a stronger set of fixed effects (e.g., district-by-product), and the results remain the same. Fact 2: Equilibrium unit values rose, and the number of transactions fell in more severe lock- down zones. We now report the results from our seller-by-buyer-level specification. In Figures 2 and 3, we report the estimates for βtvz from Equation (2), where the estimates are in comparison to cases when both sellers and buyers were located in Green districts in February 2020. In the first row of Figure 2, we plot the coefficients from regression (2) where the seller is in the Red zone, and the buyer is in Red , Orange, and Green zones respectively. Similarly, in the second row of Figure 2, we plot the coefficients from regression (2) where the seller is in the Orange zone, and in the third row, we plot the coefficients from regression (2) where the seller is in the Green zone (and the buyer is in Red and Orange zones respectively). There are two main takeaways from these figures. First, even after controlling for bilateral resistance terms, trade costs, and additional covariates, unit values rose, and the number of trans- actions fell with respect to the base category (both buyer and seller in Green zones). The rise in unit values was as much as 45 pp, and the fall in transactions was as high as 12 pp. Second, these changes seem proportional to the severity of the lockdowns for both sellers and buyers. Once again, there is no evidence of differential pre-trends across zones leading up to the shock. Our two facts jointly imply that prices, where either sellers or buyers were located in Red districts, were higher during the lockdown compared to districts where the lockdowns were mild (Green zones). As such, the lockdown induced variation in prices that we will later leverage to estimate elasticities of substitution across intermediates. 13 F IGURE 2: Unit Value, Seller-Buyer-Level Regressions Notes: This figure consists of 8 plots. Each plot shows estimates for βtvz from Equation (2). The values of the estimates are all in comparison to sellers and buyers both in Green districts in February 2020. The vertical line in January 2020 splits the period into pre and post-lockdown periods. The dependent variable for the plots is the log unit value. A product is a 4-digit HS code. Regressions include product-by-time, origin, and destination-district fixed effects. Standard errors are two-way clustered at the origin and destination district levels. All controls mentioned in the paper are included. The color of the line denotes the color of the seller’s district, while the color of the shaded 95% confidence interval denotes the color of the buyer’s district. 4 M ODEL We build a quantitative general equilibrium model of firm-to-firm trade (Baqaee and Farhi, 2019). Our model is intentionally parsimonious, and only modifies standard frameworks by including the possibility that suppliers of the same product are not perfect substitutes. This simplicity allows our 14 F IGURE 3: Number of Transactions, Seller-Buyer Level Regressions Notes: This figure consists of 8 plots. Each plot shows estimates for βtvz from Equation (2). The values of the estimates are all in comparison to sellers and buyers both in Green districts in February 2020. The vertical line in January 2020 splits the period into pre and post-lockdown periods. The dependent variable for the plots is the log number of transactions. A product is a 4-digit HS code. Regressions include product-by-time, origin, and destination- district fixed effects. Standard errors are two-way clustered at the origin and destination district levels. All controls mentioned in the paper are included. The color of the line denotes the color of the seller’s district, while the color of the shaded 95% confidence interval denotes the color of the buyer’s district. estimates to be used in other contexts.19 The production sector is perfectly competitive.20 We adapt the general nested CES structure to reflect the possibility that suppliers within the same product category could be substitutes or complements. Firms combine inputs in a CES fashion in each of its three tiers. In the first tier, firms combine labor and aggregated intermediate inputs to produce 19 However, the parsimony also implies that we should not think of the elasticity of substitution across suppliers to be a fixed exogenous parameter, but rather depend on various aspects of market structure, institutional quality, firm types, and inventories. We document this heterogeneity empirically below but focus on the average value of the parameter for much of the analysis. 20 We abstract from market power (Edmond et al., 2023; Alviarez et al., 2023) since the evidence from the data suggests that the market structure in this Indian state is highly competitive. The median HHI across 4-digit HS product categories is 0.1041, which implies a low level of market concentration within a product category. 15 output. In the second tier, firms combine aggregated intermediate inputs of a product category. In the third tier, firms combine suppliers within a product category. The model yields estimating equations we then use to estimate the firm-level elasticity of substitution between suppliers of the same product category. We consider a fixed set of firms F and of product categories I , where N = |F| is the total number of firms in the economy, Ni is the number of firms producing a good of product category i, and I = |I| is the number of product categories. Each firm produces according to its technology α α− 1 α− 1 α−1 yn j = An wnl (ln ) α + (1 − wnl ) xn j α , (3) where yn j is the output produced by firm n in product j, An is the productivity of firm n, ln is the labor used by firm n, xn j is the composite intermediate input used by firm n in product category j, α is the elasticity of substitution between labor and the composite material input, and wnl is the intensity of labor in production. The composite material input, in turn, consists of inputs from the I different product categories in the economy and is: ζ I 1 ζ −1 ζ −1 ζ xn j = wi,n j xi,n j ζ , (4) i=1 where ζ is the firm-level elasticity of substitution across products i,21 and wi,n j is the importance of inputs of product category i for firm n of product j. xi,n j are intermediate inputs from product i going to firm n producing product j, which are constructed as:22 Ni −1 1 −1 xi,n j = µm,i,n j xm,i,n j , (5) m=1 where xm,i,n j are intermediate inputs from supplier m of product i sold to firm n producing product j. µm,i,n j is the unobserved importance of input from supplier m of product i in the production of firm n of product j, and is the firm-level elasticity of substitution across different suppliers within the same product category. This is the key elasticity we want to estimate. The above production functions work for reproducible factors. For non-reproducible factors (in our case, labor), the production function is an endowment: Y f = 1. 21 Previous work has estimated different related versions of this elasticity; e.g., elasticities of substitution across industries (Atalay, 2017), across goods from different countries (Boehm et al., 2019), or across product categories (Carvalho et al., 2021; Peter and Ruane, 2022). 22 We exclude foreign intermediate goods since they are not exposed to Indian Covid-19 lockdown shocks. 16 Product 0 represents the final consumption of the household and is given by σ N σ −1 σ −1 C= w0i (ci ) σ , (6) i where i woi = 1 and σ is the elasticity of substitution in consumption. Model in standard form. To write the economy in standard form as in Baqaee and Farhi (2020), we construct an input-output matrix Ω with dimension 2 + N + I , where 2 dimensions come from the household’s consumption aggregator, and a factor of production (labor), N dimensions come from the N firms in the economy, and I dimensions come from the I product categories in the economy. We explicitly distinguish between labor and intermediate inputs since labor is non-reproducible. Consider the vector of elasticities by θ, where θ = (σ , α, ζ , ). Formally, a nested-CES economy in standard form is defined by the tuple (Ω, θ). The input-output matrix Ω of size p x (2 + N + I ) × (2 + N + I ) is a matrix where element (i, j) equals the value of Ωi j = pji yiij , which is the expenditure share of the ith firm on inputs from the jth supplier as a share of the total revenue of firm i. Note that every supplier is a CES aggregate. The Leontief inverse is ψ = (1 − Ω)−1 . Intu- itively, the (i, j)th element of ψ (the Leontief inverse) measures i s total reliance on j as a supplier. That is, it captures both the direct and indirect ways i uses j in its production. Let us also denote pi yi the sales of producer i as a fraction of GDP by λi , where λi = N p c . j j j The input-output covariance operator is 2+N +I 2+N +I 2+N +I CovΩk (ψ(i) , ψ( j) ) = Ωkl ψli ψl j − Ωkl ψli Ωkl ψl j . (7) l =1 l =1 l =1 This operator measures the covariance between the ith and the jth columns of the Leontief inverse using the kth row of the input-output matrix as distribution. The second-order macroeco- nomic impact of microeconomic shocks in this economy is given by: d 2 logY d λi = = (θk − 1)λkCovΩ(k) (Ψ(i) , Ψ( j) ). (8) dlogA j dlogAi dlogA j k For a detailed derivation of Equation (8), see the appendix of Baqaee and Farhi (2019). To inspect how firm-level shocks can propagate through supply chains, consider the following example. Firm j, which is located in a Red zone, suffers a negative productivity shock, given by d log A j < 0. The second order term captures the reallocation effect: In response to a negative 17 shock to product category j, all products k downstream of j may readjust their demand for all other inputs. Crucially, the impact of such readjustments by any given k on the output of product i depends on the size of product k as captured by its Domar weight λk , the elasticity of substitution θk in k s production function, and the extent to which the supply chains that connect i and j to k coincide with one another, as given by the covariance term. Importantly, i and j in equation 8 could be firms, and in that case, these shock propagation equations determine how shocks to individual firms ( j) can propagate through the entire network depending on the sizes of the downstream sector (k) that j supplies inputs to, the number of input suppliers to the affected downstream sector k, and how these suppliers’ supply chains overlap with j’s supply chain. 4.1 Equations to estimate firm-level elasticity of substitution across suppliers The model yields estimating equations we use to estimate firm-level elasticities of substitution across suppliers within a product. We introduce a notation change to facilitate the exposition: a firm n can be either a buyer b ∈ F or a seller s ∈ F . A firm b that sells product j ∈ I maximizes profits subject to its technology and to a CES bundle of intermediate inputs: max pb j yb j − wb j lb j − ps,i,b j xs,i,b j {lb j ,xs,i,b j } i s subject to Equations (3), (4), and (5). Details about the optimization problem are in Appendix D.1. The maximization problem yields the following expression: PMs,i,b j ps,i,b j log = (1 − ) log + log µs,i,b j , (9) PMi,b j pi,b j 1 1− where pi,b j = s ps ,i,b j µs ,i,b j 1− is a CES price index, PMs,i,b j ≡ ps,i,b j xs,i,b j , and PMi,b j ≡ s PMs,i,b j . log µs,i,b j is the error term. This is the underlying basis for our estimation of the firm-level elasticity of substitution parameter , as will be described in detail in Section 5. Note that the results of this estimation procedure hold with any CES production function with an arbi- trary number of nests as long as the lowest nest consists of suppliers within the same product.23 23 In the empirical section, we estimate the elasticities of substitution across suppliers within the same HS-4, HS-6, and HS-8 products 18 4.2 Equations to estimate firm-level elasticity of substitution across products In this section, we derive conditions from the model to estimate the firm-level elasticity of substi- tution across products, as in some previous work (Atalay, 2017; Peter and Ruane, 2022; Boehm et al., 2019; Carvalho et al., 2021). We rewrite the maximization problem of the firm such that it maximizes max pb j yb j − wb j lb j − pi,b j xi,b j {lb j ,xi,b j } i 1 1− subject to Equations (3) , (4), and pi,b j = s µsi,b j psi,b j . Details on the optimization problem 1− are in Appendix D.2.1. The maximization problem yields the following expression: PMi,b j pi,b j log = (1 − ζ ) log + log wi,b j , (10) PMb j pb j 1 −ζ 1− ζ where pb j = i p1 i ,b j wi ,b j is a CES price index, PMi,b j ≡ pi,b j xi,b j , and PMb j ≡ i PMi,b j . log wi,b j is the error term. This is our estimating equation for the firm-level elasticity of substitu- tion ζ , which we take to the data, as described in Section 5. 5 E STIMATION S TRATEGY This section discusses how we estimate the primary elasticities in our model, the vector of param- eters θ = (σ , α, ζ , ). We set the elasticity of substitution between different consumption varieties σ = 4 (Broda and Weinstein, 2006), and the elasticity of substitution between labor and the compos- ite intermediate input α = 0.5 (Baqaee and Farhi, 2019). We now estimate the firm-level elasticity of substitution across suppliers ( ) and the firm-level elasticity across products (ζ ) leveraging vari- ation in the lockdown zones. 