Estimating Unequal Gains across U.S. Consumers with Supplier Trade Data

K.7 Estimating Unequal Gains across U.S. Consumers with Supplier Trade Data Hottman, Colin J. and Ryan Monarch Please cite paper as: Hottman, Colin J. and Ryan Monarch (2018). Estimating Unequal Gains across U.S. Consumers with Supplier Trade Data. International Finance Discussion Papers 1220. https://doi.org/10.17016/IFDP.2018.1220 International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 1220 January 2018

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1220 January 2018 Estimating Unequal Gains across U.S. Consumers with Supplier Trade Data Colin J. Hottman Ryan Monarch NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Estimating Unequal Gains across U.S. Consumers (cid:3) with Supplier Trade Data Colin J. Hottman Board of Governors of the Federal Reserve System(cid:142) Ryan Monarch Board of Governors of the Federal Reserve System(cid:143) January 17, 2018 Abstract Using supplier-level trade data, we estimate the effect on consumer welfare from changes in U.S. importsbothintheaggregateandfordifferenthouseholdincomegroupsfrom1998to2014. Todothis, we use consumer preferences which feature non-homotheticity both within sectors and across sectors. After structurally estimating the parameters of the model, using the universe of U.S. goods imports, we constructimportpriceindexesinwhichavarietyisdefinedasaforeignestablishmentproducinganHS10 product that is exported to the United States. We find that lower income households experienced the most import price inflation, while higher income households experienced the least import price inflation during our time period. Thus, we do not find evidence that the consumption channel has mitigated thedistributionaleffectsoftradethathaveoccurredthroughthenominalincomechannelintheUnited States over the past two decades. JEL CLASSIFICATION: D12, E31, F14 KEYWORDS: import price index, non-homotheticity, real income inequality, product variety, markups (cid:3)We are grateful to Mary Amiti, Pol Antràs, Dave Donaldson, Ana Cecilia Fieler, Doireann Fitzgerald, GordonHanson, RalphOssa, PeterSchott, InaSimonovska, DavidWeinstein, DanielYiXu, MingzhiXu, and seminarparticipantsattheUniversityofMichigan,theBureauofLaborStatistics,theFederalReserveBoard, the Federal Reserve Bank of New York, the Fed System Conference on International Economics, the Fed System Conference on Applied Microeconomics, the 2018 American Economic Association meeting, the 2017 RockyMountainEmpiricalTradeConference,the2017GeorgetownCenterforEconomicResearchConference, the 2017 Mid-Atlantic Trade Conference, and the 2016 Duke Trade Conference. The views expressed in this paper are solely those of the authors and do not represent the views of the U.S. Census Bureau, the Board of Governors of the Federal Reserve System or any other person associated with the Federal Reserve System. All results have been reviewed to ensure that no confidential information is disclosed. (cid:142)20th Street and Constitution Avenue N.W. Washington, D.C. 20551. Email: colin.j.hottman@frb.gov. (cid:143)20th Street and Constitution Avenue N.W. Washington, D.C. 20551. Email: ryan.p.monarch@frb.gov. 1

1 Introduction How has the cost of living in the United States been affected by changes in import prices over the past two decades? How have these changes been distributed across income groups? These important questions bear directly on current public policy debates over the effects of globalization and international trade on U.S. consumers, as well as the evolution of real income inequality. Recent research has emphasized using models in which different income groups can consume goods in different proportions (non-homotheticity), with the consequence that price indexes are income group-specific1. The literature has also highlighted four majorchannelsthatcontributetochangesinthecostofliving(i.e.,priceindexes): changesinaverageprices (consisting of marginal cost movements and markup adjustment), changes in the dispersion of prices (i.e., changing opportunities for substitution), product quality changes, and an expansion (or contraction) in the set of available varieties2. In this paper, we develop a new framework based on non-homothetic preferences that allows each of the four channels to contribute to changes in the price index, and, using detailed trade transaction data for the United States from 1998 to 2014, estimate the import price index based on that framework for different income deciles. The model permits both cross-sector and within-sector non-homotheticity, the first ofwhichcapturesdifferencesinsectoralexpendituresharesacrossconsumersandthesecondwhichcaptures differences in product quality. Exact linear aggregation over consumers is preserved in our framework, even with entry and exit of varieties. Our framework also nests the standard, homothetic, Constant Elasticity of Substitution (CES) monopolistic competition model as a special case. To estimate the model parameters, we develop an extension of the Generalized Method of Moments (GMM) estimator of Feenstra (1994), which exploits the relationship between income elasticities and price elasticities for separable demand functions, and apply it to data on the universe of foreign establishments exporting to the United States from 1998 to 2014. We define a variety as the combination of a foreign establishment and a Harmonized System (HS) ten-digit product code, and, consistent with the literature around this estimator, distinguish between “continuing ” varieties– those found in all years 1998-2014– and non-continuingvarieties. Overthistimeperiod,thenumberofuniqueimportedvarietiesrisesfrom2million in 1998 to 2.9 million in 2006, before falling again afterward to about 2.2 million by 2014.3 We further disciplinethemodelusingincome-decilespecificexpendituredatafromtheBureauofLaborStatistics(BLS) Consumer Expenditure Survey. The data show that even though the share of total expenditure spent on imports is fairly constant across income groups, there are large differences in the composition of imported 1SeeHunter(1991),Neary(2004),Choietal.(2009),Fajgelbaumetal.(2011),Fieler(2011),Li(2012),Handbury(2013), Markusen(2013),Caronetal.(2014),Faber(2014),FeenstraandRomalis(2014),AguiarandBils(2015),Simonovska(2015), Fajgelbaum and Khandelwal (2016), Jaravel (2016), Borusyak and Jaravel (2017), Cravino and Levchenko (2017), Faber and Fally(2017),andAtkinetal.(Forthcoming). 2For example, Feenstra (1994), Boskin et al. (1997), Bils and Klenow (2001), Hausman (2003), Lebow and Rudd (2003), Broda and Weinstein (2006), Broda and Weinstein (2010), Khandelwal (2010), Hallak and Schott (2011), Handbury and Weinstein(2014),Hsiehetal.(2016),Aghionetal.(2017),Amitietal.(2017),Feenstra(2017),FeenstraandWeinstein(2017), andFeldstein(2017). 3 Thishump-shapedpatternforthenumberofimportedvarietiesovertimeisnewtotheliteraturebutisrobusttodifferent definitionsofavariety. 2

consumption across income groups. Estimatingtheparametersofthemodelyieldsseveralnewresults. First, weestimatesectoralelasticities ofsubstitutionsquarelyinlinewiththeliteraturebutfindthattheoverallaggregateelasticityofsubstitution (across consumer goods) is close to 2.8, higher than the typically assumed value of 1. Second, we find that non-continuing, small foreign producers are often well approximated by the CES model, but continuing varieties deviate from the behavior implied by the standard CES benchmark model. In particular, the markups of these large foreign suppliers are often declining in their quantity sold, a relationship precluded bystandardmodels4. Third,wefindthattheestimatedmarkupofthemediansuppliertotheUnitedStates fell from 1998 to 2014. The median markup among continuing varieties fell from 23% in 1998 to about 13% in 2006 and remained about constant thereafter. Next, we use our model estimates and the variety-level universe of goods trade data to construct the aggregate U.S. import price index. Taking 1998 as the reference year, import prices fell nearly 12% by 2006 (thesameperiodthatthenumberofavailablevarietieswasincreasing,andthatmarkupsfell). Itisusefulto compareourindextotheLaspeyres-basedBLSAll-CommodityImportPriceIndex,whichisbasedonsurvey data and does not capture substitution effects or changes in the set of imported varieties. That index shows a 25% increase in import prices by 2006, the opposite of our finding. By 2014, our aggregate import price indexhadrisenabout8%fromits1998level,comparedwithanincreaseof48%fortheBLSAll-Commodity ImportPriceIndexfrom1998to2014. Therefore,weestimateanupwardbiasintheLaspeyresimportprice indexoverourtimeperiodofabout37%,orabout2percentagepointsperyear. Decomposingouraggregate importpriceindexintoitsdifferentcomponents,wefindthatthemaincontributortothedifferencebetween our index and the Laspeyres-based one is that we include substitution effects. Finally, we exploit the non-homothetic nature of our preferences to ask whether different income groups experienced different levels of import price inflation over our time period. Supplementing our model estimates with information from the BLS Consumer Expenditure Survey, we determine the basket of imported consumer goods for different income deciles and calculate the associated price index for each. The U-shaped pattern observed for the aggregate import price index over time is replicated for each individual income group. However, we find that lower income households experienced the most import price inflation, while higher income households experienced the least import price inflation during our time period. For example, in our baseline results the 1st income decile experienced import price inflation of about 24% from 1998 to 2014, or about 1.33 percent per year. For comparison, the 9th income decile only experienced import price inflation of about 15% over that time period, or about 0.90 percent per year. Given our finding that each income decile’s share of expenditure on total imported goods was about the same, we do not find evidence that the consumption channel has mitigated the distributional effects of trade that have occurred through thenominalincomechannelasdocumentedinAutoretal.(2016), PierceandSchott(2016), andtherelated literature. Instead, changesinimportpricesappearto be exacerbatingincreasesinnominal incomeinequal- 4This relationship between markups and quantities implies that increased competition may raise the markups charged by theselargeforeignfirms,whichisthe"anti-competitive"effectoftradediscussedinthetheoreticalliterature. 3

