metadata: authors: 0: author: Fujiy,Brian C.
1: author: Ghose,Devaki
2: author: Khanna,Gaurav
count: World
docna: 0: docna: Production Networks and Firm-level
            Elasticities of Substitution
docty: Policy Research Working Paper
abstracts: cdata!: This paper provides one of the first
            estimates of elasticities of substitution across suppliers
            within the same product. This paper estimates these
            elasticities using new real-time administrative tax data on
            firm-to-firm transactions, with product-level prices and
            quantities, leveraging geographic and temporal variation
            from India's Covid-19 lockdowns to derive causal
            estimates of these elasticities. Suppliers are highly
            complementary even at this granular level, with an estimated
            elasticity of 0.55. The paper shows that the quality of
            institutions, input specificity, inventories, and time
            horizons explain the low elasticity. These firm-level
            complementarities amplify the propagation of negative shocks
            through production networks, and make connected firms
            important for shock propagation. In policy counterfactuals,
            the paper shows that given these complementarities, allowing
            more connected firms to operate in the face of shocks
            mitigates output declines non-linearly with the size of the
            productivity shock.
display_title: Production Networks and Firm-level
            Elasticities of Substitution
listing_relative_url: /research/2024/05/34322856/production-networks-firm-level-elasticities-substitution
topicv3: Trade m1326924 698,Coronavirus (COVID-19) m1939478 4828
publishtoextweb_dt: 2024-05-22T00:00:00Z
dois: 10.1596/1813-9450-10782
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
no_of_pages: 80
author_keys: 0: author_key: NA
1: author_key: 1947054
2: author_key: 1068231
model_extractions: 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 of substitution across suppliers within the same product. This paper estimates these elasticities using new real-time administrative tax data on firm-to-firm transactions, with product-level prices and quantities, leveraging geographic and temporal variation from India’s Covid-19 lockdowns to derive causal estimates of these elasticities. Suppliers are highly complementary even at this granular level, with an estimated elasticity of 0.55. The paper shows that the quality of institutions, input specificity, inventories, and time horizons explain the low elasticity. These firm-level complementarities amplify the propagation of negative shocks through production networks, and make connected firms important for shock propagation. In policy counterfactuals, the paper shows that given these complementarities, allowing more connected firms to operate in the face of shocks mitigates output declines non-linearly 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<sup>*</sup>

Brian C. Fujiy<sup>†</sup> Devaki Ghose<sup>‡</sup> Gaurav Khanna<sup>§</sup> _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 Globalisation, 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, Carolina 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 INTRODUCTION

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.<sup>1</sup> In this paper, we quantify the importance of firm-level elasticities of substitution across suppliers (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 supplyside shocks induced by the Indian government’s massive lockdown policy. We explore explanations 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 mitigates output declines.

We pose two main research questions. First, are suppliers of intermediate inputs within a product category complements or substitutes?<sup>2</sup> The degree of complementarity or substitutability 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 complements, 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

> 1https://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.

> 2Complementarity 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.

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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 estimate these elasticities. We begin by presenting reduced-form evidence that leverages the nationwide, 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 fundamentals and induced strong variation in transactions between firms across India.<sup>3</sup> 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 suppliers 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 analytical 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

> 3See the BBC (link), and The Wire (link)

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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 specification 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.<sup>4</sup> 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 counterfactual 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.<sup>5</sup> 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 buyers 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 intermediate 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-

> 4We 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.

> 5For 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.

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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 shortlived, 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 estimates 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 enforcement 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.<sup>6</sup> 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 ( _ε_ = 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 ( _ε_ = 2) and 0 _._ 99 _pp_ more when they are almost Leontief ( _ε_ = 0 _._ 001).<sup>7</sup> 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 connected 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

> 6As 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.

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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 connected firms (counting only the number of direct buyers) to operate.<sup>8</sup> 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 networks (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).<sup>9</sup> 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.<sup>10</sup> 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 values (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

> 8A 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.

> 9See Bernard and Moxnes (2018) and Carvalho and Tahbaz-Salehi (2019) for a comprehensive literature review. 10Carvalho 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.