5.1 Estimating equations for We face two main challenges when estimating from Equation (9). The first challenge is that we need data on price indices pi,b j , which are a function of the unobservable importance of input suppliers µs,i,b j . To address this challenge, we follow two strategies. In the first one, we include a strong buyer-product-time fixed effect that subsumes the unobserved terms. In the second one, we follow Redding and Weinstein (2020) and assume that the overall importance of a product in a 19 buyer’s input use remains constant, such that we can construct pi,b j solely as a function of prices ps,i,b j . The second challenge arises if sellers and buyers decide to trade due to unobservable reasons (i.e., extensive margin due to endogenous selection). To account for this extensive margin for the estimation, we construct and leverage variation in counterfactual prices (i.e., prices of transactions that did not happen in a particular period because the seller did not trade with a certain buyer that period but could have traded with the same buyer before or after). Addressing unobservable importance of input suppliers. To construct changes in aggregate prices pi,b j,t , we follow two alternative strategies: First, we follow Redding and Weinstein (2020) and assume that the overall importance of a product in a buyer’s input use does not change between two consecutive periods, even though the importance of inputs from suppliers within a product category can change.24 This enables us to construct changes in price indices that are not dependent on µsi,b j,t , but are directly observed in the data (details in Appendix D.1.2). Under this assumption, we derive Equation 11, which links the overall expenditure share on a certain supplier s (as a share of total expenditure on product i) to the corresponding relative prices: PM s,i,b j,t ps,i,b j,t ∗ log = ωb j,t + ωi,t + ωb j,i + ωs,i + (1 − ) log + log λi,b j,t si,b j,t + Xδ + ξs,i,b j,t , PM i,b j,t pi,b j,t (11) where vt = vv t t −1 are variables in changes to the previous month. ωb j,t , ωi,t , ωb j,i , ωs,i is a set of fixed effects, including buyer-by-time, product-by-time, buyer-by-product, and seller-by-product fixed 1 N∗ i,b j,t effects. pi,b j,t = s∈Ω∗ ps,i,b j,t is a geometric mean of unit values across common suppliers, where i,b j,t Ω∗i,b j,t ≡ Ωi,b j,t ∩ Ωi,b j,t −1 is the set of common suppliers for buyer b that appear in both the current and previous month, and Ni∗ ∗ ,b j,t ≡ Ωi,b j,t is the number of common suppliers for buyer b in month t . This strategy has three main advantages. First, it allows us to construct price indices pi,b j,t solely as a function of prices ps,i,b j,t , independent of the unobservable importance of input suppliers µs,i,b j,t . Second, under the same assumption, the constructed price indices pi,b j,t naturally generates PMsi,b j,t two controls to be included in the estimation: the change in expenditure share s∗ i,b j,t ≡ ∗ PMsi,b j,t s∈Ω i,b j,t s∈Ω∗ PMsi,b j,t i,b j,t and a Feenstra (1994) correction term λi,b j,t ≡ PMsi,b j,t . The key control is the latter since it s∈Ωi,b j,t accounts for the entry and exit of sellers in product i for buyer b in period t . More details are in Appendix D.1.3. Finally, as described in more detail in Section 5.2, this strategy allows us to estimate the elasticity of substitution between products (ζ ). Standard errors are two-way clustered 24 This assumption requires that, for instance, a shoemaker’s overall preference for leather in shoe manufacturing does not change, although its preference for leather from certain suppliers can change. Demand shocks may change µsi,b j,t (e.g., the demand for leather from certain suppliers), but the geometric mean of µsi,b j,t across suppliers within a product is stable between t and t − 1. 20 at the origin and destination state levels. X are controls, including exposure to foreign demand and supply shocks, the number and severity of Covid-19 cases, and geographic and cultural distance. Second, we consider an even more conservative strategy to estimate by including a product- buyer-time fixed effect ωi,b j,t , as in Equation 12 below. The advantage of this strategy is that we do not need to make any assumptions about the importance of a product in a buyer’s input use as we did previously, as it automatically absorbs the other terms, including the Feenstra (1994) correction term λi,b j,t . This is a demanding specification that, for instance, absorbs any demand shocks at the buyer-by-product level. This includes, for example, shocks to retail demand and retail distribution costs for the buyer’s products (Crucini and Davis, 2016). log PM s,i,b j,t = ωi,b j,t + ωs,i + (1 − ) log ps,i,b j,t + X δ + ξs,i,b j,t (12) Across the various specifications, the wide array of high-dimensional fixed effects helps con- trol for demand shocks (buyer-by-time fixed effects or buyer-by-product-by-time fixed effects), product-level changes in demand or supply (product-by-time fixed effects or buyer-by-product-by- time fixed effects), and buyer-by-seller and product-specific time-invariant characteristics (buyer- by-product and seller-by-product fixed effects). The remaining variation isolates time-varying changes across sellers within a product category. Yet, as we explain in Section 5.3, we strengthen this framework by leveraging the mosaic of Covid-19 lockdowns to derive exogenous policy- induced variation in relative prices. Accounting for the extensive margin of trade. In extensions, we further account for the exten- sive margin of trade in additional ways. Our main estimating Equation 11 is estimated based on a firm’s existing suppliers, and the elasticity of substitution dictates a buyer’s ability to substitute inputs between its suppliers. Nevertheless, sellers and buyers may decide to trade for unobservable idiosyncratic reasons (i.e., extensive margin due to endogenous selection). To account for this ex- tensive margin for the estimation, we introduce the notion of a counterfactual supplier for a buyer. A counterfactual supplier of buyer b at time t is defined as a supplier who has ever traded with that buyer in our sample period and supplies to other buyers at month t , but not to that particular buyer at month t . We expand our estimating equation to include all counterfactual suppliers of a buyer. A counterfactual supplier, therefore, accounts for potential suppliers of a buyer b at time t – suppliers who at some point traded or could trade with b and supplied inputs to other buyers in period t . A counterfactual supplier also accounts for supplier links of a buyer b that no longer exist in period t , even though this supplier continues supplying to other buyers. While the expenditure share by 21 buyer b on inputs from a particular seller s of product i can be zero in periods that the trade does not happen, in those periods, we still need the notion of a counterfactual price. The advantage of our data is that it uniquely reports transaction-level prices. As a result, we can construct counterfactual prices in those time periods as the average price that seller s would charge to other buyers, as given by Equation 13 below. s∈Ωi,b j,t b∈Φs∈Ω PMs,i,b j,t i,b j,t ps,i,b j,t ≡ , (13) s∈Ωi,b j,t b∈Φ Qs,i,b j,t s∈Ωi,b j,t where Ωi,b j,t ≡ Ωi,b j − Ωi,b j,t . Ωi,b j,t is the set of sellers in product i that traded with buyer b in period t , Ωi,b j is the set of potential sellers with buyer b in product i.25 Therefore, Ωi,b j,t is the set of counterfactual sellers as defined above. Φs∈Ωi,b j,t is the set of buyers that traded with seller s in product i in period t . For the con- struction of these counterfactual prices to be relevant, the prices charged by a seller at a particular time should be approximately the same across all buyers; that is, the counterfactual price will be difficult to measure if the seller charges different prices across buyers within the same year-month. In Figure A3, we plot the residuals from regressing prices on seller-by-product-by-time fixed ef- fects. The figure shows that these residuals are concentrated around 0, suggesting that sellers do not charge different prices to different buyers for the same product within a year-month. The estimating equation, after substituting the Feenstra (1994) term with the stronger buyer time fixed effects and accounting for the extensive margin of trade, is given below: log(PMsi,b j,t ) = ωi,b j,t + ωs,i + (1 − ) log ps,i,b j,t + X δ + ξs.i,b j,t . (14) To account for the zeros on the left-hand side of the estimating equation, we estimate via Poisson Pseudo-Maximum Likelihood (PPML). 5.2 Estimating equations for ζ Now, to estimate ζ from Equation (10), there are two issues to address. First, notice that the price index pi,b j,t is a function of (unobservable) demand shocks µsi,b j,t , and , such that pi,b j,t ≡ 1 1− s µs,i,b j,t ps,i,b j,t . Second, the price index pb j,t is also a function of unobservable product-level 1− demand shocks wi,b j,t , making their computation challenging. 1 1− First, we construct price indices as pi,b j,t ≡ s µsi,b j,t psi,b j,t 1− , where is estimated previ- ously, psi,b j,t comes directly from the data, and demand shocks µsi,b j,t are constructed recursively. 25 The set of potential sellers of buyer b is constructed based on the buyer’s purchase record between April 2018 and October 2020. 22 This recursive construction of demand shocks comes from predicting residuals from Equation (11) and setting an initial value for shocks µsi,b j,0 (Appendix D.2.2). Second, we construct buyer-level price indices pb j,t following Redding and Weinstein (2020). We assume that the overall importance of the composite intermediates at the product-level in the production function does not change between consecutive months. As such, we can construct this price independent of product-level demand shocks wi,b j,t after controlling for buyers’ expenditure shares by product. More details are in Appendix D.2.1. We then derive the following expression we take directly to the data: log PM i,b j,t = ωb j,t + ωi,t + ωb j,i + (1 − ζ ) log pi,b j,t + X δ + ξi,b j,t , (15) where ωb j,t , ωi,t , ωb j,i are a set of buyer-by-time, product-by-time, and buyer-by-product fixed effects, which again account for a wide array of demand shocks, product shocks, and buyer-product 1 Nb j,t N b j,t specific characteristics. X are the same set of controls used before. pb j,t ≡ i=1 pi,b j,t is the 1 b j,t b j,t N N geometric mean of unit values across products that buyer b purchases, and sb j,t ≡ i=1 si,b j,t is the geometric mean of expenditure shares across products. Detailed derivations are in Appendix D.2. 5.3 Addressing endogeneity concerns Despite the wide range of fixed effects, Ordinary Least Squares (OLS) estimates of may still be biased if additional unobserved demand-side shocks (changing µs,i,b j,t ) drive changes in prices and expenditure shares. The firm-level elasticity of substitution is a function of the slope of the buyer’s input demand curve, and hence, simultaneous shifts in the demand and supply curves induced by the Covid-19 shock would bias our estimates. For example, if Covid-19 induced demand shocks led to contractions in buyers’ income and, at the same time, supply shocks led to contractions in the sellers’ supply, the demand curves will look flatter (estimated higher) compared to the unbiased value of . Additionally, measurement error in input prices, proxied by unit values, may induce attenuation biases. Our estimation strategy, therefore, involves using the sudden demarcations of lockdown zones that restrict economic activity in certain Indian districts as an instrumental variable when estimating this equation in two-stage least squares (2SLS). We use the disruptions in prices caused by sudden lockdowns that made it costlier for sellers in Red and Orange zones to produce and send their intermediate goods. The idea is that, after controlling for the wide array of fixed effects, the lockdown zones the buyers are located in, exposure to international demand and supply shocks, 23 and the number and severity of regional Covid-19 cases, the remaining variation in prices facing a buyer are driven by supply shocks induced by policy-mandated sudden changes in the seller’s lockdown zones. In addition, since the goods from the seller to the buyer have to transit through several districts located in different lockdown zones facing different severity in the movements of trucks and border controls, changes in the costs of transportation induced by these lockdowns provide another source of exogenous variation to estimate the firm-level elasticity of substitution. To formalize the intuition behind our identification strategy, suppose that prices can be sep- arated between prices at the origin and a trade cost, such that log ps,i,b j,t = log τs,b j,t + log ps,i,t . Here we can see the types of variation that drive the two types of instruments we use. First, exogenous shifters to prices at the seller level ps,i,t , such as economic restrictions induced by the lockdown zone the seller is located in. Second, exogenous shifters at the seller-buyer level, for example, changes in transportation costs τs,b j,t driven by the lockdown zones of the districts the goods pass through. We now describe each of these instruments and then implement them within our estimation strategy. Seller-level instruments. We derive supply-side shifters to obtain unbiased elasticities of sub- stitution. Shocks induced by the Covid-19 lockdown policies that only impact sellers provide this variation. In Equation (16) below, we formalize this intuition. log( ps,i,t ) = β R, p Redo(s) Lockt + β O, p Orangeo(s) Lockt + νsp,i,t , (16) where Lockt is an indicator variable that equals 1 for the months from March to May of 2020, which are the months when the lockdown policies were implemented, 0 otherwise, and Redo(s) and Orangeo(s) are indicator variables that equal 1 whenever seller s was located in Red or Orange districts, respectively. Seller-Buyer-level instruments. The transportation of supplies from the location of the supplier to the buyer requires going through different districts, each of which is affected by lockdown policies in different ways. Intuitively, a route containing more Red districts should increase the cost of transportation compared to a route with no Red districts. We construct instruments that capture that idea. We allow trade costs to change over time so that we can leverage the Covid-19 lockdown policy. In particular, as we describe in Appendix D.1.4, we assume 24 log(τs,b j,t ) = σ log(TravelTimes,b j,t ). We leverage the Covid-19 lockdown as an exogenous shifter that only influences travel time between locations of seller s and buyer b, as reflected in Equation (17) below. log(τs,b j,t ) = β R,τ Redo(s) d(b) Lockt + β O,τ Orangeo(s) d(b) Lockt + νsτ,b j,t . (17) Detailed derivations and estimating equations are in Appendix D.1.4. Redo(s) d(b) and Orangeo(s) d(b) are the share of districts designated as Red and Orange, respectively, along the route between seller s and buyer b. We construct these variables using the Dijkstra algorithm for least-cost routes. De- tails about the implementation of this algorithm are in Appendix C. Finally, we instrument the changes in relative prices in Equation (15) to estimate ζ . Potential unobservable product-level demand shocks could again induce an upward bias to OLS estimates of ζ . To construct our instruments, we leverage the seller-level and seller-buyer-level instruments we used to estimate and calculate