ityoverthistimeperiod. Ourresultsalsoindicatethatcross-sectornon-homotheticityisthekeymechanism driving the differences in import inflation across import groups. The rest of this paper is structured as follows. Section 2 reviews related literature. Section 3 outlines the model. Section 4 explains our identification strategy. Section 5 discusses our estimation results. Section 6 concludes. 2 Related Literature Mostoftheinternationaleconomicsliteraturehasstudiedimportpriceindexesusinghomotheticpreferences (e.g., Feenstra (1994), Broda and Weinstein (2006), Hsieh et al. (2016), Amiti et al. (2017), Feenstra and Weinstein (2017)). Of course, homothetic preferences preclude any focus on distributional issues across consumers5. We use non-homothetic preferences, which allow us to quantify the effect of changes in U.S. imports on different household income groups. The international trade literature has highlighted two forms of non-homotheticity. First, there is sectorlevelnon-homotheticity(e.g.,Caronetal.(2014),FajgelbaumandKhandelwal(2016)),whichreflectsdifferences in sectoral expenditure shares across income groups. Second, there is variety-level non-homotheticity (e.g., Feenstra and Romalis (2014)), which the literature has associated with differences in product quality. Ourframeworkallowsustotakeintoaccountbothformsofnon-homotheticity. Weexactlymatchsector-level non-homotheticity from data on sectoral expenditure shares by income decile from the U.S. Consumer ExpenditureSurvey. Inordertoidentifyvariety-levelnon-homotheticity,weestimateourmodelonvariety-level microdata, exploiting the relationship between income elasticities and price elasticities (i.e. markups). A growing body of theoretical work shows that how the price elasticity of demand changes with firm sales determines the nature of market distortions in monopolistic competition (Dhingra and Morrow (forthcoming)), the competitive effects of opening to international trade (Zhelobodko et al. (2012), Bertoletti and Epifani (2014), Arkolakis et al. (forthcoming)), and the pass-through of cost shocks to firms’ profit margins (Mrázová and Neary (2017)). However, most existing research hard-wires the sign of the relationship between the price elasticity of demand and sales, and thus implies “pro-competitive" effects of trade: reduced markups of incumbents as a result of more foreign entry (and thus lower incumbent market shares). This is thecasewithstandardpreferencesandmarketstructuressuchasCESwitholigopoly(AtkesonandBurstein (2008),Edmondetal.(2015)),LinearDemand(MelitzandOttaviano(2008)),Logitdemand(Fajgelbaumet al. (2011)), Constant Absolute Risk Aversion, or CARA, preferences (Behrens and Murata (2007), Behrens and Murata (2012)), and Almost Ideal Demand/Translog preferences (Feenstra and Weinstein (2017)). In contrast, our new framework based on the S-branch utility tree allows us to directly test, instead of impose, how markups move with quantities sold. The markup flexibility of our model is important because the empirical literature on markup adjustment 5RecentpaperssuchasAntràsetal.(2017)andGalleetal.(2017)studythedistributionaleffectsoftradeonthenominal incomesofdifferenttypesofworkers,butinthesemodelsworkers-as-consumersstillhavehomotheticpreferences. 4

in response to trade shocks is mixed. Domestic markups have been found to decline in countries with dramatictradeliberalizations(Levinsohn(1993),Harrison(1994),KrishnaandMitra(1998)). Antidumping cases that protect domestic firms also appear to raise markups (Konings and Vandenbussche (2005)). This evidence suggests that import penetration may have pro-competitive effects. However, other recent papers have found that increased international competition may also raise domestic markups, providing estimates of anti-competitive effects (Chen et al. (2009), De Loecker et al. (2014), De Loecker et al. (2016)). None of these papers present evidence on the U.S. case. In contrast, our paper directly estimates how the price elasticity of demand varies with sales for U.S. import suppliers. The literature has shown that markup adjustment and changes in the set of varieties available might not contribute anything to gains from trade in standard general equilibrium trade models with Pareto distributed firm productivity (Arkolakis et al. (2012), Costinot and Rodríguez-Clare (2014), Arkolakis et al. (forthcoming)). However,wedonotneedtoimposethefullgeneralequilibriumstructureofthesemodelson thedatainordertoestimatethechangeintheU.S.importpriceindex. Nordoweneedtomakeassumptions aboutthedistributionoffirmproductivityinordertoestimatemarkups. Infact, ourframeworkliesoutside the class of models that these papers consider6. Recent papers have used scanner data to study related questions (e.g., Jaravel (2016), Borusyak and Jaravel (2017), Faber and Fally (2017)). However, standard scanner datasets capture only about 40 percent of goods expenditures in the U.S. Consumer Price Index (Broda and Weinstein (2010)), and do not include most consumer durable products (e.g., cars, cellphones, computers, furniture, apparel). In contrast, the trade data we use captures the universe of U.S. goods imports. ThemostcloselyrelatedpaperstooursareFajgelbaumandKhandelwal(2016)andBorusyakandJaravel (2017). Fajgelbaum and Khandelwal (2016) use non-homothetic Almost Ideal Demand and aggregate trade data from many countries to estimate how different income groups in these countries would gain or lose from a counterfactual move to autarky. They find that U.S. consumers in the 1st decile of income would face a much larger price index increase from moving to autarky than consumers in the 9th decile of income. Borusyak and Jaravel (2017) study how a counterfactual 10% reduction in U.S. trade barriers would affect thewagesandconsumerpriceindexesofcollegegraduatesandthoseworkerswithoutacollegedegree, using a log-linear approximation approach. They find that the counterfactual’s effect on prices is biased in favor of college graduates, but small enough in magnitude that they conclude that this channel is distributionally neutral. In contrast with these papers, our contribution to the literature is to use supplier-level trade data toprovidethefirstestimatesofhowU.S.importpriceindexesfordifferentincomegroupshavechangedover time in the observed data. The next section outlines the theoretical framework we use to do this. 6RelativetoArkolakisetal.(forthcoming),weallowforthepossiblepresenceofsomevarietiesforwhichareservation(choke) pricedoesnotexist. 5

3 Theoretical model 3.1 Consumers InordertostudytheeffectofchangesinU.S.importsontheconsumerwelfareofdifferentincomegroups,we develop a new theoretical framework that builds on the non-homothetic S-Branch utility tree representation of consumer preferences in Brown and Heien (1972). These preferences have not been widely used in the international trade literature7. However, the S-Branch utility tree nests as special cases preferences such as Nested CES and Generalized CES that have been used recently in the literature8. We consider a world of many producers, indexed by v. Each v should be thought of as a unique variety– the data equivalent to any individual variety v will be a supplier-HS10 product pair. The product made by each producer is classified into a broad sector s, which in our empirical application will be an HS4 code. U.S. consumers have ordinary CES preferences over sectors, such that the utility of household h at time t is given by V ht DŒ X ' h (cid:27) s (cid:27) (cid:0) t 1 Q h (cid:27) s (cid:27) (cid:0) t 1 (cid:141)(cid:27) (cid:27) (cid:0)1 (1) s2S where V is the constant elasticity of substitution aggregate of real consumption of tradable consumer ht goods sectors for household h at time t, Q is the consumption index of sector s for household h at time hst t, ' is a demand shifter for sector s for household h at time t, (cid:27) is an elasticity parameter, and S is the hst set of tradable consumer goods sectors. However, within each sector, households have some minimum quantity ˛ of each variety v that must v be consumed. Importantly, ˛ is a household-level parameter, but it is not indexed by h because we do not v allow variation across households9. In particular, the consumption index of sector s for household h at time t is Q hst DŒ X ' v (cid:27) t (cid:27) s(cid:0) s 1 .q hvt (cid:0)˛ v / (cid:27) (cid:27) s(cid:0) s 1 (cid:141)(cid:27) (cid:27) s(cid:0) s 1 (2) v2Gs where q is the real consumption of variety v in sector s for household h at time t, ' is a demand hvt vt shifter for variety v at time t, (cid:27)s is an elasticity parameter for sector s, ˛ is the household subsistence v quantity required of variety v, and G is the set of varieties in sector s. s This complete utility function– known as the S-Branch utility tree– satisfies the regularity conditions of microeconomic theory when the direct utility function is well-behaved, and satisfies the continuity, monotonicity, and curvature conditions implied by utility maximization when 7However,foranearlyapplicationtoconsumerimportdemand,seeBerner(1977). 8For examples of papers that use the former, see Atkeson and Burstein (2008), Edmond et al. (2015), Hsieh et al. (2016), andAmitietal.(2017). Forthelatter,seeArkolakisetal.(forthcoming),MrázováandNeary(2017),andDhingraandMorrow (forthcoming) 9Inprinciple,wecouldallowvariationinsubsistencequantitiesofeachvarietyacrosshouseholds(˛hv),butwelackdatato disciplinethisvariation. 6