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(Baqaee and Farhi, 2019).<sup>11</sup> 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 computational 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 quantify the non-linear effects of productivity shocks _directly_ using the network without relying on approximations using final sales.<sup>12</sup> 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.<sup>13</sup>

# 2 DATA AND CONTEXT

**Firm-to-firm trade.** Our primary data source is daily establishment-level transactions with distinct information on establishment locations.<sup>14</sup> This data is provided by the tax authority of a large Indian state with a diversified production structure, around 50% urbanization rates, and high levels 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

> 11Prior 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).

> 12As 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. 13We may hesitate to use these elasticities for exercises on long-term structural transformations.

> 14While 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.

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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 establishments, 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,<sup>15</sup> 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 nationwide. 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 essential 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 individuals 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

> 15The 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.

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in _Green_ zones.<sup>16</sup> 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 fractionalization (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

> 16See 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.

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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 comparison to baseline days. The baseline day is the median value from the 5-week period Jan 3 – Feb 6, 2020.<sup>17</sup> 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 centralized lockdown was over, these differences reduced, and by December 2020, these differences, especially in workplace mobility, become negligible.

# - 3 REDUCED 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.<sup>18</sup> 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.

> 18To 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).

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# _3.1 Empirical specifications_

Our reduced-form specifications implement a difference-in-differences approach where we compare 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:

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 Covid19 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:

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.

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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). _γo_<sup>_v_</sup> ( _s_ )<sup>and</sup><sup>_γ_</sup> _d_<sup>_z_</sup> ( _b_ )<sup>are thus indicator variables that equal 1 when seller</sup><sup>_s_is located in district</sup><sup>_o_located in</sup> lockdown zone _v_ , and when buyer _b_ is located in district _d_ located in lockdown zone _z_ , respectively. The estimates of interest are _βt_<sup>_vz_.Our base time category is February 2020, just before lockdowns</sup> 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), representing 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.

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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 lockdown zones.** We now report the results from our seller-by-buyer-level specification. In Figures 2 and 3, we report the estimates for _βt_<sup>_vz_from Equation (2), where the estimates are in comparison</sup> 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 transactions 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.

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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

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_ , _xnj_ 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: where _ζ_ is the firm-level elasticity of substitution across products _i_ ,<sup>21</sup> 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:<sup>22</sup> 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 _ε_ 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: _Yf_ = 1.

> 21Previous 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).

> 22We exclude foreign intermediate goods since they are not exposed to Indian Covid-19 lockdown shocks.

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Product 0 represents the final consumption of the household and is given by where<sup>�</sup> _i_<sup>_woi_= 1 and</sup><sup>_σ_is the elasticity of substitution in consumption.</sup>