weighted averages across suppliers to instrument for the change in relative prices for buyers. The intuition is that buyers that purchased inputs either from a larger share of sellers in Red zones, or from sellers located in districts such that the trading routes com- prised of manyRed zone districts were more exposed to supply disruptions induced by the Covid-19 lockdowns. More details of the instruments are in Appendix D.2.3. Discussion of instruments. The instruments induce buyers of certain types to be more affected than others based on their production networks. The Local Average Treatment Effect (LATE) may not represent the Average Treatment Effect (ATE) if buyers in Red , Orange, and Green zones already traded intensively with sellers in certain lockdown zones, and there is heterogeneity in responses. For instance, if buyers in Red traded mostly with sellers in Red , then our instrument may estimate effects on firms induced by having more Red sellers, thus upweighting effects on buyers in Red . In Figure A4, we run two distributional checks to investigate these patterns. These figures show that, in general, sellers from Red , Orange, and Green zones had similar interactions with buyers from Red , Orange, and Green zones. We also consider whether certain products are sourced intensively from firms located in certain zones. For instance, if all the rubber supply of firms in this production network comes from suppliers in Red zones, then buyers of rubber would find it increasingly difficult to find suppliers. Once again, if there is heterogeneity in responses by product category, our estimated LATE elasticity would weigh rubber products higher than non-rubber products. While not a source of bias, it does affect the interpretation of the estimated parameter. In Figures A4g and A4h, we 25 plot the shares of total purchases of each industry (HS Section) that are sourced from firms in Red , Orange, and Green zones. Except for the small HS industry 19 (arms and ammunition), there is no noticeable degree of concentration of suppliers from any particular zone. 6 E LASTICITY ESTIMATES In this section, we show the results of the estimation of both firm-level elasticities of substitution across suppliers within a product category, and then across product categories. 6.1 Firm-level elasticities of substitution across suppliers First, we report OLS estimates in Table 1. In columns (1)-(3), we progressively include stronger sets of fixed effects. In column (1), we include both buyer-by-time and product-by-time fixed effects, as well as the Feenstra term. In column (2), we also include buyer-by-product and seller-by- product fixed effects. In column (3), we include buyer-by-product-by-time and seller-by-product fixed effects. Notice that in column (3), the former set of fixed effects absorbs the Feenstra term. The implied elasticities exhibit values between 0.70 and 0.78 across the different specifications. To test whether our estimates vary by product aggregation, in columns (4)-(7), the estimates are based on 6-digit and 8-digit HS codes. The elasticities range between 0.66 and 0.78, so the estimates do not meaningfully change. Since these elasticities are below 1, these estimates sug- gest that, at the firm level, suppliers act as complements rather than substitutes for buyers. This is important for aggregate incomes since, from Equation (8) we can see that, once we consider second-order effects, an elasticity of substitution less than 1 implies that the aggregate impacts of negative shocks are amplified. Nevertheless, as we described previously, the OLS estimates may be contaminated by simul- taneous demand and supply shocks that happened during Covid-19. In Table 2, we report 2SLS estimates based on our proposed instruments. We find evidence that inputs across different sup- pliers of a firm within the same 4-digit HS product category are highly complementary, ranging from 0.50 − 0.66, depending on the set of fixed effects and instruments we use.26 Our preferred specifications are column (3) and (5) with an elasticity of 0.55, where we use both the seller and the seller-buyer level instrument, essentially deriving variation from both sellers’ production costs and transportation costs. We include buyer-by-time and product-by-time fixed effects that account for time-varying demand shocks and also account for firm entry and exit with the Feenstra (1994) 26 In Table A11, we show similar patterns for HS-6 and HS-8 categories as described in Section 7. 26 TABLE 1: OLS, firm-level elasticity of substitution across suppliers (1) (2) (3) (4) (5) (6) (7) ˆ p log ˆ ˜ p 0.2171 0.2222 0.3045 0.2114 0.3030 0.2441 0.3370 (0.0133) (0.0147) (0.0151) (0.0192) (0.0196) (0.0352) (0.0365) 0.7828 0.7777 0.6954 0.7885 0.6969 0.7558 0.6629 Obs 2028039 1966591 991187 2021334 816122 993583 341903 R2 0.417 0.460 0.404 0.478 0.411 0.495 0.420 HS digits 4 4 4 6 6 8 8 Feenstra term Y Y Y Y Buyer-time FE Y Y Y Y Product-time FE Y Y Y Y Buyer-product FE Y Y Y Buyer-product-time FE Y Y Y Seller-product FE Y Y Y Y Y Y Notes: OLS estimates come from estimating Equation (11). Time is monthly frequency. The first row reports the estimates associated with changes in log relative unit values. Standard errors are two-way clustered at the origin and destination state level and are reported in parentheses below each estimate. The third row reports the implied value for , which is 1 minus the estimate on the first row. The fourth row reports the number of observations. The fifth row reports the goodness of fit (R2 ). The table contains seven columns: Each corresponds to different specifications on how we define a product (4-digit, 6-digit, or 8-digit HS codes), which fixed effects are included, and whether the Feenstra (1994) term is included. These combinations are reported by the last seven rows of the table. All specifications include the controls mentioned in the paper. term. Each specification reports a high Kleibergen-Paap F-statistic, indicating that our instruments are statistically relevant. In columns (1) and (2), we use the seller-level and seller-buyer-level instruments separately. The elasticities are 0.50 and 0.61, respectively, which also reflect com- plementarity. In column (4), we also include buyer-by-product and seller-by-product fixed effects, and the elasticity rises to 0.66. Columns (5)-(6) show that our complementarity results remain after including stronger sets of fixed effects. All our estimates provide evidence for < 1. The 2SLS estimates for are smaller than the OLS estimates. As discussed in Section 5.3, the bias is in the expected direction if we expect the Covid-19 shock to also induce negative demand shocks, thereby biasing up OLS estimates of . We may expect that our estimated elasticity will be lower for the sub-sample of buyers who did not have more than one supplier to source inputs from. In Table A5, we restrict our sample to cases when a buyer traded with at least two sellers in two consecutive periods. Column (5), again yields an elasticity of substitution of 0.56, very close to the estimate from our main specification. To account for the extensive margin, we show the results from estimating Equation 14 in Table A6. Again, we find evidence of high levels of complementarity in the short-run, with the estimated elasticities ranging from 0.23-0.81 depending on the range of fixed effects used. 27 TABLE 2: 2SLS, firm-level elasticity of substitution across suppliers (1) (2) (3) (4) (5) (6) ˆ p log ˆ ˜ p 0.5042 0.3945 0.4538 0.3409 0.4418 0.3535 (0.2129) (0.0933) (0.1389) (0.1068) (0.1982) (0.1701) 0.4957 0.6054 0.5461 0.6590 0.5581 0.6464 Obs 2854292 2028039 2028039 1966591 1020362 991187 K-PF 48.232 133.688 143.413 248.977 69.827 62.215 Seller IV Y Y Y Y Y Bilateral IV Y Y Y Y Y Feenstra term Y Y Y Y Buyer-time FE Y Y Y Y Product-time FE Y Y Y Y Buyer-product FE Y Seller-product FE Y Y Buyer-product-time FE Y Y Notes: 2SLS estimates come from estimating Equation (11). Time is monthly frequency. The set of common suppliers of buyer b is Ω∗i,b j,t = Ωi,b j,t ∩ Ωi,b j,t −1 . That is, a supplier s of buyer b is considered common if they traded in both the current and previous month. The first stage uses either bilateral or seller-level instruments, as reported in rows six and seven. Bilateral instruments correspond to Equation (17), and seller-level instruments correspond to Equation (16). The first row reports estimates associated with changes in relative unit values in logs. Standard errors are two-way clustered at the origin and destination state level and are reported in parentheses below each estimate. The third row reports the implied value for , which is 1 minus the estimate on the first row. The fourth row reports the number of observations. The fifth row reports the Kleibergen-Paap F statistic from the first stage. A product is a 4-digit HS code, and the treatment period is March-May 2020. The table contains six columns. Each column corresponds to different combinations of instruments, fixed effects, and whether the Feenstra (1994) term is included. These combinations are reported in the last eight rows. All specifications include the controls mentioned in the paper. 6.2 Firm-level elasticities of substitution across products In Table 3, we report our estimates for the firm-level elasticity of substitution across products. In column (1), we show the OLS estimate of ζ = 0.92, which reflects complementarity between product categories. In columns (2) and (3), we define products more granularly. In this case, the elasticities are around 0.80, which also reflects complementarity between products. In columns (4)-(6), we report our estimates of ζ under 2SLS estimation after using a weighted average of instruments across buyers’ sellers as discussed in Section 5.3. Our specification in column (4) reports a value of 0.69, reflecting that simultaneous negative demand and supply shocks during Covid-19 led to an underestimation of ζ under OLS. This elasticity is higher than the 2SLS elasticity of substitution across suppliers for the same product ( = 0.55), reflecting a lower degree 28 TABLE 3: Firm-level elasticity of substitution across products (1) (2) (3) (4) (5) (6) ˆ p log ˆ ˜ p 0.0842 0.2014 0.1996 0.3136 0.1712 0.1996 (0.0039) (0.0045) (0.0048) (0.1060) (0.0040) (0.0048) ζ 0.9157 0.7985 0.8003 0.6863 0.4368 0.4721 Obs 1292329 794376 766804 1292329 794376 766804 K-PF . . . 27.284 17.950 15.868 Estimator OLS OLS OLS 2SLS 2SLS 2SLS HS digits 4 8 8 4 8 8 Product-time FE Y Y Y Y Y Y Buyer-time FE Y Y Y Y Y Y Buyer-product FE Y Y Notes: The estimates come from estimating Equation (10). Time is monthly frequency. Price indices are constructed by recovering the residuals when estimating and the estimates of . The first three columns are OLS estimates of ζ . The last three columns are 2SLS of estimates of ζ using weighted averages of both bilateral and seller-level instruments across sellers. Bilateral instruments correspond to Equation (17), and seller-level instruments correspond to Equation (16). Each column corresponds to a different combination of fixed effects and HS codes. Columns (1)-(2) and (4)-(5) correspond to our preferred specification when estimating for 4-digit and 8-digit HS codes. Additionally, in columns (3) and (6), we also include buyer-by-product fixed effects. The first row reports the estimates associated with changes in log relative unit values. Standard errors are clustered at the buyer’s district level and are reported in parentheses below each estimate. The third row reports the implied value for ζ , which is 1 minus the estimate on the first row. The fourth row reports the number of observations. The fifth row reports the Kleibergen-Paap F statistic from the first stage. The sixth row denotes whether the estimates are obtained through OLS or 2SLS. The seventh row reports the HS code. The last three rows indicate the combination of fixed effects. of complementarity across products compared to suppliers.27 In columns (5) and (6), similar values for this elasticity hold when we define a product as an 8-digit HS code, and after the inclusion of buyer-by-product fixed effects. Finally, first-stage F-stats are high, which reflects the statistical relevance of our weighted averaged instruments. Unlike the elasticity of substitution across suppliers within a product category, there have been previous attempts in the literature to estimate the elasticity of substitution across products or industries. In particular, other work has estimated a wide range of values for parameters akin to ζ depending on the aggregation of the industry and the research question. Our elasticity is close to Boehm et al. (2019), who estimate an elasticity across HS-10 products that lies between 0.42 − 0.62 for non-Japanese affiliates and 0.20 for Japanese affiliates. Atalay (2017) finds an estimate of around 0.10 for 30 aggregated industries using US data. 27 This is consistent with the macroeconomics literature (Houthakker, 1955; Bachmann et al., 2022; Lagos, 2006). 29 7 W HY ARE SUPPLIERS COMPLEMENTARY IN THE SHORT- RUN ? Our estimates indicate that a buyer’s expenditure share for a given supplier rises with the input price of that same supplier in the short run. This suggests that buyers cannot easily substitute away from the input supplied by this specific supplier. Note that this does not necessarily mean that when the price of a seller’s input rises, a buyer purchases less from a different seller. Instead, it indicates that the buyer is unable to replace a sufficient amount of the input from the high-cost supplier with one from the lower-cost substitute to lower the expenditure share attributed to the high-cost supplier. The inability of the firm to substitute suppliers in the short run goes in line with other studies arguing that it is hard for firms to substitute across different products in the short-run (Boehm et al., 2019; Barrot and Sauvagnat, 2018). We show that this inability to substitute exists even when we consider individual suppliers of the same product. In this section, we explore different possibilities that can explain this complementarity result. In the process of doing so, we document relevant heterogeneity in the elasticity estimates. Institutional Quality and Input Specificity. Our complementarity result may arise if inputs are often sourced from specific suppliers and customized, rather than bought “off-the-shelf".28 Relatedly, specialized inputs have a greater need for contract enforcement in trade, and so rely on stronger court institutions. We test these two hypotheses. First, we study whether intermediate inputs that are differ- entiated (i.e., relationship-specific) display higher complementarity. To test this hypothesis, we estimate our firm-level elasticities of substitution across suppliers conditional on whether the seller offers an intermediate input classified as either “Differentiated" or “Homogeneous" according to Rauch (1999). Second, we study whether intermediate inputs that require highly-specified contracts and, therefore, are likely to be traded in locations with better institutional quality display higher com- plementarity (Boehm, 2022; Boehm and Oberfield, 2020). To test this hypothesis, we use the geographic variation in institutional quality across Indian districts. We classify districts into ter- ciles (Low, Medium, and High) according to the average time it takes for courts in each Indian district until a first hearing from the DevData Lab database, and we estimate our firm-level elas- 28 For instance, in Elliott and Golub (2022), firms need specific knowledge about input requirements and capabilities from their trading partners and therefore need an ongoing relationship to overcome moral hazard problems. 