(cid:27) >0; ' >0; (cid:27)s >0; ' >0; k <˛ <q ; (3) hst vt hv v hvt where k < 0 is defined in the appendix and is required to ensure a regular interior solution to the hv utility maximization problem. Allowing for ˛ < 0 extends the parameter region considered in Brown and v Heien (1972)10. The regularity region is defined by the set of prices and sector expenditures such that P Y > ˛ p . Thus, this utility function is effectively globally regular in the sense of Cooper and hst v2S v vt McLaren (1996), because the regularity region grows with real expenditure. 3.1.1 Variety-Level Demand Maximizing household utility can be done as a two-stage budgeting process, where we first maximize the utility from sector s given a preliminary sectoral expenditure allocation. Taking sector expenditure Y as hst given, the utility maximizing quantity demanded of variety v in sector s for household h at time t is !0 1 q hvt D˛ v C p vt P (cid:0)(cid:27) 1 s (cid:0) ' (cid:27) v (cid:27) s t s(cid:0)1 @Y hst (cid:0) X ˛ j p jtA; (4) st j2Gs where P is a sectoral price aggregate given by st 0 1 1(cid:0) 1 (cid:27)s P st D @ X p j 1 t (cid:0)(cid:27)s ' jt (cid:27)s(cid:0)1 A ; (5) j2Gs Y is the expenditure on sector s for household h at time t, and p is the variety-specific price at hst vt time t. For any variety v that does not have a positive quantity sold in all time periods t, we require that ˛ (cid:20) 0. The possibility of “negative subsistence" quantities is also a feature of Stone-Geary preferences. v One interpretation of a negative ˛ , as can be seen in Equation 4, is that it lowers the utility–maximizing v quantity demanded of a particular variety relative to the CES case, which would correspond to ˛ D0. v This variety-level demand system is known as the Generalized CES demand function (Pollak and Wales (1992)) or the Pollak demand function (Mrázová and Neary (2017)) and nests well-known functions as special cases. For example, the Constant Absolute Risk Aversion (CARA) demand function (Behrens and Murata(2007))andtheLinearExpenditureSystem(Stone-Geary)arelimitingcasesoftheGeneralizedCES demand function as (cid:27)s approaches zero and one, respectively. If all the ˛ terms are zero, the Generalized v CESdemandfunctionreducestothestandardCESdemandfunction,whichitselfcontainstheCobb-Douglas and Leontief demand systems as limiting cases as (cid:27)s approaches one and zero, respectively. Note that in the CES case, varieties are substitutes if (cid:27)s > 1 and complements if (cid:27)s < 1. Finally, Mrázová and Neary (2017)notethatthisdemandfunction,asperceivedbyamonopolisticallycompetitivefirm,reducestoLinear demandas(cid:27)s approachesnegativeone. However,theLineardemandcaseisruledoutherebytheparameter restrictions required for integrability. 10Theadmissibilityof˛v<0wasnotedbyBlackorbyetal.(1978),page280. 7

The variety-level demand function has important advantages for the purposes of quantifying the gains from new varieties. First, this demand system does not feature symmetric substitution patterns. Differentiating Equation 4 with respect to the price of another variety j in the same sector s and multiplying by pjt qhvt gives the following cross-price elasticity of demand11: @q p q (cid:0)˛ p hvt jt D. hvt v /. jt /Œ.(cid:27) (cid:0)1/.q (cid:0)˛ /(cid:0)˛ (cid:141) (6) @p q q Y (cid:0)P ˛ p s hjt j j jt hvt hvt hst j2Gs j jt Itimmediatelyfollowsthat, ingeneral, @qhvt pjt ¤ @qhkt pjt , wherek isathirdvarietyinsectors12. Haus- @pjt qhvt @pjt qhkt man(1996) arguesthatsymmetricsubstitution patternsleadsCESandlogit demandsystems topotentially overstate the gains from new varieties, because, upon entry, new varieties gain sales symmetrically from all other existing varieties. Related to this symmetry issue, Ackerberg and Rysman (2005) note that CES and logit may overstate the gains from variety because they do not feature crowding in the product space (i.e., products do not become closer substitutes as the number of products grows). It is clear from Equation 6 that the substitutability of varieties changes as the number of varieties G changes. s Second, the consumer gain from the availability of a new variety, holding all else fixed, is the change in the indirect utility function (or price index) when the price of the new variety changes from its reservation price to the price at which it is sold in positive quantities. Unlike other standard demand functions used in the trade literature, this demand function allows for reservation prices to be finite or infinite depending on the sign of ˛ . Note from Equation 4 that if ˛ D 0, then the reservation price at which variety v is v v demanded in zero quantity is infinite, as in the case of standard CES demand. Reservation prices are also infinite in logit-based models (Bajari and Benkard (2003)). However, if ˛ < 0, then the reservation price v at which variety v is demanded in zero quantity is finite, as in the case of Translog demand. In the finite reservation price case, the reservation price is decreasing in the number of available varieties. 3.1.2 Sector-Level Demand Having solved for the household utility-maximizing quantity demanded for each variety given a preliminary sectoral allocation, we can now solve for the utility-maximizing sectoral expenditures. First substitute Equation 4 into Equation 2, and then substitute the result into Equation 1. As in Brown and Heien (1972)13, maximizing this resulting expression, subject to the constraint that P Y D Y , yields the s hst ht utility maximizing expenditures of household h on sector s: 0 1 Y hst D P ' h (cid:27) s (cid:0) ' t 1 (cid:27) P (cid:0) s 1 t P 1(cid:0)(cid:27) 1(cid:0)(cid:27) @Y ht (cid:0) X X ˛ v p vtA C X ˛ v p vt ; (7) r2S hrt rt r2Sv2Gr v2Gs 11Note that in this derivation we are holding sector expenditure fixed. However, as will be seen later in the paper, except in the case of a Cobb-Douglas upper tier of utility, sector expenditure will in general respond to changes in the sectoral price aggregate. 12However,notethatSlutskysymmetryissatisfied. 13Note that the Brown and Heien (1972) expression for the utility-maximizing expenditure allocation had an error. Our expressionfortheexpenditurescorrectsthiserror. 8

where Y is the total expenditure of household h at time t, S is the number of sectors, and P is the ht st sectoral price aggregate. 3.1.3 Within-Sector and Cross-Sector Non-Homotheticity These preferences feature non-homotheticity both within and across sectors. To demonstrate within-sector non-homotheticity, multiply both sides of Equation 4 by the price of variety v, then divide both sides by the sectoral expenditure Y . This procedure yields the following variety-level share equation: hst p vt q hvt (cid:17)s D˛ p vt C. p vt 1(cid:0)(cid:27)s' v (cid:27) t s(cid:0)1 /. Y hst (cid:0)P j2Gs ˛ j p jt /; (8) Y hst hvt v Y hst P j2Gs . p 'j j t t/1(cid:0)(cid:27)s Y hst where s is the expenditure share of variety v for household h at time t. It is clear from Equation 8 hvt thatGeneralizedCESpreferencesarenon-homothetic: theshareofexpenditurespentonaparticularvariety v is different for households of different incomes.14 The fact that different households can spend different shares of their income on any variety v can also be seen from the elasticity of variety demand with respect to sectoral expenditure: @q Y Y q (cid:0)˛ hvt hst D. hst /. hvt v / (9) @Y q Y (cid:0)P ˛ p q hst hvt hst j2Gs j jt hvt In fact, the Generalized CES expenditure function exhibits the Gorman Polar Form (Deaton and Muellbauer (1980)). Thus, its budget shares are not independent of the level of income, and its Engel curves are not lines through the origin. Instead, the Generalized CES demand function has linear Engel curves that are shifted to have intercepts equal to the subsistence requirements ˛ . Depending on the value of ˛ , the v v expenditure elasticity can be greater than one (luxury goods), equal to one, less than one (necessity goods), ornearlyequaltozero,butnotlessthanzero(i.e.,noinferiorgoods). Thesepreferencessatisfythenecessary and sufficient conditions for the existence of a representative consumer. To demonstrate the presence of cross-sector non-homotheticity, we divide Equation 7 by total household expenditure, which gives the following sector-level expenditure share equation: S D P v2Gs ˛ v p vt C P st 1(cid:0)(cid:27)' hst (cid:27)(cid:0)1 ! (cid:18)Y ht (cid:0)P r2S P v2Gr ˛ v p vt (cid:19) (10) hst Y P P 1(cid:0)(cid:27)'(cid:27)(cid:0)1 Y ht r2S rt hrt ht where S the share of total expenditure for household h spent on sector s. This equation shows that hst these preferences feature non-homotheticity at the sector-level as well, for two reasons. The first reason is because we allow the sector-level demand shifters (' ) to be different across income groups, which will hst generate sector-level non-homotheticity even if all ˛ terms are zero. Second, even if all income groups have the same sector-level demand shifters, the sectoral expenditure shares will differ with income long as the ˛ terms are not all zero. 14Inthemodel,thereisnosavingortransfers,sohouseholdincomeishouseholdexpenditure. Wewilldealwiththisdistinction in detail in the empirical section using U.S. Census income deciles together with implied expenditure patterns for households inthatdecilefromtheConsumerExpenditureSurvey. 9

3.2 Import Price indexes In this section, we first derive an expression for the household-level import price index. We then use the aggregation properties of the model to build up the aggregate import price index. 3.2.1 Combining Sector-Level and Variety-Level Demand Combining Equations 4 and 7, we can write the utility-maximizing quantity demanded of variety v in sector s in terms of aggregate expenditure Y for household h as ht q D˛ C. p vt (cid:0)(cid:27)s' v (cid:27) t s(cid:0)1 /. ' h (cid:27) s (cid:0) t 1P st 1(cid:0)(cid:27) /.Y (cid:0) X X ˛ p /: (11) hvt v P . pjt/1(cid:0)(cid:27)s P '(cid:27)(cid:0)1P 1(cid:0)(cid:27) ht j jt j2Gs 'jt r2S hrt rt r2Sj2Gr 3.2.2 Household Import Price indexes In this section, we derive expressions for the household-specific import price indexes. As shown in Blackorby et al. (1978) (pages 280 - 284), we can substitute Equation 11 into Equation 2, then substitute the subsequent expression into Equation 1, to write the indirect utility function dual to our complete direct utility function as ! 1 (cid:27)(cid:0)1 V D X '(cid:27)(cid:0)1P 1(cid:0)(cid:27) .Y (cid:0) X X ˛ p /; (12) ht hst st ht v vt s2S s2Sv2Gs where V is indirect utility for household h at time t and P is the same sectoral price aggregate given ht st earlier. Re-arranging Equation 12 gives the following household expenditure function: ! 1 1(cid:0)(cid:27) Y DV X '(cid:27)(cid:0)1P 1(cid:0)(cid:27) C X X ˛ p : (13) ht ht hst st v vt s2S s2Sv2Gs Picking a reference indirect utility V to hold constant defines the following price index for household hk imported consumption: ! 1 1(cid:0)(cid:27) P (cid:17)V X '(cid:27)(cid:0)1P 1(cid:0)(cid:27) C X X ˛ p ; (14) ht hk hst st v vt s2S s2Sv2Gs where the change in the import price index for household h from time t to time t Ci can be written as P htCi D V hk (cid:0)P s2S ' h (cid:27) s (cid:0) t 1 Ci P stCi 1(cid:0)(cid:27)(cid:1) 1(cid:0) 1 (cid:27) CP s2S P v2Gs ˛ v p vtCi : (15) P ht V (cid:0)P '(cid:27)(cid:0)1P 1(cid:0)(cid:27) (cid:1) 1(cid:0) 1 (cid:27) CP P ˛ p hk s2S hst st s2S v2Gs v vt Using the expenditure function to substitute in for the reference utility level V , we can alternatively hk express the change in the import price index for household h from time t to time t Ci as 10