**Model in standard form.** To write the economy in standard form as in Baqaee and Farhi (2020), we construct an input-output matrix Ω<sup>�</sup> 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_<sup>�</sup> , where _θ_<sup>�</sup> = ( _σ, α, ζ, ε_ ). Formally, a nested-CES economy in standard form is defined by the tuple (Ω<sup>�</sup> _, θ_<sup>�</sup> ). The input-output matrix Ω<sup>�</sup> of size<sup>_<u>jxij</u>_</sup> (2 + _N_ + _I_ ) _×_ (2 + _N_ + _I_ ) is a matrix where element ( _i, j_ ) equals the value of Ω _i j_ =<sup>_<u>p</u>_</sup> _piyi_<sup>,whichis</sup> the expenditure share of the _i_ th firm on inputs from the _j_ th 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)<sup>−1</sup> . Intuitively, the ( _i, j_ ) _th_ element of _ψ_ (the Leontief inverse) measures _i_<sup>_′_</sup> 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 the sales of producer _i_ as a fraction of GDP by _λi_ , where _λi_ = _Npiyi_ <u>�</u> _j_<sup>_p jcj_.</sup> The input-output covariance operator is 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 macroeconomic impact of microeconomic shocks in this economy is given by: 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_<sup>_′_</sup> _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: subject to Equations (3), (4), and (5). Details about the optimization problem are in Appendix D.1. The maximization problem yields the following expression: <u>1</u> where _pi,b j_ = �� _s_<sup>_′_</sup> � _p_<sup>1</sup> _s_<sup>_′_−</sup> _,i_<sup>_ε_</sup> _,bj_<sup>_μs′,i,bj_</sup> �� 1− _ε_ is a CES price index, _PMs,i,b j ≡ ps,i,b jxs,i,b j_ , and _PMi,bj ≡_ � _s_<sup>_PMs,i,b j_.log</sup> � _μs,i,bj_ � is the error term. This is the underlying basis for our estimation of the firm-level elasticity of substitution parameter _ε_ , 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 arbitrary number of nests as long as the lowest nest consists of suppliers within the same product.<sup>23</sup> 23In 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 substitution 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 <u>1</u> subject to Equations (3) , (4), and _pi,bj_ = �� _s_<sup>_μsi,b jp_</sup> _si_<sup>1−</sup> _,b j_<sup>_ε_</sup> � 1− _ε_ . Details on the optimization problem are in Appendix D.2.1. The maximization problem yields the following expression: <u>1</u> 1− _ζ_ where _pb j_ = �� _i_<sup>_′_</sup> � _p_<sup>1</sup> _i_<sup>_′_−</sup> _,bj_<sup>_ζwi′,bj_</sup> �� is a CES price index, _PMi,b j ≡ pi,b jxi,b j_ , and _PMb j ≡_<sup>�</sup> _i_<sup>_PMi,bj_.</sup> log � _wi,b j_ � is the error term. This is our estimating equation for the firm-level elasticity of substitution _ζ_ , which we take to the data, as described in Section 5.

# 5 ESTIMATION STRATEGY

This section discusses how we estimate the primary elasticities in our model, the vector of parameters _θ_<sup>�</sup> = ( _σ, α, ζ, ε_ ). 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 composite intermediate input _α_ = 0 _._ 5 (Baqaee and Farhi, 2019). We now estimate the firm-level elasticity of substitution across suppliers ( _ε_ ) and the firm-level elasticity across products ( _ζ_ ) leveraging variation in the lockdown zones.

## _5.1 Estimating equations for ε_

We face two main challenges when estimating _ε_ from Equation (9). The first challenge is that we need data on price indices _pi,bj_ , which are a function of the unobservable importance of input suppliers � _μs,i,bj_ �. 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.<sup>24</sup> 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:

where _v_ � _t_ = _vvt_ − _<u>t</u>_ 1<sup>are variables in changes to the previous month.</sup> � _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 <u>1</u> effects. _p_ � _i,b j,t_ =<sup>�</sup> _s∈_ Ω<sup>_∗_</sup> _i,bj,t_<sup>_p_</sup> _sN,ii_<sup>_~~∗~~_</sup> _,,bjbj,t,t_<sup>is a geometric mean of unit values across common suppliers, where</sup> Ω<sup>_∗_</sup> _i,b j,t_<sup>_≡_Ω</sup><sup>_i,bj,t ∩_Ω</sup><sup>_i,bj,t_−1is the set of common suppliers for buyer</sup><sup>_b_that appear in both the current</sup> and previous month, and _Ni_<sup>_∗_</sup> _,bj,t_<sup>_≡_Ω</sup><sup>_∗_</sup> _i,bj,t_<sup>is the number of common suppliers for buyer</sup><sup>_b_in month</sup><sup>_t_.</sup> This strategy has three main advantages. First, it allows us to construct price indices _pi,bj,t_ solely as a function of prices _ps,i,bj,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 two controls to be included in the estimation: the change in expenditure share � _s_<sup>_∗_</sup> _i,b j,t_<sup>_≡_</sup> <u>�</u> _s∈_ Ω _PM_<sup>_∗_</sup> _i,bjsi,t,b_<sup>_PM_</sup> _<u>j,t</u>_<sup>_si,b j,t_</sup> <u>�</u> _s∈_ Ω<sup>_∗_</sup> _i,b_ _<u>j,t</u>_<sup>_PMsi,bj,t_</sup> and a Feenstra (1994) correction term _λi,b j,t ≡_ <u>�</u> _s∈_ Ω _i,bj,t_<sup>_PMsi,bj,t_.The key control is the latter since it</sup> 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

> 24This 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.