30 ticities of substitution across suppliers conditional on whether buyers are located in each of these terciles of institutional quality. In Table A7, we report 2SLS estimates of our elasticity by input specificity and institutional quality In the first two columns, we focus on input specificity. As expected, we find an elasticity of substitution across suppliers for differentiated inputs of 0.62, and for homogeneous inputs of 1.07. The higher degree of complementarity of input suppliers for differentiated goods, in compar- ison to homogeneous goods, reflects the importance of customized products that typically require durable relationships between buyers and sellers. Instead, for homogeneous inputs, the elasticity of substitution is greater than 1. The last three columns focus on institutional quality. We find an elasticity of substitution across suppliers located in low-quality districts of 0.80, which is higher than the elasticity for suppliers located in high-quality districts of 0.37. The elasticity of substitution across suppliers lo- cated in medium-quality districts is 0.58, which is just in between the previous two estimates. This positive relationship between the degree of complementarity and the quality institutions reflects the fact that places with better contract enforcement facilitate trade with more specialized inputs that are harder to substitute. Industry-Specific Elasticities. An alternative way to test for the role of input-specificity for our complementarity result is to check whether firms that source highly specific intermediate inputs (e.g., processed foods or art) should report a lower elasticity of substitution across suppliers than firms that source from more general inputs (e.g., textiles or jewelry). We analyze whether the degree of substitution across suppliers varies by industry (HS section). In Table A8 and Figure A5, we show the estimates of this elasticity of substitution across suppliers by twenty-one broad industries. We find that the OLS elasticity of substitution across suppliers by industry lies in the range of 0.41 − 0.87. Once we instrument for the unit values with the Covid-induced lockdown variation, we find that there is wider heterogeneity across industries. These reflect industry-specific ease of finding alternatives. For instance, Art yields the lowest elasticity of 0.149, perhaps as it reflects seller-specific products, whereas Jewelry yields the highest elasticity of 1.372, perhaps as gold and silver are not particularly seller-specific. Longer-run elasticities of substitution across suppliers. Our estimates and identification strat- egy are only applicable in the short run since the lockdowns only lasted for three months. But the elasticity of substitution is necessarily tied to the time horizon (Ruhl et al., 2008; Peter and Ruane, 2022). We estimate a short-run elasticity (monthly), so we allow firms a month to adjust expen- ditures in response to price shocks. A potential avenue of exploration is to understand how this 31 elasticity changes with the frequency of the measurement. As such, we estimate our elasticities after aggregating the data at the quarterly level, so we allow firms a quarter to adjust expenditures in response to price shocks. In Table A9, we report OLS and 2SLS estimates of the firm-level elasticity of substitution across suppliers at both frequencies. The first two columns show our baseline estimates of at a monthly frequency. They correspond to column 1 from Table 1 and column 3 from Table 2. The last two columns show the estimates of at a quarterly frequency. As expected, the estimates at quarterly frequency are slightly higher than the monthly frequency ones: the OLS estimates go up from 0.78 to 0.86, and the 2SLS estimates go up from 0.55 to 0.79. Our quarterly estimates still exhibit complementarity in production since they are below 1. Inventories. Inventories of intermediate inputs allow firms to absorb unforeseen shocks to input deliveries without an impact on production (Boehm et al., 2019). Do firms in industries that typi- cally hold higher inventories have an easier time substituting across inputs from different suppliers? To test this hypothesis, we estimate the elasticity of substitution between suppliers conditional on sellers or buyers belonging to industries of “Low" or “High" inventory. In Table A10, we show the results disaggregated by four types of industries: Industries where sellers typically hold high and low inventories (above or below median inventory-level across all industries) and industries where buyers typically hold high and low inventories. We find that, indeed, when buyers are in industries with high levels of inventory, inputs across suppliers of the same product are much less complementary (column 4). This is because the buyer can temporarily substitute a costly input from a seller with its input inventory holdings. From column (2), we find that the level of inventory of a seller’s industry matters much less: Inputs remain highly complementary and close to our baseline estimate of 0.55 irrespective of the inventory level of the seller’s industry. Elasticities by Product Aggregation. Finer product classifications (e.g., HS-8) may imply that there are fewer suppliers one may be able to source from, and so we may expect a lower elasticity of substitution between suppliers. To examine these differences, we re-estimate our main spec- ification in Table A11 using HS-6 and HS-8 as product definitions. In columns (1) and (3), we replicate our main specifications, with elasticities of 0.43 (for HS-6) and 0.06 (for HS-8), respec- tively. These numbers reflect higher degrees of complementarity when we consider a more granular notion of product. Overall, these patterns suggest that inputs are highly specific for buying firms. 32 8 Q UANTIFICATION AND COUNTERFACTUALS In this section, we use both data from our production network and our newly estimated elasticities to quantify the role of these elasticities in the propagation of shocks. To do this, we need to write down the Leontief matrix in standard form. Given the production structure of our economy, we need four submatrices: (i) firm purchases of 4-digit HS products, (ii) firm sales of 4-digit HS products, (iii) labor employed by each firm, and (iv) final sales by each firm. The first two submatrices are directly constructed from the firm-to-firm trade data from the pre-Covid-19 period of March 2019 to February 2020. Labor employed and final sales by firms are obtained by merging in firm-level data from IndiaMART 29 , which contains information on firm-level employment and final sales. For more details on this, see Appendix C. The economy is comprised of N = 93, 260 firms, I = 1, 293 different HS-4 products, labor, and a composite final good. The average firm buys 10 distinct products as a buyer and sells 5 distinct products as a seller. The most connected buyer and seller buys and sells over 500 distinct products. We use this 94, 555 × 94, 555 input-output matrix to understand how complementarities at the firm level affect the propagation of shocks through production networks. While recent work also quantifies the effect of firm-level shocks on aggregate GDP up to the second order, they mostly rely on changes in firm-level final sales rather than the full production network. Instead, we identify the most connected firms, deriving the Leontief inverse from the en- tire production network. Without relying on any approximation, we use the full network to quantify the importance of firm connectivity separately from firm size. This exponentially increases com- putational complexity from the order of (N + I + 2) to (N + I + 2)2 . As such, we use computational innovations in big data to implement this procedure. For more details on the derivation of the shock propagation equation and its numerical implementation, see Appendix E. Note that our quantification exercises in this section are conditional on the products that firms buy or sell being given at the extensive margin, even though a firm can change its set of buyers or suppliers (Baqaee et al., 2023; Khanna et al., 2022). We, therefore, need to empirically assess whether the set of HS-4 products a buyer buys and the set of HS-4 products that a seller sells changes between the pre and the post-Covid-19 period. We do this by inspecting whether both sellers and buyers of each product continued to trade in their corresponding product categories after the Covid-19 lockdowns. In Figure A6, we show the product-level distribution of the share of sellers that sold and buyers that purchased goods of that product during both periods t and t − 1, where t is a 6-month window before and after the lockdowns. In the figure, we see that, for both sellers and buyers, these two distributions are very similar to each other. The overall stability in 29 See IndiaMART (link). 33 Figure A6 shows that the assumption that the products that firms buy or sell do not change is tenable when analyzing the impact of negative productivity shocks. 8.1 How important is the firm-level elasticity of substitution across suppliers? We assess the importance of the estimated firm-level elasticity of substitution across suppliers for the same product by studying how this elasticity determines the impacts of negative firm-level productivity shocks on aggregate GDP. In this counterfactual, we shock the productivity of firms located in Red zones by 25%. We find that this productivity shock reduces GDP by 10.95%. As an empirical benchmark, the state’s annual GDP fell by 11.3% in 2020/21. This fall would be 2.68 pp less in a model where firms in the same HS-4 product are considered substitutes ( = 2) and 0.99 pp more when firms in the same HS-4 product are considered almost Leontief ( = 0.001). Given the quarterly GDP of this state in 2020-2021, the additional losses due to firm-level complementarities translate into 870 million USD, which is about 25 USD per capita per quarter, compared to the case when firms are substitutes.30 Note that the differences in GDP that arise from changing the values of firm-level elasticities of substitution across suppliers only change the second-order effects on GDP, not the first-order. Then, how important are these second-order effects that we have estimated? In Figure 4, we simulate different levels of negative productivity shocks for four different values of the elasticity and plot the second-order percentage point change in GDP due to these shocks. The blue and red lines show these differences for high levels of complementarity between suppliers: 0.001 and our estimated elasticity 0.55, respectively. The green and yellow lines show the additional second- order change in GDP for high levels of substitution across suppliers: 1.75 and 1.25, respectively. These plots provide two main lessons. First, for a given negative productivity shock, the second-order effects intensify with the degree of complementarity between suppliers. Second, given the same value of , the second-order effects intensify with the magnitude of the productivity shocks. Finally, as suppliers exhibit higher substitutability, the second-order effects dampen the negative first-order effects, and more so, for higher values of productivity shocks. When suppliers instead exhibit complementarity, the second-order effects magnify the negative first-order effects. That is, unlike the first-order effects, which only depend on firm size, complementarities at the firm level non-linearly amplify the effects of negative productivity shocks. This reflects similar amplification patterns that (Baqaee and Farhi, 2019) document, but at the industry level. These graphs illustrate the importance of second-order effects largely driven by firm complementarities, 30 To put these numbers into perspective, Baqaee and Farhi (2019) showed that complementarities at the industry level, with an elasticity of substitution 0.001, amplify the effect of a negative 13% shock in the oil industry on GDP by around 0.61%. 34 especially for large, short-lived, negative productivity shocks such as Covid-19. F IGURE 4: How important are second-order effects? Notes: The horizontal axis is the percentage change in productivity for firms in Red districts. The vertical axis is the second-order change in GDP in percentage points for different values of the firm-level elasticity of substitution across suppliers ( ). Different values of the elasticity ( = 0.001, = 0.55, = 1.25, and 1.75) are plotted with different colors. 8.2 How important is a firm’s connectivity in its network? Since Hulten (1978), policy-makers and researchers have emphasized the importance of firm sizes in the propagation of shocks. Now, we investigate the importance of firm connectedness, given fixed firm sizes, in the propagation of shocks. We conduct these counterfactuals for different values of elasticities of substitution when suppliers are complementary, as our empirical analysis points to strong complementarities between suppliers. In this counterfactual, we explore the importance of a firm’s connectivity in its network. We measure firm connectivity by its value within the Leontief inverse matrix, which measures firms’ direct and indirect connections to other firms.31 Firm size is measured by its Domar weight. Since firm sizes and connectivity are highly correlated with a correlation coefficient of 0.75, we vary the firms’ connectivity for a given level of firm size to tease out the pure effect of connec- tivity. To implement this, we choose firms with Domar weights equal in size for up to 5 decimal 31 The (i, j)th entry of the Leontief is a measure of firm i’s total reliance on j as a supplier. Summing across all i’s yields a measure of the connectivity of each supplier j or its importance in the firm network in terms of connectivity. 