P htCi D Œ P s2S ' hstCi (cid:27)(cid:0)1P stCi 1(cid:0)(cid:27)(cid:141)1(cid:0) 1 (cid:27) (cid:18)Y hk (cid:0)P s2S P v2Gs ˛ v p vt (cid:19) C (cid:18)P s2S P v2Gs ˛ v p vtCi (cid:19) (16) P ht Œ P s2S ' hst (cid:27)(cid:0)1P st 1(cid:0)(cid:27)(cid:141)1(cid:0) 1 (cid:27) Y hk Y hk This expression for the household-specific price index change clearly shows that households of different incomes will experience different import price inflation rates if either 9.˛ ¤0/ or ' ¤' 8h. v hst st 3.2.3 Aggregate Market Demand In this section, we show how our household-level demand functions can be consistently aggregated up to market-level demand functions. In doing so, we primarily utilize the Gorman polar form of our variety-level householddemandfunctions. Crucially,weshowthatasimplerestrictionontheparametersofourfunctional form, which will be supported by our unrestricted empirical estimates, allows us to extend the exact linear aggregation results of Gorman (1961) to the case where varieties enter and exit the data over time. To our knowledge, this is the first discussion of this issue and this theoretically consistent solution in the literature. Specifically, we retain exact linear aggregation for our functional form with variety entry and exit under two conditions.15 The first condition is that non-continuing varieties all have ˛ D 0. We will not initially v impose this constraint on our estimation, but our unconstrained estimates will provide support for this condition. The second condition that we require, a standard condition with Gorman polar form, is that ˛ v forcontinuing varieties issuchthat each householdincome groupbuys some positiveamount of eachvariety in each time period t that has positive sales in the aggregate at that time t.16. In other words, we require P that Y > ˛ p and ˛ > k , which is the condition defined in the appendix that ensures regular hst v2S v vt v hv interior solutions.17 Under these simple parameter restrictions, the exact linear aggregation over households’ variety-level demand is preserved, and the market demand for variety v at time t is given by q D P q . Market vt h hvt demand can be written as q D.˛ n /C. p vt (cid:0)(cid:27)s' v (cid:27) t s(cid:0)1 /.Y (cid:0) X .˛ n /p /; (17) vt v t P . pjt/1(cid:0)(cid:27)s st j t jt j2Gs 'jt j2Gs where Y is aggregate U.S. expenditure on imports in sector s and demand is aggregated over n housest t holds. The representative U.S. consumer’s sector-level demand will take the following form: 0 1 Y st D P ' s (cid:27) t (cid:0) ' 1 (cid:27) P (cid:0) s 1 t P 1(cid:0)(cid:27) 1(cid:0)(cid:27) @Y t (cid:0) X X .˛ v n t /p vtA C X .˛ v n t /p vt ; (18) j2S jt jt s2Sv2Gs v2Gs 15Incontrast,onlyexactnonlinearaggregationisretainedintheimplicitlyadditivenon-homotheticCESdemandsystemof Hanoch(1975). 16Each of our household income groups will represent multiple households that fall within the same income decile, so the property that each income group buys positive quantities of all goods with positive sales in the aggregate is not completely unrealistic. Further, these preferences can also be given a discrete choice microfoundation with a random utility model from theGeneralizedExtremeValuedistribution(ThisseandUshchev(2016)). 17Weimposethisconditionasaconstraintonourparameterestimation. 11

where Y is aggregate U.S. expenditure on imports and ' is the representative agent’s sectoral demand t st shifter for sector s, which can be thought of as the average ' across households h.18 hst 3.2.4 Aggregate Import Price Index When we analyze household price indexes, we will focus on consumer goods sectors. However, in our baseline aggregate price index results we will treat all sectors as if they were consumer facing, as in Broda and Weinstein (2006) and the subsequent literature.19 The expenditure function of the representative U.S. consumer is given by Y t DV t Œ X ' s (cid:27) t (cid:0)1P st 1(cid:0)(cid:27)(cid:141)1(cid:0) 1 (cid:27) C X X .˛ v n t /p vt : (19) s2S s2Sv2Gs Picking a reference indirect utility V , the change in the price index for aggregate imports from time t k to time t Ci can be written as P tCi D V k (cid:0)P s2S ' s (cid:27) t (cid:0) C 1 i P stCi 1(cid:0)(cid:27)(cid:1) 1(cid:0) 1 (cid:27) CP s2S P v2Gs .˛ v n t /p vtCi (20) P t V (cid:0)P '(cid:27)(cid:0)1P 1(cid:0)(cid:27) (cid:1) 1(cid:0) 1 (cid:27) CP P .˛ n /p k s2S st st s2S v2Gs v t vt 3.2.5 Price Index Decomposition A useful property that we will exploit is that the CES subcomponent of our aggregate import price index, following Hottman et al. (2016), can be linearly decomposed as below: ln.Œ X ' s (cid:27) t (cid:0)1P st 1(cid:0)(cid:27)(cid:141)1(cid:0) 1 (cid:27)/D N 1 S X . N 1 v X lnp vt / s2S t s2S st v2Gst 1 X 1 X 1 X (cid:0) . ln' /(cid:0) ln' NS Nv vt NS st t s2S st v2Gst t s2S 1 1 X 1 (cid:0) lnNS (cid:0) lnNv (21) (cid:27) (cid:0)1 t NS (cid:27)S (cid:0)1 st t s2S (cid:0) 1 ln. 1 X . 3 P 's s t t/1(cid:0)(cid:27) /(cid:0) 1 X 1 ln. 1 X . 4 p 'v v t t/1(cid:0)(cid:27)S /: (cid:27) (cid:0)1 N t S s2S .P 's s t t/1(cid:0)(cid:27) N t s s2S (cid:27)S (cid:0)1 N s v t v2Gst .p 'v v t t/1(cid:0)(cid:27)S 3 whereN t S isthenumberofsectorsattimet, N s v t isthenumberofvarietiesins4ectors attimet, .P 's s t t/1(cid:0)(cid:27) is the geometric average of the sector-level quality-adjusted prices at time t, and .pvt/1(cid:0)(cid:27)S is the geometric 'vt average of the variety-level quality-adjusted prices in sector s at time t. In this decomposition of the CES subcomponent of the import price index, each set of terms captures different economic forces. The first term on the right-hand side is a geometric average of prices, which 18Wedonotrelyonexactlinearaggregationholdingatthesectoraldemandlevel,althoughforthepurposesofconstructing anaggregateimportpriceindexweutilizetheformgiveninthetext. 19SeeCaliendoandParro(2015)andOssa(2015)forpapersthatinsteaduseaninput-outputstructuretotreatintermediate goodsdifferently. 12

captures the prices of varieties available at time t as in a standard price index. The terms in the second row capture the geometric average sector quality and the geometric average variety quality at time t, which are often absent from standard price indexes and thus give rise to a quality adjustment bias in these indexes. Thetermsinthethirdrowcapturethenumberofsectorsattimet andthenumberofvarietiesintheaverage sector at time t, which, when absent from standard price indexes, give rise to a new goods bias. The terms on the final row of the decomposition above capture the dispersion in quality-adjusted prices across sectors at time t and the dispersion in quality-adjusted prices across varieties within sectors at time t, which, when absent from standard price indexes, give rise to a substitution bias. Importantly, this decomposition allows us to attribute changes in the price index to changes in quality or changes in the set of available varieties, two of the main channels the literature attributes to changes in import prices. The decomposition also separates the contribution of the geometric average of prices from the contribution of dispersion in (quality-adjusted) prices. Therefore, our model can capture all four of the major channels responsible for changing import prices, and identify which ones are most important via estimation of the model. 3.3 Firms Firms in each sector are assumed to engage in monopolistic competition and treat the sector price index and expenditure parametrically20. Under this assumption, and given our demand structure, firms that sell multiple varieties in the same sector will still behave like single-variety firms. That is, equilibrium markups will vary across the varieties within a multivariety firm as if each variety was sold by a different firm in the same sector. The exposition that follows refers to firms and varieties interchangeably, as would be the case for single-variety firms. In specifying the firm’s cost structure, we allow marginal costs to be variable, (weakly) increasing in output, and given by c Dı .1C! /q!s (22) vt vt s vt where ! (cid:21) 0 parameterizes the convexity of the cost function in sector s and ı > 0 is a variety-level s vt shifter of the cost function. Using Equation 17 (and as in Mrázová and Neary (2017)), the monopolistically competitive own-price elasticity of demand perceived by each firm is given by21 @q p q (cid:0)˛ n " (cid:17)(cid:0) vt vt D. vt v t /.(cid:27)s/ (23) vt @p q q vt vt vt Importantly,eveninthecaseof˛ >0afirmcanperceiveitselftobefacingprice-elasticmarketdemand v as long as ˛ n is not too large relative to q , holding (cid:27)s fixed. v t vt 20Technically,firmsdonotinternalizetheireffectonthesectorpriceindexbecausetheyareassumedtobemeasurezerowith regardtothemarketinwhichtheyoperate. 21Ifsingle-productfirmsinternalizetheireffectonthepriceindex,theperceivedelasticitybecomes"vt D.qvt(cid:0) qv ˛ t vnt/Œ.(cid:27)s/C ˛vnt . p Y v s t t (cid:0) (cid:0) . P (cid:27)s j (cid:0) 2G 1/ s p ˛ v j t. n q t v p t j (cid:0) t ˛ / vnt/(cid:141). 13