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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 _ε_ by including a productbuyer-time fixed effect � _ωi,bj,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).

Across the various specifications, the wide array of high-dimensional fixed effects helps control 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-bytime fixed effects), and buyer-by-seller and product-specific time-invariant characteristics (buyerby-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 policyinduced variation in relative prices.

**Accounting for the extensive margin of trade.** In extensions, we further account for the extensive 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 _ε_ 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 extensive 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.

where Ω _i,b j,t ≡_ Ω _i,bj_ − Ω _i,bj,t_ . Ω _i,bj,t_ is the set of sellers in product _i_ that traded with buyer _b_ in period _t_ , Ω _i,b j_ is the set of potential sellers with buyer _b_ in product _i_ .<sup>25</sup> Therefore, Ω _i,b j,t_ is the set of Φ<sup>thesetofbuyersthattradedwithseller</sup><sup>_s_inproduct</sup><sup>_i_inperiod</sup><sup>_t_.Forthecon-</sup> _s∈_ Ω _i,bj,t_<sup>is</sup> 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 effects. 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: To account for the zeros on the left-hand side of the estimating equation, we estimate _ε_ 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,bj,t_ is a function of (unobservable) demand shocks _μsi,b j,t_ , and _ε_ , such that _pi,bj,t ≡_ <u>1</u> 1− _ε_ �� _s_<sup>_μs,i,b j,t p_</sup> _s_<sup>1</sup> _,_<sup>−</sup> _i,_<sup>_ε_</sup> _bj,t_ � . Second, the price index _pb j,t_ is also a function of unobservable product-level demand shocks _wi,bj,t_ , making their computation challenging.

<u>1</u> First, we construct price indices as _pi,b j,t ≡_ �� _s_<sup>_μsi,b j,t p_</sup> _si_<sup>1−</sup> _,b j_<sup>�</sup><sup>_ε_</sup> _,t_ � 1− _ε_ �, where � _ε_ is estimated previously, _psi,b j,t_ comes directly from the data, and demand shocks _μsi,b j,t_ are constructed recursively.

25The 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: where � _ωb j,t, ωi,t, ωbj,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 <u>1</u> � _Nbj,t_ specific characteristics. **X** are the same set of controls used before. _pb j,t ≡_<sup>�</sup><sup>_N_</sup> _i_ =1<sup>_bj,tp_�</sup> _i,b j,t_<sup>isthe</sup> <u>1</u> _Nbj,t_ geometric mean of unit values across products that buyer _b_ purchases, and � _sb j,t ≡_<sup>�</sup><sup>_N_</sup> _i_ =1<sup>_bj,ts_�</sup> _i,b j,t_<sup>is the</sup> 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 _ε_ 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 _ε_ higher) compared to the unbiased value of _ε_ . 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 separated between prices at the origin and a trade cost, such that 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 substitution. Shocks induced by the Covid-19 lockdown policies that only impact sellers provide this variation. In Equation (16) below, we formalize this intuition.

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.

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. Details 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 _ε_ 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 comprised of many _Red_ 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 ELASTICITY 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-byproduct 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 suggest 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 simultaneous 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 suppliers 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.<sup>26</sup> 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)

> 26In 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)|
|---|---|---|---|---|---|---|---|
|log<br>�<br>ˆ_p_<br>ˆ ̃_p_<br>�|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)|
|_ε_|0.7828|0.7777|0.6954|0.7885|0.6969|0.7558|0.6629|
|Obs|2028039|1966591|991187|2021334|816122|993583|341903|
|R<sup>2</sup>|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 _ε_ , 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 ( _R_<sup>2</sup> ). 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 complementarity. 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 _ε <_ 1.