35 F IGURE 5: Second order GDP effects when firms with the same size but different connectivity are affected (a) = 0.001 (b) = 0.55 (c) = 0.98 Notes: The figure comprises three panels. In each panel, the horizontal axis is the percentage change in productivity for firms in our state, and the vertical axis is the second-order change in GDP, in percentage points for different values of firm-level elasticity of substitution across suppliers ( ). Each panel corresponds to different values of the firm-level elasticity of substitution across suppliers ( ). Each panel contains three scenarios. The first scenario is the blue line, where the most connected firms are shocked. The second scenario is the green line, where random firms are shocked. The third scenario is the red line, where the least connected firms are shocked. places. The first set consists of the most connected firms, the second set is a random draw of firms, and the third set consists of the least connected firms. Since firm sizes are given, the first-order effects are the same irrespective of how connected the firms are.32 In Figure 5, we only plot the second-order effects on GDP under these three experiments. In the first scenario, only the most connected firms are affected by negative productivity shocks (blue line). In the second, a random draw of firms is affected (green). Finally, in the third scenario, only 32 This is the only counterfactual where we draw firms from the entire state rather than only firms located in Red zones to maximize the number of firms that vary in connectivity for a given firm size. 36 the least connected firms are affected (red). We perform these experiments under three different elasticities of substitution: an elasticity of substitution amounting to near perfect complementarity ( = 0.001) in the left panel, our estimated complementarity ( = 0.55) in the right panel, and near Cobb-Douglas ( = 0.98) in the bottom panel. All these experiments are conditional on given firm sizes; that is, we vary the connectivity of firms after matching on firm sizes. These counterfactuals show that the fall in GDP is much larger if the most connected firms are affected compared to the least connected firms or a random set of firms for a given firm size (Domar weight).33 The importance of the most connected firms increases non-linearly with the negative productivity shocks: as the shock gets larger, it becomes increasingly important to give attention to the most connected firms. Our experiment suggests that for our baseline value of elasticity of substitution ( = 0.55) and a negative productivity shock of 45%, if the better-connected firms are allowed to operate, given the same firm sizes, compared to randomly targeting firms, the fall in GDP would be 0.20 pp less, and 0.31 pp less compared to targeting the least connected firms. We notice three patterns: First, in the near Cobb-Douglas case ( = 0.98), the differences in GDP when allowing the least or the most connected firms to operate are negligible because the second-order effects are negligible. Second, as the level of productivity shock increases, it becomes more important to save the most connected firms. While for a low productivity shock of 5%, the differences in GDP are negligible (0.001 pp and 0.002 pp), for a productivity shock of 25%, these differences are 0.05 pp and 0.07 pp compared to saving randomly connected and the least connected firms. Third, the effects of these non-linearities are more pronounced when suppliers are highly complementary. For near-perfect complementarity ( = 0.001) and a high negative productivity shock (−45%), the gains from saving the most connected firms in an economy compared to saving randomly targeted and least connected firms are 0.38 pp and 0.60 pp, which is almost double the gains if instead suppliers were moderately complementary ( = 0.55). 8.3 How important is measuring a firm’s total (direct plus indirect) connectivity? Existing work has shown that shocks to a firm’s suppliers affect the buyer firm and its suppliers (Barrot and Sauvagnat, 2018). But shocks to a firm can affect not just its direct but also other indirect connections (Carvalho et al., 2021). In this counterfactual, we quantify how important it is to take into account a firm’s indirect connectivity in understanding how firm-level shocks affect aggregate GDP. To be precise, a firm’s indirect connections measure not only the number of direct 33 Relatedly, Liu and Tsyvinski (2020) show that negative shocks to upstream sectors can have more adverse effects on GDP despite having identical Domar weights. While our effects materialize from the second-order propagation through a roundabout production network with firm complementarity, the results in Liu and Tsyvinski (2020) stem from adjustment costs in a vertical production network. 37 buyers of a supplier but also the buyers’ buyers and their buyers and so on.34 F IGURE 6: How important is a firm’s connectivity in its network? Notes: The horizontal axis is the percentage change in productivity for firms in Red districts. The vertical axis is the second-order change in GDP in percentage points for = 0.55. The blue line corresponds to the baseline case when all firms in the Red districts are affected. The red line corresponds to the case when the government only bails out the 10% most directly connected firms. The green line corresponds to the case when the government bails out the 10% firms with the most (direct+indirect) total connections. We conduct two experiments. In the first, the most directly connected 10% firms in Red zones are allowed to operate, where direct connectivity is measured by the number of buyers a supplier directly supplies (red line in Figure 6). In the second, the most connected 10% firms in Red zones are allowed to operate, where the total connectivity of a firm is measured by all its direct and indirect connections (green line in Figure 6). Note that, unlike the previous counterfactual, we do not fix firm sizes. We are interested in understanding if the most directly connected firms are allowed to operate as opposed to the most connected firms, irrespective of size, and how that would affect aggregate GDP. We report the total effect on GDP under these two sets of experiments and the baseline results (shock to all firms in Red zones). We find that, under our estimated elasticity of = 0.55 and a negative productivity shock of 25%, the fall in GDP would be 2.56 pp less if firms were allowed to operate based on total connectivity as opposed to direct connectivity. We see that as the level of the negative productivity shock increases, the difference in aggregate GDP between these two sets of experiments rises, emphasizing the importance of measuring a firm’s indirect connections as well. 34 As a reminder, we measure the total connectivity of a firm by its value within the Leontief inverse matrix, which measures firms’ direct and indirect connections to other firms. 38 9 C ONCLUSION In this paper, we use highly disaggregated firm-to-firm transaction-level data from a large Indian state and provide one of the first estimates of firm-level elasticities of substitution across suppliers within the same product category. We provide new estimation strategies and estimates for these elasticities by leveraging regional variation in supply-side shocks induced by the Indian govern- ment’s massive lockdown policy. We find that suppliers of inputs are highly complementary even at this very granular level. We explore explanations behind this low elasticity, showing that con- tract specificity of inputs, the level of inventory of the buyer’s industry, and the time horizon matter. This elasticity crucially determines aggregate impacts and the transmission of shocks across the network but is especially difficult to pin down (Baqaee and Farhi, 2019). The combined advantage of having product-level unit values and quasi-experimental variation in supply-side shocks allows us to overcome previous challenges in the literature, and credibly estimate this elasticity across suppliers of a particular product. Since inputs are complementary, adverse shocks to even a small subset of firms that are highly linked in the supply chain can negatively affect the aggregate economy by propagating through firm networks. When we conservatively shock only the productivity of firms located in Red zones by 25%, we find that if suppliers of the same product were substitutes instead of comple- ments, the fall in aggregate quarterly GDP in the state under study would be about 870 million USD lower, or about 25 USD per capita lower per quarter. Using big data computational techniques, we quantify this decline directly using information on the economy-wide firm-to-firm network with- out relying on first-order approximations. Our methods thus provide new techniques to quantify shocks through large and complex production networks. Using data on the entire production net- work in the state, we measure the full connectivity of firms in the network and show that as the level of complementarity and the magnitude of the negative productivity shock increase, it is more effective to save the more connected firms, after controlling for firm size. 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The International Elasticity Puzzle. unpublished paper, NYU , 703–726. 43 Appendix for online publication only A A PPENDIX TABLES TABLE A1: Summary statistics Panel A: 2019 Jan-March April-June July-September Number of sellers 135,849 131,996 133,897 Number of buyers 193,660 188,708 189,219 Total sales (mln. rupees) 962,688 908,361 1,036,831 Number of transactions 7,772,883 7,808,325 7,934,706 Panel B: 2020 Jan-March April-June July-September Number of sellers 113,121 69,171 86,696 Number of buyers 164,153 114,353 135,056 Total sales (mln. rupees) 811,755 369,645 775,478 Number of transactions 7,362,508 3,201,081 4,782,336 Notes: This table consists of two panels. Panel A contains information about the number of sellers, buyers, trans- actions, and total sales for the periods January-March, April-June, July-September for the year 2019. Panel B is the same as Panel A, but for 2020. i TABLE A2: Indegree summary statistics # suppliers per industry-by-buyer-by-time Mean St. Dev. P1 P50 P99 4-digit HS 6.134 16.719 1 2 60 4-digit HS, monthly 1.432 1.662 1 1 7 4-digit HS, quarterly 1.574 2.122 1 1 9 6-digit HS 5.168 12.733 1 2 45 6-digit HS, monthly 1.300 1.275 1 1 6 6-digit HS, quarterly 1.399 1.652 1 1 7 8-digit HS 5.078 10.331 1 2 40 8-digit HS, monthly 1.207 .958 1 1 4 8-digit HS, quarterly 1.266 1.073 1 1 5 Notes: This table reports number of observations, mean, standard deviation, and percentiles 1/50/99 for the number of suppliers per product-by-buyer-by-time. Each row reports these statistics by different definitions of a product (4, 6, and 8 digit HS codes) and by time frequency (atemporal, monthly, and quarterly). TABLE A3: Outdegree summary statistics # buyers per seller-by-time Mean St. Dev. P1 P50 P99 4-digit HS 64.182 532.814 1 5 985 4-digit HS, monthly 9.211 44.908 1 2 124 4-digit HS, quarterly 12.035 63.022 1 2 169 6-digit HS 69.082 598.099 1 6 1047 6-digit HS, monthly 9.904 50.472 1 2 134 6-digit HS, quarterly 13.041 71.208 1 2 185 8-digit HS 73.293 768.552 1 5 1120 8-digit HS, monthly 11.097 66.348 1 2 151 8-digit HS, quarterly 13.893 86.029 1 2 199 Notes: This table reports number of observations, mean, standard deviation, and percentiles 1/50/99 for the number of buyers per seller-by-time. Each row reports these statistics by different definitions of a product (4, 6, and 8 digit HS codes) and by time frequency (atemporal, monthly, and quarterly). ii TABLE A4: Distribution of economic activity by industry and type of transaction HS section Sales share Purchase share Animals 1.5034 0.7723 Vegetables 15.2982 11.2945 Fats 2.2934 2.6251 Processed foods 4.2172 5.5548 Minerals 13.1241 10.2353 Chemicals 9.8288 9.0791 Plastics 13.1516 9.1410 Leather 0.1618 0.1677 Wood 2.5110 1.2130 Wood derivatives 1.0783 1.3598 Textiles 3.6342 6.4576 Clothing 1.3428 0.9107 Handicrafts 1.0190 1.9337 Jewelry 1.7005 1.4980 Metal 10.4473 12.1969 Machinery 10.9909 13.5771 Transport equipment 4.7124 8.4147 Surgical instrum. 