The curvature (convexity) of demand perceived by each monopolistically competitive firm is given by (cid:16) (cid:17)(cid:0) q vt @2p @2 v q t. v q t vt/ D (cid:0) pvt @q @ v 2 @p t q @ . v 2 v p t p t v . v p t t / vt/ D. (cid:27)s C1 /. q vt / (24) vt @pvt.qvt/ " vt (cid:27)s q vt (cid:0)˛ v n t @qvt The first-order condition for profit maximization implies that firms set prices as a markup over marginal cost according to " p D vt c (25) vt " (cid:0)1 vt vt where c is the marginal cost of variety v at time t. Each firm’s first order condition is satisfied when vt " >1, and each firm’s second-order condition is satisfied when (cid:16) <2.22 vt vt Combining the first-order condition and marginal cost equations gives the following pricing equation " p D vt ı .1C! /q!s (26) vt " (cid:0)1 vt s vt vt Using the equation for the own-price elasticity of demand, we can write the markup term as " .q (cid:0)˛ n /.(cid:27)s/ vt D vt v t (27) " (cid:0)1 .q (cid:0)˛ n /.(cid:27)s/(cid:0)q vt vt v t vt Thereareafewthingstonoteaboutthismarkupterm. First,markupsarebothvarietyspecificandtime specific. Second, even though the markup is variety specific, the only observable data needed to calculate the markup (after estimating ˛ and (cid:27)s) are the quantity sold at time t- no production function estimation v is required. Finally, the markup reduces to the constant markup of (cid:27)s if ˛ D0. (cid:27)s(cid:0)1 v Taking the ratio of markups for two different varieties in the same sector and subtracting one gives "v " t v (cid:0) t 1 (cid:0)1D .q kt (cid:0)˛ k n t /q vt (cid:0).q vt (cid:0)˛ v n t /q kt ; (28) "kt .q (cid:0)˛ n /.q (cid:0)˛ n /.(cid:27)s/(cid:0).q (cid:0)˛ n /q "kt (cid:0)1 kt k t vt v t kt k t vt where it can immediately be seen that a higher (cid:27)s, all else equal, lowers the scope for differences in markups across varieties in a sector. Differentiating the markup term in Equation 27 with respect to quantity gives @. "v " t v (cid:0) t 1 / D (cid:0).˛ v n t /.(cid:27)s/ ; (29) @q Œ.q (cid:0)˛ n /.(cid:27)s/(cid:0)q (cid:141)2 vt vt v t vt which is negative if and only if ˛ is positive 23. v Tradeisdescribedas“pro-competitive"ifanincreaseincompetitionviaincreasedentryfromforeignfirms results in reduced market shares or quantity sold for incumbent producers and decreased markups for these incumbents. Most preference structures used in the literature either deliver a positive partial derivative of 22Insection6.3oftheappendixwebrieflydiscusstheequilibriumofthismodel. 23Intheoligopolycaseoffootnote21,Equation29canstillbeeitherpositiveornegativedependingonthesignandmagnitude of˛v. 14

markupswithrespecttoquantityor(asintheCEScase)nochangeinmarkupsatall. However,asEquation 29shows,ratherthanassumingtheseeffects,estimating˛ allowsustotestwhethertradeispro-competitive, v anti-competitive, or neither.24 All of these cases are possible in general, as highlighted in recent theory (Krugman (1979), Zhelobodko et al. (2012), Mrázová and Neary (2017), Parenti et al. (2017), Dhingra and Morrow (forthcoming)). As can be seen from differentiating the markup term, firm v’s markup is increasing in its quantity sold if ˛ < 0, v is decreasing in its quantity sold if ˛ > 0, and is constant if ˛ D 0. Standard monopolistic competition v v models used in the trade literature based on demand systems such as Almost Ideal Demand or Translog (FeenstraandWeinstein(2017)),Lineardemand(MelitzandOttaviano(2008)),Logitdemand(Fajgelbaum etal.(2011)),andCARAdemand(BehrensandMurata(2007))donotfeatureanti-competitiveeffectsfrom openingtotrade. Additionally, trademodelsbased onCESdemand witholigopolisticcompetition(Atkeson and Burstein (2008), Holmes et al. (2014), De Blas and Russ (2015), Edmond et al. (2015)) also have the feature that larger market share implies larger market power, and thus successful import penetration must have pro-competitive effects in these models. 4 Empirical strategy In this section, we describe our strategy for recovering the deep parameters of the model laid out above, usingU.S.importdatafrom1998to2014. Importantly,wecanestimatetheparametersoftheabovepartial equilibriummodelwithoutrelyingonassumptionsaboutthedistributionoffirmproductivityorafullgeneral equilibrium framework. Our estimation will proceed in two stages. In the first stage, we will estimate the parameters of the variety demand functions at the aggregate market level. To address endogeneity concerns, we extend the Feenstra (1994) approach of identification via heteroskedasticity to estimate the model directly from the aggregatedataontheuniverseofgoodssuppliersexportingtotheUnitedStates. Aswillbemadeclear, the only observable data needed are variety-level prices and sales. In the second stage, we will estimate the parameters of the sectoral demand functions at the household level. We do this by supplementing the import data with additional data on the sectoral composition of the consumption baskets of different income groups from the BLS Consumer Expenditure Survey. To deal with possible endogeneity, we develop an instrumental variables strategy for this setting. These two stages of estimation will recover all of the parameters of our model, which, when combined with the microdata, allow us to construct household-level and aggregate import price indexes. 24Intheempiricalsection, weareonlyusingforeignproducerdata, so, inourcontext, incumbentsshouldbeinterpretedas producersalreadyexportingtotheUnitedStates. Thisisadataconstraint,notatheoreticalone. 15

4.1 Estimation Stage One While the theory is written to accommodate arbitrarily different ˛’s for every variety, we will specify the following empirical specification for this parameter: 1 h i ˛ D ˇC min.q j q >0/ for v 2GC; (30) v n 1998 s t vt vt s and 1 h i ˛ D ˇEmin.q j q >0/ for v 2GE; (31) v n 1998 s t vt vt s where ˛ is the subsistence quantity required for variety v, n is the number of households in 1998, v 1998 GC is the set of varieties in sector s that constantly have positive quantities sold throughout the sample s period, GE is the set of varieties in sector s that enter or exit (e.g., have zero quantity sold) at some point s during the sample period, and ˇj is the parameter to be estimated that potentially differs across the two s sets of varieties for sector s. We require that ˇE (cid:20) 0 and ˇC < 1. These restrictions satisfy the parameter s s restrictions on ˛ required for regularity to hold for all our observations of prices and expenditure. v Asasimpleexampletofixtheintuitionforthisspecification,considerthecaseofˇC D1. Inthiscase,the s totalsubsistencequantityofvarietyv soldin1998(˛ n )issimplytheminimumquantityobservedinthe v 1998 import data, and therefore each household consumes mint.qvt j qvt>0/ for subsistence. The total subsistence n1998 quantity sold in any other year t, again in the case of ˇC D1, is given by ˛ n D nt Œmin .q j q >0/(cid:141). s v t n1998 t vt vt For each sector, there are four deep parameters to be estimated: (cid:27)s, ˇC, ˇE, and ! . Conditional on s s s estimating these parameters, the remaining variety-level unobservables of ˛ (subsistence quantities), ' v vt (demand shifters), ı (cost shifters), "vt (markups), and c (marginal costs) can be recovered from the vt "vt (cid:0)1 vt model’s structure given the data on prices and sales.25 Toestimatetheseparameters,weextendtheFeenstra(1994)approachofidentificationviaheteroskedasticitytoourframework. OurextensionissimilarinspirittotheapproachinFeenstraandWeinstein(2017). Identification via heteroskedasticity has also been proposed in other recent papers (Rigobon (2003), Lewbel (2012)). Start from the variety-level demand expression in Equation 17. Multiplying both sides by p , taking vt logs, taking the time difference and difference relative to another variety k in the same sector s gives (cid:129)k;tln.p q (cid:0)˛ n p /D.1(cid:0)(cid:27)s/(cid:129)k;tln.p /C(cid:29) ; (32) vt vt v t vt vt vt where(cid:129)k;t referstothedoubledifferenceandtheunobservederrortermis(cid:23) D.1(cid:0)(cid:27)s/ (cid:2)4tln' (cid:0)4tln' (cid:3) , vt kt vt where 4t refers to a single difference across time periods. Next, we work with the variety-level pricing expression in Equation 26. Multiplying both sides by p!s, vt taking logs, and double-differencing as before gives 25ThenumberofU.S.householdsinagivenyear,nt,isavailableinpublicdatafromtheU.S.CensusBureau. 16