The 2SLS estimates for _ε_ 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 _ε_ . 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)|
|---|---|---|---|---|---|---|
|log<br>�<br>ˆ_p_<br>ˆ ̃_p_<br>�|0.5042|0.3945|0.4538|0.3409|0.4418|0.3535|
||(0.2129)|(0.0933)|(0.1389)|(0.1068)|(0.1982)|(0.1701)|
|_ε_|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<sup>_∗_</sup> _i,bj,t_<sup>= Ω</sup><sup>_i,bj,t ∩_Ω</sup><sup>_i,bj,t_−1.That is, a supplier</sup><sup>_s_of buyer</sup><sup>_b_is considered</sup><sup>_common_if they traded in both the</sup> 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 _ε_ , 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 ( _ε_ = 0 _._ 55), reflecting a lower degree 28
TABLE 3: Firm-level elasticity of substitution across products

||(1)|(2)|(3)|(4)|(5)|(6)|
|---|---|---|---|---|---|---|
|log<br>�<br>ˆ_p_<br>ˆ ̃_p_<br>�|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 _ε_ and the estimates of _ε_ . 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 _ε_ 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.

of complementarity across products compared to suppliers.<sup>27</sup> 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.

27This is consistent with the macroeconomics literature (Houthakker, 1955; Bachmann et al., 2022; Lagos, 2006).

29
# 7 WHY 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".<sup>28</sup> 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 differentiated (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 complementarity (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 terciles ( _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-

> 28For 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 comparison 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 located 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 expenditures 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 _ε_ 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 _ε_ 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 typically 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 specification 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), respectively. 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 QUANTIFICATION 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_<sup>29</sup> , 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 entire 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 computational complexity from the order of ( _N_ + _I_ + 2) to ( _N_ + _I_ + 2)<sup>2</sup> . 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 ( _ε_ = 2) and 0 _._ 99 _pp_ more when firms in the same HS-4 product are considered almost Leontief ( _ε_ = 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<sup>30</sup> 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 _ε_ 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 secondorder 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 _ε_ , 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,

> 30To 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%.

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<!-- Start of picture text --> % productivity change<br>-50 -40 -30 -20 -10 0<br>a<br>+Q<br>a<br>—<br>n<br>is)<br>og<br>(0)<br>a<br>2<br>fe)<br>to<br>c<br>fe)<br>Oo<br>oO<br>7p)<br>9<br>oo —o—-_ —s—-_ —-— — 4 —<br>e=0.001 6=0.55 ¢=1.25 €=1.75<br><!-- End of picture text -->
<!-- Start of picture text --> % productivity change<br>-50 -40 -30 -20 -10 0<br>Oo<br>_—— _———__, Qa<br>ZZ aw<br>+Sne}<br>ro<br>no}fe)c<br>ce)<br><!-- End of picture text --> <!-- Start of picture text --> % productivity change<br>-50 -40 -30 -20 -10 0<br>oO<br>a ~~ Qroe<br>2<br>oo<br>o<br>ne}oo<br>+O<br>me}cfe)<br>ie)<br>2”o<br>eo<br><!-- End of picture text --> <!-- Start of picture text --> % productivity change<br>-50 -40 -30 -20 -10 0<br>Q<br>a of&<br>#<br>3)<br>a@®<br>eS<br>©<br>=<br>3<br>ae)<br>ne}<br>Cc<br>fe)<br>(s)<br>°2®<br>mop)<br>—e— —o— ——<br>Most connected firms Random firms Least connected firms<br><!-- End of picture text -->
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 ( _ε_ = 0 _._ 001) in the left panel, our estimated complementarity ( _ε_ = 0 _._ 55) in the right panel, and near Cobb-Douglas ( _ε_ = 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).<sup>33</sup> 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 ( _ε_ = 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 ( _ε_ = 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 ( _ε_ = 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 ( _ε_ = 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

> 33Relatedly, 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.

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# 9 CONCLUSION

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 government’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 contract 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 complements, 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 without 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 network 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.

Our findings have implications for policymakers worldwide, who often face difficult tradeoffs in crisis regarding which firms to bail out. Given the underlying variation used, these estimates are relevant for other crises that are expected to remain short-lived, such as natural disasters, temporary trade wars and sanctions, and supply-chain disruptions.

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