1.4478 1.6478 Arms and ammo 0.0057 0.0095 Miscellaneous 1.2263 1.4936 Art 0.3043 0.4166 Type of transaction Within-state 72.6822 52.2224 Inter-state 23.2183 44.5151 Foreign 4.0994 3.2623 Notes: The table consists of an upper panel and a lower panel. In the upper panel, we show the share of sales to and purchases from our Indian state of analysis by industry (HS Section). In the lower panel, we show the share of sales to and purchases from our Indian state, by whether the buyer or seller is within the state, in another state of India, or abroad. Statistics were calculated using data for 2019. iii TABLE A5: 2SLS, firm-level elasticity of substitution across (at least two) suppliers (1) (2) (3) (4) (5) (6) ˆ p log ˆ ˜ p 0.2383 0.3381 0.4121 0.3688 0.4418 0.4048 (0.1206) (0.0627) (0.1236) (0.1146) (0.1982) (0.2223) 0.7616 0.6618 0.5878 0.6311 0.5592 0.5952 Obs 851120 599918 599918 544819 1020362 751411 K-PF 58.989 97.958 233.084 527.534 69.827 98.916 Seller IV Y Y Y Y Y Bilateral IV Y Y Y Y Y Feenstra term Y Y Y Y Buyer-time FE Y Y Y Y Product-time FE Y Y Y Y Buyer-product FE Y Seller-product FE Y Y Buyer-product-time FE Y Y Notes: 2SLS estimates are obtained from estimating Equation (11). Time is monthly frequency. The set of common suppliers of buyer b is Ω∗i,b j,t = Ωi,b j,t ∩ Ωi,b j,t −1 . That is, a supplier s of buyer b is considered common if they traded during both the current and previous month. We only consider the cases when a buyer traded with at least two common suppliers in a given period. The first stage uses either bilateral or seller-level instruments, as pointed out by rows six and seven. Bilateral instruments correspond to Equation (17), while seller-level instruments correspond to Equation (16). The first row reports the estimates associated with changes in log relative unit values. Standard errors are two- way clustered at the origin and destination state level, and are reported in parentheses below each estimate. The third row reports the implied value for , which is 1 minus the estimate on the first row. The fourth row reports the number of observations. The fifth row reports the Kleibergen-Paap F statistic from the first stage. A product category is a 4-digit HS code and the treatment period is March-May 2020. The table contains six columns. Each column corresponds to different combinations of instruments fixed effects, and whether the Feenstra is included, as pointed out by the last eight rows. All specifications include the controls mentioned in the paper. iv TABLE A6: 2SLS, extensive margin: firm-level elasticity of substitution across suppliers (1) (2) (3) (4) log ( p) 0.7460 0.1905 0.2503 0.7707 (0.7568) (0.5827) (2.118233) (0.9545) 0.2539 0.8094 0.7496 0.2292 Obs 719375 519777 502828 502828 K-PF 56.776 118.441 27.993 24.290 Seller IV Y Y Y Bilateral IV Y Y Y Buyer-product-time FE Y Y Y Y Seller FE Y Notes: The first row reports 2SLS estimates from Equation (14). Time is monthly frequency. The first stage uses either bilateral or seller-level instruments, as pointed out by rows six and seven. Bilateral instruments correspond to Equation (17), while seller-level instruments correspond to Equation (16). The first row reports the estimates associated with changes in log unit values. Standard errors are two-way clustered at the origin and destination state level, and are reported in parentheses below each estimate. The third row reports the implied value for , which is 1 minus the estimate on the first row. The fourth row reports the number of observations. The fifth row reports the Kleibergen- Paap F statistic from the first stage. A product category is a 4-digit HS code and the treatment period is March-May 2020. Rows eight and nine report whether buyer-by-product-time or seller fixed effects are included. v TABLE A7: 2SLS, firm-level elasticity of substitution across suppliers, by institutional quality and input specificity Input specificity Institutional quality Differentiated Homogeneous Low Mid High ˆ p log ˆ ˜ p 0.3766 -0.0741 0.1961 0.4186 0.6637 (0.1139) (0.2652) (0.1089) (0.3300) (0.1114) 0.6233 1.0741 0.8038 0.5813 0.3362 Obs 1267668 457219 628824 784589 515548 K-PF 708.404 299.327 72.520 246.447 61.899 Notes: The first row reports 2SLS estimates from Equation (11). Time is monthly frequency. For the 2SLS estimates, the set of common suppliers of buyer b is Ω∗ i,b j,t = Ωi,b j,t ∩ Ωi,b j,t −1 . That is, a supplier s of buyer b is considered common if they also traded during the previous month. The first stage uses both bilateral and seller-level instruments. Bilateral instruments correspond to Equation (17), while seller-level instruments correspond to Equation (16). Standard errors are two-way clustered at the origin and destination state level, and are reported in parentheses below each estimate. The third row reports the implied value for . The fourth row reports the number of observations. The fifth row reports Kleibergen-Paap F statistics from the first stage. The table is comprised of five columns. The first two columns report estimates conditional on the seller’s input classified as “Differentiated" or “Homogeneous" according to Rauch (1999). The last three columns report estimates conditional on the institutional quality of buyer’s location. The locations of buyers are categorized in terciles according to their institutional quality from the DevData Lab database. Districts in the first tercile of institutional quality are categorized as “Low", in the second tercile are categorized as “Medium", and districts in the third tercile of institutional quality are categorized as “High". This variable is measured as the average number of days a district’s courts until the first hearing. All specifications include buyer-by-time, product-by-time, buyer-by-product, and seller-by-product fixed effects. vi TABLE A8: Firm-level elasticities of substitution across suppliers, by HS section Section Name OLS elast. 2SLS elast. 1 Animals 0.6892 0.1648 2 Vegetables 0.7799 0.7149 3 Fats . . 4 Processed foods 0.7125 0.1917 5 Minerals 0.8326 0.3974 6 Chemicals 0.7735 0.5828 7 Plastics 0.7179 0.9796 8 Leather . . 9 Wood 0.8728 0.6154 10 Wood derivatives 0.7812 0.8915 11 Textiles 0.8249 0.8103 12 Clothing 0.8232 0.3360 13 Handcrafts 0.6737 . 14 Jewelry 0.8104 1.3721 15 Metal 0.8145 0.8142 16 Machinery 0.6072 0.8691 17 Transport equipment . . 18 Surgical instruments 0.5954 0.3799 19 Arms and ammo 0.4140 . 20 Miscellaneous 0.6903 0.8383 21 Art 0.5514 0.1486 Notes: Each row corresponds to an industry, which is defined as an HS section. Time is monthly frequency. The second column contains the name of the industry. The third and fourth columns report the estimated elasticities by OLS and 2SLS from Equation (11). Both OLS and 2SLS estimators include product-by-time, buyer-by-time, buyer- by-product, and seller-by-product fixed effects. Standard errors are two-way clustered at both origin and destination states. All specifications include the controls mentioned in the paper. Elasticities were not reported if there was low statistical power or a weak first stage. vii TABLE A9: Firm-level elasticity of substitution across suppliers, monthly vs. quarterly Normal FEs Monthly Quarterly OLS 2SLS OLS 2SLS ˆ p log ˆ ˜ p 0.2171 0.4538 0.1436 0.2116 (0.0133) (0.5461) (0.8564) (0.7883) 0.7828 0.5461 0.8564 0.7883 Obs 2028039 2028039 1518102 1518102 K-PF . 143.413 . 11.501 Strong FEs ˆ p log ˆ ˜ p 0.3062 0.5238 0.1924 0.5326 (0.0099) (0.2273) (0.0082) (0.3737) 0.6937 0.4761 0.8075 0.4673 Obs 1712307 1106739 891790 961465 K-PF . 87.134 . 17.253 Notes: The first row report the OLS estimates from Equation (11), and 2SLS estimates from Equation (11). For the 2SLS estimates, the set of common suppliers of buyer b is Ω∗ i,b j,t = Ωi,b j,t ∩ Ωi,b j,t −1 . That is, a supplier s of buyer b is considered common if they also traded during the previous period. The first stage uses both bilateral and seller-level instruments. Bilateral instruments correspond to Equation (17), while seller-level instruments correspond to Equation (16). Periods are either months or quarters depending on the frequency of the data. The first two columns are monthly estimates, and the last two columns are quarterly estimates. Standard errors are two-way clustered at the origin and destination state level, and are reported in parentheses below each estimate. The third row reports the implied value for . The fourth row reports the number of observations. The fifth row reports Kleibergen-Paap F statistics from the first stage. “Normal FEs" include buyer-by-period and product-by-period fixed effects, while “Strong FEs" also include buyer-by-product and seller-by-product fixed effects. viii TABLE A10: 2SLS, firm-level elasticity of substitution across suppliers, by levels of inventory Sellers’ industry Buyers’ industry inventories inventories Low High Low High ˆ p log ˆ ˜ p 0.4120 0.3847 0.4504 -0.0800 (0.1881) (0.4584) (0.1984) (0.4108) 0.5879 0.6152 0.5495 1.080026 Obs 700938 698996 319318 293203 K-PF 274.5876 161.5413 750.8763 192.9017 Mean Inventories/Sales 0.181 1.766 0.200 2.523 Notes: The first row reports 2SLS estimates from Equation (11). Time is monthly frequency. For the 2SLS estimates, the set of common suppliers of buyer b is Ω∗i,b j,t = Ωi,b j,t ∩ Ωi,b j,t −1 . That is, a supplier s of buyer b is considered common if they also traded during the previous month. The first stage uses both bilateral and seller-level instruments. Bilateral instruments correspond to Equation (17), while seller-level instruments correspond to Equation (16). Standard errors are two-way clustered at the origin and destination state level, and are reported in parentheses below each estimate. The third row reports the implied value for . The fourth row reports the number of observations. The fifth row reports Kleibergen-Paap F statistics from the first stage. The first two columns report estimates conditional on seller’s level of inventory, and the last two columns report estimates conditional on buyer’s level of inventory. A seller or a buyer are “Low" inventory when they belong to a 4-digit HS code that reports a below-median ratio of inventories to total sales. A seller or a buyer are “High" inventory when they belong to a 4-digit HS code that reports an above-median ratio of inventories to total sales. All specifications include buyer-by-time, product-by-time, buyer-by-product, and seller-by-product fixed effects. ix TABLE A11: Alternative specifications: 2SLS, firm-level elasticity of substitution across suppliers (1) (2) (3) (4) (5) (6) (7) (8) ˆ p log ˆ ˜ p 0.5687 0.5476 0.4927 0.4531 0.9371 0.8063 0.7084 0.7809 (0.2086) (0.1818) (0.1894) (0.1778) (0.3856) (0.3305) (0.4308) (0.3307) 0.4312 0.4523 0.5072 0.5468 0.0628 0.1936 0.2915 0.2190 Obs 879997 851483 843965 816122 1026381 993583 351397 341903 K-PF 37.629 121.309 38.680 102.746 42.335 87.990 83.737 50.489 HS digits 6 6 6 6 8 8 8 8 Seller IV Y Y Y Y Y Y Y Y Bilateral IV Y Y Y Y Y Y Y Y Feenstra term Y Y Y Y Buyer-time FE Y Y Y Y Product-time FE Y Y Y Y Buyer-product FE Y Y Seller-product FE Y Y Y Y Buyer-product-time FE Y Y Y Y Notes: 2SLS estimates are obtained from estimating Equation (11). Time is monthly frequency. The set of common suppliers of buyer b is Ω∗ i,b j,t = Ωi,b j,t ∩ Ωi,b j,t −1 . That is, a supplier s of buyer b is considered common if they also traded during both the current and previous month. In all specifications, the first stage uses both bilateral and seller- level instruments as pointed out in rows seven and eight. Bilateral instruments correspond to Equation (17), while seller-level instruments correspond to Equation (16). The first row reports the estimates associated with changes in relative unit values in logs. Standard errors are two-way clustered at the origin and destination state level, and are reported in parentheses below each estimate. The third row reports the implied value for , which is 1 minus the estimate on the first row. The fourth row reports the number of observations. The fifth row reports the Kleibergen- Paap F statistic from the first stage. A product category is either a 6-digit or 8-digit HS code, as pointed out by the sixth row, and the treatment period is March-May 2020. The table contains eight columns. Each column corresponds to different combinations of HS codes, fixed effects, and whether the Feenstra is included, as pointed out by the last nine rows. All specifications include the controls mentioned in the paper. x B A PPENDIX F IGURES F IGURE A1: Variation over time in aggregate outcomes (a) Number of sellers (b) Number of buyers (c) Number of transactions (d) Total sales Notes: This figure consists of 4 panels. In each panel, the horizontal axis is time, and the vertical axis is a different aggregate outcome. In the first panel, we show the number of sellers that reported a transaction by period. In the second panel, we show the number of buyers that reported a transaction by period. In the third panel, we show the number of transactions that were reported in a given period. In the fourth panel, we show the total sales for a given period. xi F IGURE A2: Google mobility trends by lockdown zone (a) Retail and recreation (b) Grocery and pharmacy (c) Parks (d) Transit stations (e) Workplaces (f) Residential Notes: These plots are constructed using Google Mobility Trends data, which shows how visits and length of stay at different places change compared to a baseline. The baseline is the median value, for the corresponding day of the week, during January 3rd - February 6th 2020. The raw data is at the daily frequency for each district in India. We collapse the data at a weekly frequency, and at the zone level. Each panel corresponds to mobility in different places. xii ˆ p F IGURE A3: Residuals of log ˆ ˜ p on seller-by-HS-by-time fixed effects ˆ Notes: These figures show densities of residuals from regressing log p ˆ ˜ p on seller-by-HS-by-time fixed effects, by HS-digit codes. The figure on the left shows the density for values from −0.5 to 0.5. xiii F IGURE A4: Distribution of links and sales across lockdown zones Share distributions of colors (a) Sellers in Red (b) Sellers in Orange (c) Sellers in Green (d) Buyers in Red (e) Buyers in Orange (f) Buyers in Green Share of sales and purchases by color zone of destination districts (g) Sales (h) Purchases Notes: This figure comprises two sets of panels. The first six figures are the first panel, and the last two figures are the second panel. For the first panel, in the three upper sub-figures, each panel plots the distribution of the share of buyers located in Red , Orange, or Green districts. Each sub-figure corresponds to sellers located in their corresponding color district. In the middle three sub-figures, each sub-figure plots the distribution of the share of sellers located in Red , Orange, or Green districts. Each sub-figure corresponds to buyers located in their corresponding color district. The time period is April 2018 - February 2020. For the lower panel, on the left sub-figure, for each HS section (horizontal axis), we plot the share of total sales of firms located in our large Indian state by the zone of selling districts. In the lower right sub-figure, for each HS section (horizontal axis), we plot the share of total purchases of firms located in our large Indian state by the zone of buying districts. The time period for this data is 2019. xiv F IGURE A5: Elasticities by seller’s industry Notes: The vertical axis is the OLS estimate of , and the horizontal axis is the 2SLS estimate of . Time is monthly frequency. These estimates come from estimating Equation (11). For the 2SLS estimate, the set of common suppliers of buyer b is Ω∗i,b j,t = Ωi,b j,t ∩ Ωi,b j,t −1 . That is, a supplier s of buyer b is considered common if they traded in both the current and previous month. The first stage uses both bilateral and seller-level instruments. Bilateral instruments correspond to Equation (17), and seller-level instruments correspond to Equation (16). An industry is an HS section. The size of each bubble is determined by total sales in the corresponding industry. See Table A8 for industry-specific numbers. xv F IGURE A6: Change in Product Category