! 1 " (cid:129)k;tlnp D s (cid:129)k;tln.p q /C (cid:129)k;tln. vt /C(cid:20) ; (33) vt 1C! vt vt 1C! " (cid:0)1 vt s s vt where the unobserved error term is (cid:20) D 1 (cid:2)4tlnı (cid:0)4tlnı (cid:3) . vt 1C!s vt kt As in Feenstra (1994), the orthogonality condition for each variety is then defined as G.ˇ s /DE TŒx vt .ˇ s /(cid:141)D0 (34) 0 (cid:27)s 1 where ˇ s DB B @ ˇ ˇ s E C C C A and x vt D(cid:23) vt (cid:20) vt . s ! s This condition assumes the orthogonality of the idiosyncratic demand and supply shocks at the variety level after variety and sector-time fixed effects have been differenced out. This orthogonality is plausible becauseinadditiontothefixedeffects,wehavealsoremovedvariationinpricesduetomarkupvariationand movementsalongupward-slopingsupplycurves. Theremainingsupplyshockstaketheformofidiosyncratic shifts in the intercept of the variety-level supply curve and are unlikely to be correlated with idiosyncratic shifts in the intercept of the variety-level demand curve26. For each sector s, stack the orthogonality conditions to form the GMM objective function ˇ O Dargmin ˚ G (cid:3) .ˇ / 0 WG (cid:3) .ˇ / (cid:9) (35) s s s ˇs whereG(cid:3).ˇ /isthesamplecounterpartofG.ˇ /stackedoverallvarietiesinsectors andW isapositive s s definite weighting matrix. Following Broda and Weinstein (2010), we give more weight to varieties that are present in the data for longer time periods and sell larger quantities27. As can be seen from the estimating equationabove,thenecessaryobservablestoestimatethemodelaresupplier-productquantitiesq andsales vt p q . Supplier-product prices p can be obtained by dividing revenues by quantities to form unit values. vt vt vt To see how the parameters are identified, rewrite the orthogonality condition that holds for each variety and rearrange to get ! E TŒ.(cid:129)k;tlnp vt /2(cid:141)D .1C s ! / E TŒ(cid:129)k;tln.p vt q vt /(cid:129)k;tlnp vt (cid:141) s 1 (cid:0) (cid:27)s (cid:0)1 E TŒ(cid:129)k;tlnp vt (cid:129)k;tln.p vt q vt (cid:0)˛ v n t p vt /(cid:141) ! C .1C! /. s (cid:27)s (cid:0)1/ E TŒ(cid:129)k;tln.p vt q vt /(cid:129)k;tln.p vt q vt (cid:0)˛ v n t p vt /(cid:141) (36) s 1 " C .1C! / E TŒ(cid:129)k;tlnp vt (cid:129)k;tln. " v (cid:0) t 1 /(cid:141) s vt 1 " C .1C! /.(cid:27)s (cid:0)1/ E TŒ(cid:129)k;tln.p vt q vt (cid:0)˛ v n t p vt /(cid:129)k;tln. " v (cid:0) t 1 /(cid:141) s vt 26Inarobustnessexercisereportedbelow,weshowthatourestimatesdonotchangeverymuchwhenweonlyestimateusinga sampleofmulti-varietyfirms,andtakedifferencesrelativetoanothervarietywithinthesamefirm,inordertoremovefirm-time fixedeffectsfromthesupplyanddemanderrorterms. 27Varietieswithlargerimportvolumesareexpectedtohavelessmeasurementerrorintheirunitvalues. 17

This estimating equation shows the importance of heteroskedasticity. If the variances and covariances in this equation are the same across the different varieties in a given sector, then there is not identification. However, if the variances and covariances differ across varieties, then pooling the observations of these moments across varieties in a sector allows for identification of the four common sector-level parameters, so long as there more varieties in the sector than parameters. Finally,atthispointwerecoverthevariety-leveldemandshifters. InthespiritofReddingandWeinstein (2016),wenormalizethedemandshifterforthegeometricaveragefirm-producttobeoneforalltimeperiods. e Thus, we normalize ' D' D1 across firm-products, where the tilde denotes the geometric average. The kt ek demand shifters for all firm-products can be computed using Equation 17 in differences to get the following expression e e Aln.p q (cid:0)˛ n p /(cid:0)ln.p q (cid:0)˛ n p /C.(cid:27)s (cid:0)1/.lnp (cid:0)lnp / ' Dexpf vt vt v t vt kt kt k t kt vt kt g; (37) vt (cid:27)s (cid:0)1 where .p q (cid:0)˛ n p / is the geometric average of .p q (cid:0)˛ n p / across varieties in the sector at kt kt k t kt kt kt k t kt time t. 4.2 Estimation Stage Two Given the previously estimated parameters, we can now estimate (cid:27) in the following way. First, we must construct Y , the expenditure on imports by HS4 sector s for household h. We leave the specific details to hst the data section, but broadly, we construct the variable using the BLS Consumer Expenditure Survey and the trade data. Then, starting from the household sector-level demand expression in Equation 7, take the time difference and difference relative to another sector k bought by the same household h. This doubledifferencing gives X (cid:129)k;tln.Y (cid:0) ˛ p /D.1(cid:0)(cid:27)/(cid:129)k;tln.P /C(cid:29) ; (38) hst v vt st hst v2Gs where (cid:29) D .(cid:27) (cid:0)1/ (cid:2) (cid:129)k;tln' (cid:3) . We can construct the objects that enter this equation using our hst hst previous parameters and the data. We then form our estimating equation by pooling the double-differenced observations across households, sectors, and time. We expect that running Ordinary Least Squares on the above equation would probably not produce a consistent estimate of (cid:27), because of potential endogeneity bias from a possible correlation between the sectoral price index and the error term. To address this potential issue, we pursue an instrumental variables approach as in Hottman et al. (2016). Note that, as in section 3.2.5, the change in the log of the sectoral price index can be linearly decomposed into four terms as follows: (cid:129)k;tlnP D(cid:129)k;t. 1 X lnp /(cid:0)(cid:129)k;t. 1 X ln' /(cid:0)(cid:129)k;t 1 lnNv (cid:0)(cid:129)k;t 1 ln. 1 X . D p 'v v t t/1(cid:0)(cid:27)S /; st N s v t v2Gst vt N s v t v2Gst vt (cid:27)S (cid:0)1 st (cid:27)S (cid:0)1 N s v t v2Gst .p 'v v t t/1(cid:0)(cid:27)S 18

We use the fourth term on the right-hand side, which measures the change in dispersion in qualityadjusted variety-level prices within a sector, as an instrument for the change in the price index term when we estimate Equation 38. Given an estimate of (cid:27), we can then solve for the household-specific sectoral demand shifters (' ), e e hst which can be done by normalizing ' D ' D 1 across sectors and using Equation 7 in differences to hkt hk B derive e ln.Y (cid:0)P ˛ p /(cid:0)ln.Y (cid:0)P ˛ p /C.(cid:27) (cid:0)1/.lnP (cid:0)lnP / ' Dexpf hst v2Gs v vt hkt v2Gk v vt st kt g (39) hst .(cid:27) (cid:0)1/ 4.3 Data 4.3.1 Trade Data The main data come from the Linked-Longitudinal Firm Trade Transaction Database (LFTTD), which is collected by U.S. Customs and Border Protection and maintained by the U.S. Census Bureau. Every transaction in which a U.S. company imports or exports a product requires the filing of Form 7501 with U.S. Customs and Border Protection, and the LFTTD contains the information from each of these forms.28 There are typically close to 40 million transactions per year. We utilize the import data from 1998 to 2014, which includes the quantity and value exchanged for each transaction,HarmonizedSystem(HS)10productclassification,dateofimportandexport,portinformation, country of origin, and a code identifying the foreign supplier. Known as the manufacturing ID, or MID, the foreign partner identifier contains limited information on the name, address, and city of the foreign supplier.29 Monarch (2014) and Kamal and Monarch (n.d.) find substantial support for the use of the MID asareliable,uniqueidentifier,bothovertimeandinthecrosssection. PierceandSchott(2012),Kamaland Sundaram (2016), Eaton et al. (2014), Heise (2015), and Redding and Weinstein (2017) have all used this exporteridentifier,andReddingandWeinstein(2017)alsoshowthatmanyofthesalientfeaturesassociated with exporting activity (such as the prevalence of multi-product firms and high rates of product and firm turnover) are replicated for MID-identified exporters. We build on the methods of Bernard et al. (2009) for cleaning the LFTTD. Specifically, we drop all transactionswithimputedquantitiesorvalues(whicharetypicallyverylow-valuetransactions)orconverted quantities or values. We also drop all observations without a valid U.S. firm identifier. After making these reductions, the average year has close to 2.6 million imported varieties. Figure 1 presents a snapshot of the number of varieties over different years in our sample- interestingly, the number of varieties increases dramatically between 1998 and 2007 before declining, rising, and declining again in the latter half of our time period. 28Approximately80to85percentofthesecustomsformsarefilledoutelectronically(KamalandKrizan(2012)). 29Specifically,theMIDcontainsthefirstthreelettersoftheproducer’scity,sixcharacterstakenfromtheproducer’sname, uptofournumericcharacterstakenfromitsaddress,andtheISO2codeforthecountryoforigin. 19