Links, before and after lockdown (a) Sellers (b) Buyers Notes: The figure has two density plots. On the left, we study sellers; on the right, buyers. In the left plot, we show the distribution of the share of sellers that sold goods to a given product category in both periods t and t − 1, where these periods are one year apart. In the right plot, we show the distribution of the share of buyers that purchased goods from a given product category in both periods t and t − 1, where these periods are one year apart. Product categories are 4-digit HS codes. The green densities are for periods before Covid-19 lockdowns, where t is between June 2019 and October 2019, and t − 1 is between June 2018 and October 2018. The red densities are for periods after Covid-19 lockdowns, where t is between June 2020 and October 2020, and t − 1 is between June 2019 and October 2019. xvi C DATA Exposure variables. We construct two exposure variables at the firm level: EDs,i,t and IMs,i,t . EDs,i,t is the exposure of firm s selling product i to global demand shocks in month t . IMs,i,t is the exposure of firm s selling product i to global supply shocks in month t . First, we construct these exposures by country, such that Ys,i,x,0 EDs,i,x,t = Xi,x,t x Ys,i,x ,0 Ys,i,m,0 IMs,i,m,t = Mi,m,t , m Ys,i,m ,0 where Ys,i,x,0 is the value of goods of seller s of product i shipped to country x in 2018, Ys,i,m,0 is the value of goods of seller s of product i shipped from country m in 2018, Xi,x,t is the value of export demand from country x for product i in month t , excluding demand for Indian products, and Mi,m,t is the value of import demand to country x for product i in month t , excluding demand for Indian products. We then construct our exposure variables as a weighted sum of these measures across countries, such that EDs,i,t = EDs,i,x,t x IMs,i,t = IMs,i,m,t m Labor and sales. Our firm-to-firm dataset lacks data on the number of employees and final sales. We obtain data on the number of employees and total sales from an external dataset for a subset of our firms. We then estimate an OLS regression of both labor and final sales on observable variables in our firm-to-firm dataset, store the OLS estimates, and use them to predict labor and final sales for all firms. We scrape data on the number of employees and total sales from the website IndiaMART, India’s largest B2B digital platform. We scraped around 300,000-400,000 firm profiles, and then sent them to the tax authority to be matched with our firm-to-firm trade dataset. The matching procedure (conducted by the government) yielded 50,720 unique firms. Each firm reports its number of employees and annual turnover (sales), both reported in brackets. The reported brackets for sales are: up to 50 Lakh, 50 Lakh-1 Crore, 1-2 Crore, 2- 5 Crore, 5-10 Crore, 10-25 Crore, 25-50 Crore, 50-100 Crore, 100-500 Crore 500-1,000 Crore, 1,000-5,000 Crore, 5,000-10,000 Crore, more than 10,000 Crore. First, we convert each reported xvii number into rupees, since sales in the trade dataset are reported in rupees.35 Then, for each firm we assign the median value of its corresponding sales bracket. For the last bracket, we consider the upper bound to be 100,000 Crore. The reported brackets for labor are: up to 10 employees, 11-25, 26-50, 51-100, 101-500, 501-1000, 1001-2000, 2001-5000, and more than 5000 employees. For each firm, we assign the median value of its corresponding labor bracket. For the last bracket, we consider the upper bound to be 50,000 employees. We then estimate the following OLS regressions: l log (laborn ) = α0 + α1 log (salesn ) + α2 log (distancen ) + i log ( f inaln ) = β0 + β1 log (salesn ) + β2 log (distancen ) + if , where salesn are total sales of intermediates of firm n, distancen is the average distance in kilo- meters of all firms’ registered transactions, laborn is the number of employees constructed as previously explained, and f inaln is final sales. We constructed final sales by subtracting total in- termediate sales from total sales, where we construct the former directly from our firm-to-firm dataset. In almost all cases, this difference was positive, which reassures that IndiaMART indeed reports total sales. Whenever the differences were negative, we input a value of 0, which implies that all firm’s sales are of intermediates. We obtain the following estimated elasticities: (α0 , α1 , α2 ) = (−2.1138, 0.2502, 0.2853), and (β0 , β1 , β2 ) = (9.8848, 0.3665, 0.4227). They are significant at the 1% confidence level. We then use these estimates to predict labor and final sales to all firms in our dataset. Dijkstra algorithm We now list the steps of the Dijkstra algorithm we used to construct our seller-buyer-level instruments. We obtained a set of shapefiles of district administrative bound- aries for India according to India’s 2011 census. We reprojected the shapefiles into an Asian/South Equidistance Conic projection, which is the projection that best preserves the distance measure- ments. Once shapefiles are reprojected, the objective is to construct a transportation network be- tween Indian districts. First, we obtain the centroid of each district in India. Then, we construct a network structure according to the set of centroids. There are many ways to construct a network, so we need to take a stance on how to form the connections between centroids. For each centroid, we generate connections to the k closest centroids according to Euclidean distances.36 We follow Fajgelbaum 35 100,000 rupees = 1 Lakh; and 10,000,000 rupees = 1 Crore. 36 Consider the set of nodes Φ, where K ≡ |Φ| is the number of nodes. The number of connections per node k could range from 0 up to K , where each represents extreme cases of network formation. k = 0 is a network without connections, so it is not possible to run a Dijkstra algorithm since it is not possible to go from one node to another. xviii and Schaal (2020) and consider k = 8 such that we consider the main cardinal directions (i.e. north, south, east, west, north-east, south-east, north-west, south-west). We now run the Dijkstra algorithm. For all district pairs, the algorithm provides us with the list of all districts that comprise the route between the district pair, and the distance of each leg that comprise the route. Using the names of the districts, we use the lockdown data to assign a lockdown color to each district along the route, and obtain our seller-buyer-level instruments. Our first instrument is the share of districts in a route that are Red , Orange, or Green. When calculating these shares, we rule out the zone where the buyer resides so we do not consider demand-side shocks in our instrument. Using the distance of each leg, our second instrument is the share of meters of the route that are Red , Orange, or Green. We consider a leg to be of color x = Red , Orange, Green whenever the origin district was of color x. In this case, we also ignore the color of the district where the buyer resides. D D ERIVATIONS D.1 Estimation of firm-level elasticities of substitution across suppliers In this section, we describe the steps to derive the firm-level elasticity of substitution across sup- pliers for the same product. First, we describe the model and the equations we take to the data. Second, explain how we construct price indices we need to estimate this elasticity. Third, we de- scribe how we account for the entry and exit of suppliers for the estimation. Finally, we explain how we construct the seller-level and seller-buyer-level instruments we use to causally estimate our elasticity. k = K is a fully-connected network, where all nodes are connected with each other. Running a Dijkstra algorithm on this scenario is trivial since the shortest distance between any pair of nodes is their connection itself. Therefore, a feasible number of connections per node must be k ∈ (0, K ). xix D.1.1. Expression to estimate firm-level elasticities of substitution across suppliers A firm b selling product j ∈ F maximizes profits subject to its technology and to a CES bundle of intermediate inputs: max pb j yb j − wb j lb j − psi,b j xsi,b j i s s.t . α α− 1 α− 1 α−1 yb j = Ab wbl lb j α + (1 − wbl ) xb j α , ζ 1 ζ −1 ζ −1 ζ ζ xb j = wi,b j xi,b j , i −1 1 −1 xi,b j = µsi,b j xsi,b j s The first order condition with respect to xsi,b j is α −1 α − 1 αα −1 −1 xsi,b j : pb j yb j Θ1 bj (1 − wbl ) xb j α−1 α ζ −1 ζ ζ −1 −1 xb j Θ2 bj wi, j xi,ζ bj ζ −1 ζ −1 −1 1 −1 −1 −1 xi,b j Θ3 i,b j µsi,b j xsi,b j = psi,b j , −1 α− 1 −1 = pb j yb j Θ1 bj (1 − wbl ) xb α j ζ −1 −1 Θ2 bj wi, j xi,ζ bj 1 −1 −1 Θ3 i,b j µsi,b j xsi,b j = psi,b j , where {Θ1 2 3 b j , Θb j , Θi,b j } are composite terms that cancel out in the next steps. Now, consider the first order conditions with respect to xsi,b j and xs i,b j and divide them, such that xx 1 −1 µsi,b j xsi,b j psi,b j 1 1 = , − µs i,b j xs i,b j ps i,b j −1 −1 1− 1 −1 xsi,b j psi,b j psi,b j µsi,b j = , −1 −1 1− 1 −1 xs i,b j ps i,b j ps i,b j µs i,b j −1 −1 −1 −1 −1 −1 xsi,b j psi,b j ps i,b j µs i,b j = psi,b j µsi,b j xs i,b j psi,b j , xsi,b j psi,b j p1− 1− s i,b j µs i,b j = psi,b j µsi,b j xs i,b j psi,b j , PMsi,b j p1− 1− s i,b j µs i,b j = psi,b j µsi,b j PMs i,b j , PMsi,b j p1− 1− s i,b j µs i,b j = psi,b j µsi,b j PMs i,b j , s s PMsi,b j p1 − i,b j = p1− si,b j µsi,b j PMi,b j , 1− PMsi,b j psi,b j 1− 1 = µsi,b j , PMi,b j pi,b j PMsi,b j psi,b j log = (1 − ) log + log µsi,b j , PMi,b j pi,b j where PMsi,b j ≡ psi,b j xsi,b j , p1 i,b j ≡ − s p1 s i,b j µs i,b j , and PMi,b j ≡ − s PMs i,b j . This is the derivation of Equation (9). D.1.2. Constructing price indices In this section, we derive the expressions that allow us to construct price indexes based on observ- able data. First, go back to the derivation in Appendix D.1, where PMsi,b j p1 − 1− i,b j = psi,b j µsi,b j PMi,b j . In the data, we observe prices and expenditures over time, so we introduce a time dimension such that PMsi,b j,t p1 − 1− i,b j,t = psi,b j,t µsi,b j,t PMi,b j,t , where t is a month. We can now express this equation in changes, such that PM si,b j,t p1 − 1− i,b j,t = psi,b j,t µsi,b j,t PM i,b j,t , xxi where xt ≡ xx t t −1 . Our objective is for pi,b j,t not to depend on µsi,b j,t , which are not observable. To do this, we rely on Redding and Weinstein (2020). The key assumption is that the overall importance of a product category in a buyer’s input use is time-invariant. Concretely, the geometric mean of µsi,b j,t across common sellers is constant. From the maximization problem of the firm, we obtain the following expression for the CES price index at the buyer level:  1  1− pi,b j,t =  µsi,b j,t p1− si,b j,t  , s∈Ωi,b j,t where Ωi,b j,t is the set of all sellers that provided to buyer b in time t . We apply Shephard’s Lemma to this CES price function, which in turn yields an expression for expenditure share: µsi,b j,t p1 − si,b j,t ssi,b j,t = , p1 − i,b j,t PMsi,b j,t where ssi,b j,t ≡ PMsi,b j,t . We can then rewrite this expression such that s∈Ωi,b j,t 1 µsi,b j,t 1− pi,b j,t = psi,b j,t , ∀s ∈ Ωi,b j,t . ssi,b j,t This expression in changes is 1 µsi,b j,t 1− pi,b j,t = psi,b j,t . ssi,b j,t Now, common suppliers for a buyer b in time t is the set of suppliers Ω∗ i,b j,t that sold to ∗ buyer b in the current and previous period (i.e. Ωi,b j,t ≡ Ωi,b j,t ∩ Ωi,b j,t −1 ), where Ni∗ ∗ ,b j,t ≡ Ωi,b j,t is the number of common sellers for buyer b in time t . We now apply a geometric mean to this expression, such that xxii Ni∗ ,b j,t 1 Ni∗ ,b j,t µsi,b j,t 1− pi,b j,t = psi,b j,t , s=1 ssi,b j,t Ni∗ ,b j,t Ni∗ ,b j,t Ni∗ ,b j,t Ni∗ ,b j,t 1 1 pi,b j,t = psi,b j,t µsi,b j,t 1− 1 ssi−,b j,t , s=1 s=1 s=1 1 Ni∗ Ni∗ Ni∗ 1   1− ,b j,t 1 ,b j,t 1 ,b j,t 1 −1 N∗ N∗ N∗ i,b j,t i,b j,t i,b j,t pi,b j,t = psi,b j,t  µsi,b j,t  ssi,b j,t , s=1 s=1 s=1 1 Ni∗   1− 1 ,b j,t 1 −1 N∗ i,b j,t pi,b j,t = pi,b j,t si,b j,t  µsi,b j,t  . s=1 We now formally state the assumption we require to move forward, which is Ni∗ ,b j,t 1 Ni∗ ,b j,t 1 N∗ N∗ i,b j,t i,b j,t µi,b j,t = µsi,b j,t = µsi,b j,t −1 = µi,b j,t −1 . s=1 s=1 Then, the last term of our expression is Ni∗ ,b j,t 1 Ni∗ ,b j,t 1 N∗ µsi,b j,t N∗ i,b j,t i,b j,t µsi,b j,t = , s=1 s=1 µsi,b j,t −1 1 Ni∗ ,b j,t N∗ i,b j,t s=1 µsi,b j,t = 1 , Ni∗ ,b j,t N∗ i,b j,t s=1 µsi,b j,t −1 µi,b j,t = , µi,b j,t −1 = 1. So our final expression boils down to 1− pi,b j,t p1 − i,b j,t = , si,b j,t 1 N∗ i,b j,t where pi,b j,t ≡ s psi,b j,t is a geometric mean of unit values across common suppliers, and si,b j,t ≡ xxiii 1 N∗ i,b j,t s ssi,b j,t is a geometric mean of expenditure shares across common suppliers. Notice that we have reached to our objective, since now pi,b j,t is independent of µsi,b j,t . Finally, the expression we take to the data is PM si,b j,t p1 − 1− i,b j,t = psi,b j,t µsi,b j,t PM i,b j,t , 1− −1 PM si,b j,t pi,b j,t si,b j,t = p1− si,b j,t µsi,b j,t PM i,b j,t , 1− PM si,b j,t psi,b j,t = si,b j,t µsi,b j,t , PM i,b j,t pi,b j,t PM si,b j,t psi,b j,t log = (1 − ) log + log si,b j,t µsi,b j,t , PM i,b j,t pi,b j,t PM si,b j,t psi,b j,t log = (1 − ) log + log si,b j,t + log µsi,b j,t . PM i,b j,t pi,b j,t D.1.3. Addressing the entry and exit of suppliers In this section, we explain how we address the fact that seller and buyer matches do not happen in every period (i.e. entry and exit of sellers). The concern is that not taking into account the fact that sellers and buyers do not trade in every period could induce a bias in the estimation of . We address this by including a correction term by Feenstra (1994) in our regressions. First, notice we can write down the expenditure share as ssi,b j,t ≡ λi,b j,t s∗ si,b j,t , where λi,b j,t is the Feenstra correction term, and s∗ si,b j,t is the expenditure share with respect to total expenditure on common suppliers. Notice that these terms are constructed as PMsi,b j,t ssi,b j,t ≡ , s∈Ωi,b j,t PMsi,b j,t s∈Ω∗ PMsi,b j,t i,b j,t λi,b j,t ≡ , s∈Ωi,b j,t PMsi,b j,t PMsi,b j,t s∗ si,b j,t ≡ . s∈Ω∗ PMsi,b j,t i,b j,t xxiv In changes, the expression for expenditure shares is ssi,b j,t = λi,b j,t s∗ si,b j,t . Then, the geometric mean for expenditure shares is Ni∗ ,b j,t 1 N∗ i,b j,t si,b j,t = ssi,b j,t , s=1 Ni∗ ,b j,t 1 N∗ = λi,b j,t s∗ si,b j,t i,b j,t , s=1 Ni∗ ,b j,t 1 N∗ = λi,b j,t s∗ si,b j,t i,b j,t , s=1 ∗ λi,b j,t si,b j,t . So the final expression we take to the data is PM si,b j,t psi,b j,t log = (1 − ) log + log si,b j,t + log µsi,b j,t , PM i,b j,t pi,b j,t psi,b j,t ∗ = (1 − ) log + log λi,b j,t si,b j,t + log µsi,b