3,200,000 Figure 1: Number of Imported Varieties in the U.S., 1998-2014 3,000,000 2,800,000 2,600,000 2,400,000 2,200,000 2,000,000 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Thispatternisnewtotheliterature,sowecheckitagainstanumberofothersimilarmeasuresofvariety. First,wecomparethisnumbertoamoretraditionaldefinitionofavarietyinFigure2usingpubliclyavailable Census data on the number of country-HS combinations imported by the U.S.30 Although growth is more mutedthaninthemoredisaggregatedcase(astherearetypicallymultiplesupplierspercountry),theoverall contour is also very similar, with the exception of the change from 2013 to 2014. Second, motivated by the potential for changes in HS codes over time (as documented by Pierce and Schott (2009)) to affect our results, weconfirmthatthispatternholdsevenwhenonlyusingthoseHScodesthatarepresentinallyears of the data, defining a variety as a supplier in a continuing HS code (the yellow line) and a country in a continuing HS code (the orange line). Correcting for entering and exiting HS codes does cause the level of varieties to shrink by close to one-third, but does little to change the time series pattern of the respective variety measures. The first estimation stage entails estimating Equation 36 using the trade data. The estimation is performedonareducedsampleofvarieties, asalargenumberofsupplier-HS10combinationsonlyappearonce. Thus, for our cleaned sample, we use only those varieties that are present for six or more years of data. Furthermore, as Equation 36 relies on double-differenced price and sales terms as components, we winsorize by dropping double-differenced variety price and sales changes that are below the 1st percentile and above the 99th percentile. We also drop any HS4 sector that features fewer than 30 varieties over our 17 years of data. Note that with the exception of our parameter estimation sample, all our results will use the universe of goods varieties in the U.S. import data. We run our estimation routine on each HS4 sector where there are enough observations to do so, which 30These data are described in Schott (2008) and are available from http://faculty.som.yale.edu/peterschott/sub_ international.htm. 20

Figure 2: Number of Imported Varieties in the U.S., relative to 1998 1.5 1.4 1.3 1.2 1.1 1 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Supplier‐HS Varieties Supplier‐Continuing HS Varieties Country‐HS Varieties Country‐Continuing HS Varieties amounts to 980 HS4 sectors and over 95% of total U.S. goods imports. The parameters are estimated using a nonlinear solver to solve the GMM problem described above for each of 980 HS4 sectors. We directly impose constraints on this nonlinear estimation31. Our approach contrasts with the two-step process of Broda and Weinstein (2006) and the related literature, which involves estimating parameters from the unconstrained GMM problem in the first step and then conducting a grid search when the parameters from the unconstrained estimation take implausible values. 4.3.2 Consumer Expenditure Survey Data Thenextstepistomovetoestimatingthesectoraldemandequationsatthehouseholdlevel. Herewepresent thedetailedprocedureforconstructingthehousehold-specificexpenditureonimportsinanHS4sector,Y . hst TheBLSConsumerExpenditureSurvey(CE)publicdataprovidesinformationonhow10incomedeciles allocatetheirexpenditureacrossdifferentCEcategories,beginningin2014(prioryearsonlyreportexpenditurebyquintiles)32. TheCEisthedatathatunderliethecategoryexpenditureweightsintheU.S.Consumer Price Index. In order to utilize the decile expenditure information along with our parameters estimated at the HS4 level, we undertake the following steps: 31Weimposethefollowingconstraints: "vt (cid:21)1:01,!s (cid:21)0,(cid:16)vt (cid:20)1:99,ˇ s E (cid:20)0,ˇ s C <1,andkhv<˛v,thelastofwhichis theconditiondefinedintheappendix. 32Thepublic-usemicrodatahasincometop-codingabovethe6thdecile, sowecannotusethisdatatocalculatedecile-level expendituresharesforearlieryears. Inprinciple,wecouldusethenon-publicmicrodatatoproducedecile-leveltablespriorto 2014. 21

1. Begin with public U.S. Census Bureau estimates for income levels for each decile and year, 1998 to 2014. 2. Constructexpenditure-to-income(EI)ratiosforeachincomelevelestimateinStep1usingpublicdata from the CE in every year. 3. Apply the EI ratios to the Census income numbers by decile to obtain expenditure in every year for every decile. 4. Apply the 2014 decile-specific expenditure shares across CE categories to each year’s decile total expenditure to get decile-specific expenditure on each CE category. 5. Concord the CE categories to HS4 codes to get decile-specific expenditure on each HS4 category. 6. Apply the import share in domestic absorption for each year to create decile-specific imported expenditure in each HS4 category. For Step 2, we take income levels from Census data and construct expenditure-to-income ratios using the appropriate annual income-group from the BLS Consumer Expenditure Survey. For example, the first income decile had an income of $14,070 in 1998, and the CE for 1998 indicates that people who earned between $10,000 and $15,000 had expenditures equal to 1.613 of their income, on average, so we impute total expenditure in 1998 to be $22,691.64. We generate total expenditures by applying these EI ratios to the income data by decile in this manner (Step 3)- the implied expenditure numbers appear reasonably free from large year-to-year swings.33 After this, we apply the decile-specific category expenditure shares from 2014 to each of these implied total decile expenditure numbers (Step 4). This is a workaround to the fact thattheBLSonlyprovidesdecile-specifictotalexpenditureandcategoryexpendituresharesfor2014. When we look at earlier years, we find that the category expenditure shares by quintile do not change significantly from 1998 to 201334. For Step 5, we use a concordance between the categories in the CE and Harmonized System categories developed by Furman et al. (2017).35 An important fact to note here is that only about 20% of HS4 sectors can be found in consumer expenditures- the rest are intermediate inputs. In the end, we will use 228 HS4 sectors in the household price indexes. Although this may appear small, we find that 54.6% of total U.S. goods import value is in HS4 categories that can be concorded to the CE. Additionally, 361 of the 787 total expenditure categories in the CE can be linked to an imported HS4 category. Step 6 entails converting total HS4 expenditure into imported HS4 expenditure. To do this, we multiply by the sectoral import share in domestic absorption for each year, which is defined as in Feenstra and Weinstein (2017). Household-specific expenditure on imports from sector s are given by 33TheexpenditurenumbersareincludedinTable16intheAppendix. 34Forexample,usingtheexpendituresharesofthebroadCEcategories(Food,Housing,Apparel,Transportation,Healthcare, andEntertainment)wefindacorrelationbetweenthe1998quintilesharesandthe2013quintilesharesof0.9933. 35In cases where one CE category maps into multiple HS4 categories, we use the share of total U.S. import expenditure to allocatespendingacrossHS4s. 22

M Y DE . s;t /; (40) hst hst G (cid:0)X CM s;t s;t s;t where E is the sector-level expenditure by household h derived in Step 5, M is the nominal value of hst s;t U.S. imports in sector s; G is the nominal value of U.S. production in sector s; and X is the nominal s;t s;t value of U.S. exports in sector s. We use total sectoral output data from the BEA to construct G , and s;t aggregate imports and exports from the LFTTD to construct M and X .36 Also, to clarify notation, s;t s;t summing Y across sectors will result in total expenditure on imports, not total expenditure in the CE, hst which we will refer to as TotExp . Note that although Ehst will be constant over time by construction ht TotExpht (since we only have CE decile shares for 2014), sector s’s share of household h’s import basket ( Yhst ) P sYhst will not be constant over time because sectoral import shares in domestic absorption have different sectoral trends37. We find several interesting facts from our Y calculation.38 First, the share of imports in total expenhst diture (i.e., P sYhst ) does not differ much across income groups: we find that in 2014, the share of imports TotExpht inactualexpenditureaveragedabout10%acrossdeciles,withastandarddeviationof0.5percentagepoints. Thus, any differences in import price indexes across income groups can be meaningfully compared because therearesmalldifferencesinsharesofimportsinconsumption. Table1showstheshareoftotalexpenditure on imports across different deciles in 1998 and 2014. Interestingly, there is not much variation across deciles in the cross section, but the share of spending on imports does increase over our time period. Table 1: Share of Expenditure on Imports by U.S. Decile of Income, 1998 and 2014 (%) Year 1st Decile 2nd Decile 4th Decile 5th Decile 6th Decile 8th Decile 9th Decile 1998 6.49 6.58 6.35 6.99 7.49 7.01 7.05 2014 9.96 9.22 9.24 10.05 10.63 10.02 10.06 Second, non-homotheticity across broad imported sectors is evident from the data. As one example, for each household, we create the ratio of expenditure on CE-HS4 concorded imported food as a share of total P P CE-HS4concordedimports: Y = Y . Figure3showsthedifferencesacrossincomedecilesfor s2Food hst s hst spending on imported food in 2014: there are indeed large differences across income groups. 36Gs;t is constructedusing a concordance between NAICS codes and HScodes. In caseswhere one NAICScode mapsinto multipleHS4categories,weusetheshareofU.S.exportsineachsectortoallocateproductionacrossHS4s. 37Whenwecomputethepercentchangebetween1998and2014foreachsector’sshareofeachdecile’simportedconsumption (P Y s h Y s h t st ),wefindthatacrossalldecile-sectorsthe25thpercentileisa27.6%decreaseinshare,themedianisa7.1%increase inshare,andthe75thpercentileisa85.8%increaseinshare. 38Thecalculationsinsection4.3.2relyonaversionofYhst constructedusingpublictradedataforMs;t andXs;t. 23