j,t , pi,b j,t psi,b j,t ∗ = (1 − ) log + log λi,b j,t + log si,b j,t + log µsi,b j,t . pi,b j,t D.1.4. Addressing endogeneity concerns The equation from the previous section is what we take to the data. Nevertheless, there are further endogeneity issues that would contaminate our estimates for . In particular, Covid-19 lockdowns could have also induced changes in demand, which in turn would bias our estimates. For example, if Covid-19 shocks also induce negative demand shocks, our estimates would then be biased up- wards. In this section, we derive our instruments. First, we consider non-arbitrage in shipping, so prices at the origin and destination between sellers and suppliers are related as psi,b j,t = psi,t τsb,t , where psi,t is the marginal cost (MC) of production of good i for seller s in month t , τsb,t is the xxv iceberg cost of transporting the good from seller s to buyer b in month t . Now, we can then express this in changes, such that psi,b j,t = psi,t τsb,t . In logarithms, we have log psi,b j,t = log psi,t + log τsb,t . These two components of price imply two instruments. First, our seller-level instrument uses variation in MC at the seller-product level due to lockdown measures at the seller’s district. To isolate variation in marginal costs driven by seller’s lockdown zone, we interact the lockdown indicator (Lockt ) which takes the value 1 between March and May with indicator variables Redos and Orangeos that equal 1 whenever seller s was located in a district o that was either Red or Orange during the lockdown. Then, our excluded instruments are p log( psi,t ) = β R, p Redo(s) Lockt + β O, p Orangeo(s) Lockt + νsi,t . Now we explain how we construct the instrument at the seller-buyer level. We have to take a stance about the functional form of the trade cost τsb,t . We assume that trade costs are proportional to the travel time of the transportation of intermediate inputs, such that τsb,t = TravelTimeσ sb,t . If we express this in changes, we get σ τsb,t = TravelTimesb,t . We exploit variation from the Covid-19 lockdown, which induced exogenous variation in the travel time between location pairs of sellers and buyers. Given this, we assume the following difference-in-differences setup for travel time: log TravelTimesb,t = β R,T Redo(s) d(b) Lockt + β O,T Orangeo(s) d(b) Lockt + νsb T ,t , where Redo(s) d(b) and Orangeo(s) d(b) are the share of the number of districts or of distance designated as Red and Orange, respectively, along the route between seller s and buyer b. We constructed these variables using Dijkstra algorithms. Further details about this are in Appendix C. Combining the expression for changes in travel time due to the lockdown and trade costs, we get the following xxvi expression for our seller-buyer-level excluded instruments τ log(τsb,t ) = β R,τ Redo(s) d(b) Lockt + β O,τ Orangeo(s) d(b) Lockt + νsb,t , τ where β R,τ ≡ σβ R,T , β O,τ ≡ σβ O,T , and νsi T ,b j,t ≡ σνsi,b j,t . Together, all excluded instruments are such that log psi,b j,t = β R, p Redo(s) Lockt + β O, p Orangeo(s) Lockt + β R,τ Redo(s) d(b) Lockt + β O,τ Orangeo(s) d(b) Lockt + νsi,b j,t , p τ where νsi,b j,t ≡ νsi,t + νsb,t . D.2 Estimation of firm-level elasticities of substitution across products In this section, we describe the steps to derive the firm-level elasticity of substitution across prod- ucts. First, we describe the model and the equations we take to the data. Second, we describe how we construct price indices we need to estimate this elasticity. Finally, we describe the instrument we use to causally estimate our elasticity. D.2.1. Expressions to estimate firm-level elasticities of substitution across products We rewrite the initial maximization problem, so max pb j yb j − wb j lb j − pi,b j xi,b j i s.t . α α− 1 α− 1 α−1 yb j = Ab wbl lb j α + (1 − wbl ) xb j α , ζ I 1 ζ −1 ζ −1 ζ ζ xb j = wi,b j xi,b j , i 1 1− pi,b j = µsi,b j p1− si,b j s xxvii The first order condition with respect to xi,b j is α −1 α − 1 αα −1 −1 xi,b j : pb j yb j Θ4 bj (1 − wbl ) xb j α−1 α ζ −1 1 ζ ζ −1 −1 xb j Θ5 bj wiζ ,b j xi,ζ bj = pi,b j , pi,b j ζ −1 ζ −1 α− 1 −1 = pb j yb j Θ4 bj (1 − wbl ) xb α j 1 −1 −1 Θ5 bj wiζ ζ ,b j xi,b j , where {Θ4 5 b j , Θb j } are composite terms that cancel out in the next steps. Now, consider the same first-order conditions with respect to xi ,b j and divide them, such that 1 −1 wiζ ζ ,b j xi,b j pi,b j 1 −1 = , ζ ζ pi ,b j wi ,b j xi ,b j 1 −1 −1 −1 wiζ ζ ζ ,b j xi,b j pi,b j ζ pi,b j pi,b j 1 −1 1 = 1 , ζ ζ −ζ −ζ wi ,b j xi ,b j pi ,b j pi ,b j pi ,b j −ζ wi ,b j xi,b j pi,b j p1 i,b j = −ζ , wi,b j xi ,b j pi ,b j p1 i ,b j −ζ −ζ PMi,b j wi ,b j p1 i ,b j = PMi ,b j wi,b j p1 i,b j , −ζ −ζ PMi,b j wi ,b j p1 i ,b j = PMi ,b j wi,b j p1 i,b j , i i −ζ −ζ PMi,b j wi ,b j p1 i ,b j = wi,b j p1 i,b j PMi ,b j , i i −ζ −ζ PMi,b j p1 bj = wi,b j p1i,b j PMb j , −ζ PMi,b j wi,b j p1 i,b j = −ζ , PMb j p1bj 1−ζ PMi,b j 1−ζ pi,b j 1 = wi,b j , PMb j pb j PMi,b j pi,b j log = (1 − ζ ) log + log wi,b j , PMb j pb j 1 1−ζ 1−ζ where PMb j ≡ i PMi,b j , and pb j = i wi,b j pi,b j . As we did for the estimation of the elasticity xxviii of substitution across suppliers, we introduce a time dimension, apply Shephard’s lemma to this CES price function, and also assume that the overall importance of the composite intermediates is time-invariant, so −ζ wi,b j,t p1 i,b j,t si,b j,t = −ζ , p1 b j,t 1 wi,b j,t 1−ζ pb j,t = pi,b j,t , si,b j,t 1 wi,b j,t 1−ζ pb j,t = pi,b j,t , si,b j,t Nb j,t 1 N j,t wi,b j,t 1−ζ pb b j,t = pi,b j,t , i=1 si,b j,t Nb j,t Nb j,t Nb j,t 1 1 N j,t 1−ζ pb b j,t = pi,b j,t wi,b j,t siζ −1 ,b j,t , i=1 i=1 i=1 1 1 Nb j,t 1 Nb j,t 1 1−ζ Nb j,t 1 ζ −1 Nb j,t Nb j,t Nb j,t pb j,t = pi,b j,t wi,b j,t si,b j,t , i=1 i=1 i=1 1 1 1−ζ ζ −1 pb j,t = pb j,t wb j,t sb j,t , 1 ζ −1 pb j,t = pb j,t sb j,t , pb j,t pb j,t = 1 , 1−ζ sb j,t 1 Nb j,t Nb j,t where pb j,t ≡ i=1 pi,b j,t is the geometric mean of unit values across product categories that buyer 1 Nb j,t Nb j,t b sources from, and sb j,t ≡ is the geometric mean of expenditure shares across products. i=1 si,b j,t Now, if we also introduce a time dimension into our estimating equation, express it in changes, and consider our expression for unit values, we have xxix −ζ 1−ζ PMi,b j,t p1 b j,t = wi,b j,t pi,b j,t PMb j,t , −ζ 1−ζ PM i,b j,t p1 b j,t = wi,b j,t pi,b j,t PM b j,t , PM i,b j,t pi,b j,t log = (1 − ζ ) log + log wi,b j,t , PM b j,t pb j,t   PM i,b j,t p   i,b j,t  log = (1 − ζ ) log   + log wi,b j,t , PM b j,t  pb j,t  1 1− ζ sb j,t PM i,b j,t pi,b j,t log = (1 − ζ ) log + log sb j,t + log wi,b j,t . PM b j,t pb j,t D.2.2. Constructing price index pi,b j,t To estimate ζ , we need values for pi,b j,t , which are not directly observed in the data since pi,b j,t ≡ 1 1− s µsi,b j,t psi,b j,t , which is a function of and µsi,b j,t . For , we consider = , where is our 1− estimated elasticity. For µsi,b j,t , we use the fact that the residuals when estimating are a function of these shocks. Recall that PM si,b j,t psi,b j,t log = (1 − ) log + X β + φsi,b j,t , PM i,b j,t pi,b j,t µsi,b j,t where φsi,b j,t = log µsi,b j,t = log µsi,b j,t −1 = log µsi,b j,t − log µsi,b j,t −1 are the residuals of this estimating equation. By assumption, log µsi,b j,t are i.i.d and normally distributed shocks with mean µ and variance σ 2 , so the mean and variance of log µsi,b j,t − log µsi,b j,t −1 is 0 and 2σ 2 , respectively. We now construct pi,b j,t by the following steps: 1. Estimate the 2SLS regression to obtain the estimate ; 2. Recover predicted values for the error term φsi,b j,t ; 2 3. Calculate the empirical mean and variance of φsi,b j,t : µφ , σφ ; σφ2 4. Recover the values for mean and variance of log µsi,b j,t , such that: (i) µ = µφ and σ 2 = 2 ; 5. Make a random draw for log µsi,b j,0 , which is drawn from a normal distribution with mean σφ2 µφ and variance 2 ; xxx 6. For a given µsi,b j,0 , recover µsi,b j,t according to the following law of motion: µsi,b j,t log = φsi,b j,t , µsi,b j,t −1 µsi,b j,t = exp φsi,b j,t , µsi,b j,t −1 µsi,b j,t = exp φsi,b j,t µsi,b j,t −1 ; 7. We then construct unit values by 1 1− pi,b j,t ≡ µsi,b j,t p1− si,b j,t s D.2.3. Constructing instruments To obtain an exogenous shifter of relative unit values, which we use to obtain an unbiased estimate of ζ , we rely on the instruments we use to estimate . Consider the set of instruments Zsi,b j,t . Then, we consider the new set of instruments: 1 Wi,b j,t = Z si,b j,t = Zsi,b j,t . Ni,b j,t s Consider the instrument that varies across both the color zone of the seller and the buyer (i.e. the share of districts in the Red zones within the route between the location of the seller and of the buyer). Then, the new instrument is the simple average of these shares across sellers. Intuitively, a higher share of districts in the Red zone should help predict a larger positive shock on unit values. E S IMULATIONS USING QUANTITATIVE MODEL E.1 Deriving expression for shock propagation through GDP In this section, we discuss the details of the simulation using the quantitative model. First, recall the notations used in the paper. N is the number of firms, and I is the number of product categories. λk is the Domar weight of firm or sector k. θk is the elasticity of substitution corresponding to the kth reproducible sector. Ωli is the (l , i)th element of the (N + I + 2) input output matrix Ω, which captures the direct reliance of l on i as a supplier . ψli is the (l , i)th element of the (N + I + 2) Leontief inverse matrix ψ ≡ (1 − Ω)−1 , which captures the direct and indirect reliance of l on i as a supplier. xxxi The aggregate change in GDP (∆logy) in response to changes in productivity of firm j (∆logA j ) up to a second order is given by the following: N N N N ∂ logy 1 ∂ 2 logy 1 ∂ 2 logy ∆logy = (∆logA j ) + (∆logAi )(∆logA j ) + (∆logAi )2 . j=1 ∂ logA j 2 i=1 j=1,i= j ∂ logAi ∂ logA j 2 i=1 ∂ logA2i (18) Following Baqaee and Farhi (2019), after replacing second order terms, we obtain N N N N 1 = λ j (∆logA j ) + (θk − 1)λkCovΩ(k) (ψ(i) , ψ( j) ) (∆logAi ) ∆logA j j=1 2 i=1 j=1,i= j k=0 N N 1 + (θk − 1)λkVarΩ(k) ψ(i) (∆logAi )2 2 i=1 k=0 N N N N N +F 1 = λ j (∆logA j ) + (θk − 1)λk Ωkl ψli ψl j (19) j=1 2 i=1 j=1,i= j k=0 l =1 N +F N +F − Ωkl ψli Ωkl ψl j (∆logAi ) ∆logA j l =1 l =1 N N N +F N +F N +F 1 + (θk − 1)λk Ωkl ψli ψli − Ωkl ψli Ωkl ψli (∆logAi )2 2 i=1 k=0 l =1 l =1 l =1 N 1 1 = λ j (∆logA j ) + B + C. j=1 2 2 We now write down the expressions for terms B and C in matrix form to evaluate second- order effects. In terms of notation, Jm,n is a matrix of ones of size m by n, × is matrix multiplication, and · is element-by-element matrix multiplication. Term B. This term primarily captures the second-order effects on GDP that operates through changes in firm i s Domar weight in response to productivity shocks to firm j, where j ∈ N , j = i. xxxii We construct this term in matrix form by sequentially deriving the following matrices: M = ψ · (∆logA)T , N = J(N +I +2,N +I +2) · J(N +I +2,1) × ψ · (∆logA)T − ψ · (∆logA)T , Covar1 = Ω × (M · N ), Covar21 = Ω × M , Covar22 = Ω × N , Covar2 = Covar21 · Covar22, B = (θ − 1) · λ × Covar1 − Covar2 . Term C. This term primarily captures the second-order effects on GDP that operates through changes in firm i s Domar weight in response to productivity shocks to firm i itself. In matrix form, this term is C= (θ − 1) · λ × Ω × (ψ · ψ ) − (Ω × ψ ) · (Ω × ψ ) × ∆logA · ∆logA . GDP change in matrix form. In matrix form, we can rewrite Equation (18) as ∆logy = λ × ∆logA + .5 (θ − 1) · λ × Covar1 − Covar2 + .5 (θ − 1) · λ × Ω × (ψ · ψ ) − (Ω × ψ ) · (Ω × ψ ) × ∆logA · ∆logA (20) E.2 Numerical implementation in Python Numerical implementations of our simulations are challenging due to the sheer size of the firm-to- firm trade network. We have data on 93,260 firms across 1293 product categories. This generates a 94,555 by 94,555 input-output matrix. The elements inside the input-output matrix are small as the fraction of a product’s output to a single firm is small and, in turn, each product category sources from a large number of suppliers. To maintain calculations as precise as possible, we used f loat 64 variable types within these matrices. Nevertheless, this also drastically increased the amount of computer memory required to hold matrices. For instance, the Leontief inverse matrix required more than 66GB of storage/memory size. xxxiii These large matrices require many steps to perform matrix multiplication operations on them. In computer science, matrix multiplication is one of the most demanding operations in terms of computing resources. We break down these operations by leveraging state-of-the-art computing techniques in big data. These techniques provide us with scalability when applying them to arbi- trarily large input-output matrices. As detailed firm-to-firm transaction data are becoming more widely available, these techniques are promising to advance the literature on quantifying the prop- agation of shocks through firm networks. We now briefly describe these techniques. First, we fit datasets larger than RAM using Dask, which is a Python library with multi-core, distributed, and parallel execution on larger-than-memory datasets.37 We use Dask’s distributed capabilities to parallelize our calculations when computing second-order effects which require few matrix multiplication operations on large 94,555 by 94,555 matrices. Second, we use a computer powered by multiple GPUs. GPUs are essential for performing a large number of matrix multiplications. For example, computing 10 columns of the Leontief inverse matrix, which is only around 0.0001% of columns we need to compute, takes about 4 days on a powerful server with multiple CPUs, 500 GB of RAM, and 16 cores. Computing the entire Leontief inverse on a server powered with 4 GPUs takes about 1 hour. Third, we use the properties of sparse matrices to define matrix multiplications that ignore large contiguous sub-matrices full of zeros, which is a typical feature of input-output matrices. Fourth, we developed a custom matrix multiplication function to overcome the limitation of the relatively small memory size of GPUs. The custom matrix multiplication function splits the matrix into sub-matrices of full columns (typically in the order of a few 1000’s of columns), it multiplies the sparse input-output matrix by each sub-matrix, and it concatenates all result chunks to formulate the final result. 37 https://tutorial.dask.org/00_overview.html xxxiv