Figure 3: Imported Food as a Share of Imported Consumption by Income Decile in 2014 32.0 30.0 28.0 26.0 24.0 22.0 20.0 18.0 1 2 4 5 6 8 9 Imported Food's Share of Income Decile's Imported Consumption Another way to show this fact is to compare expenditure in HS4 categories as a share of total HS4 expenditures (i.e., Yhst (cid:17) Yhst) across income deciles. Table 2 presents summary statistics across HS4 P sYhst Yht categories, weighted by expenditure in an HS4. Again, we find meaningful variation across deciles. Panel (a) of Table 2 presents summary statistics for the ratio of Decile 9 expenditure shares over Decile 1 expenditure shares in 2014 (Y9st=Y9t). The 25th and 75th percentiles demonstrate a wide range of differences in Y1st=Y1t expenditure shares across these deciles. Panels (b)-(d) show similar findings for other decile comparisons in 2014. Table 2: Summary Statistics for Decile-to-Decile Expenditure Share Ratios in 2014 (a) Decile 9 to Decile 1 (Y9st=Y9t) Y1st=Y1t 10th Percentile 25th Percentile Median 75th Percentile 90th Percentile 0.6101 0.7313 1.0068 1.1271 1.7832 (b) Decile 9 to Decile 5 (Y9st=Y9t) Y5st=Y5t 10th Percentile 25th Percentile Median 75th Percentile 90th Percentile 0.7370 0.8644 0.8918 1.1753 1.4986 (c) Decile 2 to Decile 5 (Y2st=Y2t) Y5st=Y5t 10th Percentile 25th Percentile Median 75th Percentile 90th Percentile 0.2735 0.9220 1.1292 1.3089 1.4230 (d) Decile 2 to Decile 1 (Y2st=Y2t) Y1st=Y1t 10th Percentile 25th Percentile Median 75th Percentile 90th Percentile 0.6228 0.9254 1.0806 1.1438 1.3301 With Y constructed, we work with Equation 38 to obtain (cid:27) and ' , giving us everything we need to hst hst 24

make the household-specific import price indexes. 5 Estimation Results Weusethesupplier-leveldatatoestimatethesector-levelparametersofthemodelandusethemtoconstruct import price indexes in the aggregate and for different income groups. 5.1 Parameter Estimates We start with estimates of (cid:27)s, which is the sectoral-level elasticity of substitution that is comparable to estimates from Broda and Weinstein (2006). In all sectors our estimate of (cid:27)s is statistically different from zero at the 5 percent level or better. We also find that (cid:27)s > 1 in all sectors. Across sectors, our estimates oftheelasticityofsubstitutionhaveamedianof4.9,squarelyinlinewithearlierfindingsforU.S.imports39. Table 3: Summary of (cid:27)s 10% Median 90% 3.06 4.93 8.59 Table 4 reports our estimate of (cid:27), whichis the aggregate-levelelasticity ofsubstitution (across consumer goods). The first column shows the OLS result from our estimating equation, while the second column reports the Instrumental Variable (IV) estimate. As would be expected from the presence of an endogeneity bias in this setting, the OLS estimate is biased toward zero. The IV estimate of (cid:27) is about 2.8, with a 95 percent confidence interval between 2.6 and about 3. Note that most papers in the literature assume that (cid:27) D1, making upper-tier utility Cobb-Douglas. Redding and Weinstein (2017) also estimates the elasticity of substitution across U.S. HS4 import sectors from 1997-2011, and reports an estimate of 1.36. Table 4: Estimates of (cid:27) OLS estimate IV estimate IV 95% C.I. 0.82 2.78 (2.60 - 2.97) Another parameter that corresponds with earlier work is the elasticity of marginal cost with respect to output, ! . In all but a handful of sectors our estimate of ! is statistically different from zero at the 5 s s percent level or better. Again, our estimated parameters are in line with previous work40. 39For comparison, starting with the Broda and Weinstein (2006) estimates of (cid:27)s at the HS10 level for U.S. imports from 1990-2001, and collapsing to the HS4 level by taking the mean across HS10 estimates, then the 10th percentile value of (cid:27)s is 1.91,themedianis4.46,andthe90thpercentileis22.5. 40Forcomparison,Soderbery(2015)reportshybridFeenstraestimatesof!s forU.S.importsattheHS8levelfrom1993to 2007,whichrangefrom0.03atthe25thpercentileto1.06atthe75thpercentile,withamedianof0.26. Aftercollapsingtothe HS4 level by taking the median of !s across HS8 estimates in Soderbery (2015), then the 10th percentile value of !s is 0.03, themedianis0.30,andthe90thpercentileis14.09. 25

Table 5: Summary of ! s 10% Median 90% 0.16 0.44 1.59 The next objects of interest are our ˛ parameters. These are the subsistence quantities that each v household must consume. Remember that ˛ D 1 (cid:2) ˇC min .q j q >0/ (cid:3) , and analogously for v 2GE. v n1998 s t vt vt s Thus,ˇC andˇE aresector-specificcommonslopes,andtheydeterminethesignof˛ . Asasimpleexample s s v to fix ideas, consider the case of ˇC D 1. In this case, the total subsistence quantity of variety v sold in s 1998 (˛ n ) is simply the minimum quantity observed in the import data, and therefore each household v 1998 consumes mint.qvt j qvt>0/ for subsistence. n1998 Westartbydiscussingthe˛termsthatrepresentcontinuingfirms. Thesearethebiggest,mostimportant suppliers, as ˇC is only able to be estimated in sectors where there are some suppliers present for every year s of the sample. These results are reported in Table 6. The median ˇC across sectors is positive, and 91% of s HS4 sectors have positive values, meaning that in many cases, markups are decreasing in quantity sold (i.e. trade can be anti-competitive).41 Table 6: Summary of ˇC (Continuers) s 10% Median 90% 9.96 (cid:2) E-5 0.33 0.39 Remember also that ˛ D 0 would be consistent with CES preferences. In fact, 85% of sectors have v values of ˇC that are statistically significant at the 5 percent level or better, demonstrating that for this s sample of continuers, CES is not a good way of summarizing their behavior. Ontheotherhand,ˇE reflectsthebehavioroffirmswhodonottradeineveryperiod-typicallymarginal s suppliers who trade much less and are much smaller in size. As can be seen from Table 7, for these firms CES does indeed appear to be a reasonable assumption- ˇE must be (weakly) negative by definition, but s the vast majority of sectors have estimates extremely close to zero42. Table 7: Summary of ˇE (Non-Continuers) s 10% Median 90% -5.97 (cid:2) E-5 -2.55 (cid:2) E-9 -1.08 (cid:2) E-10 Before moving on to discuss how we use these parameter values, we circle back to discuss the exclusion restriction at the heart of our Feenstra (1994) style estimation approach. Recall the identifying assumption that the idiosyncratic demand shocks (which are functions of ' ) and supply shocks (which are functions of vt ı ) at the variety level are assumed to be orthogonal after variety and sector-time fixed effects have been vt 41Althoughthereislittlepriorworkwithwhichtocomparetheseestimates,Arkolakisetal.(forthcoming)reportsestimates of ˛v parameters for U.S. imports at the HS2 digit sector level of which the median estimate is negative, the 25th percentile estimateisnegative,andthe75thpercentileestimateispositive. 42Hottmanetal.(2016),usinganoligopolymodelandU.S.scannerdata,findsimilarresultsinthesensethatlargefirmsare foundtoquantitativelydeviatefromtheCESbenchmarkwhilesmallfirmsdonot. 26

differenced out. There is some potential for these shocks to be correlated, however, whereby an increase in a variety’s quality relative to another ((cid:129)k;t' ") leads to an increase in the intercept of the cost function vt for that variety relative to the same comparison ((cid:129)k;tı "). Hottman et al. (2016) avoid this issue with vt Nielsen data using reference varieties within multi-product firms in the double-difference procedure rather than simply a reference variety within the same sector. In the spirit of their approach, we re-estimate our parameters using multi-product suppliers as a robustness check. The estimation sample drops significantly, primarily because the only identifying variation within a sector now comes only from those suppliers with multiple varieties within the same HS4 sector. That said, the parameter values are quite similar to the baseline. Thus, our results are robust to this more exacting specification. Table 8: Parameter Estimates using Multi-Variety Exporters 10% Median 90% (cid:27)s 1.73 4.17 9.60 ˇC -0.74 0.33 0.43 s ˇE -0.89 -1.91 (cid:2) E-9 -1.73 (cid:2) E-10 s As another robustness check, we can re-estimate our parameters by alternatively specifying the supply equation in our Feenstra (1994) estimation to include oligopolistic market power in the markup term (so markups depend on the elasticity in footnote 21 instead of the baseline Equation 23). The results of this alternative estimation are reported in Table 9. Our results are qualitatively unchanged from the baseline. Table 9: Parameter Estimates using Oligopolistic Exporters 10% Median 90% (cid:27)s 3.02 4.62 9.18 ˇC -0.53 0.08 0.23 s ˇE -0.03 (cid:2) E-2 -1.36 (cid:2) E-9 -1.52 (cid:2) E-10 s ! 0.14 0.40 1.48 s So far, we have seen that the aggregate-level elasticity of substitution (across consumer goods) is greater than 1, markups of continuing firms are often decreasing in quantity sold (˛ > 0), and standard CES v preferences better describe the behavior of marginal firms as opposed to infra-marginal firms. 5.2 Markups Using the estimated parameter vector, we can generate the expression "vt according to Equation 27 for "vt (cid:0)1 every supplier for every time period. We first illustrate how markups vary across sectors. To summarize this statistic, within each sector, we weight the variety-specific markup by its total trade weight within that sector over all years of data and create a sector-level summary statistic. Table 10 shows the summary of this variable across HS4 sectors. As can be seen in the table below, the median markup is about 25% over marginal cost, with the low end close 27

to 13% and the high end at 48%.43 Table 10: Markup Variation across HS4 Sectors ( "vt ) "vt (cid:0)1 10% Median 90% Sales-Weighted Average 1.132 1.250 1.482 We can also how see the typical markup changes over time. Here we simply take the sales-weighted median markup over all varieties sold in each year and track how this median moves over time. As can be seeninTable11,themedianmarkupdeclinedoverthefirsthalfofthesamplebeforeflatteningoutfrom2010 onward. Given our earlier results showing that markups are often decreasing in quantity sold, these markup declines in the early part of our sample likely reflect a large increase in imports in early years, followed by a leveling off.44 Table 11: Median Markup Over Time (Sales-Weighted) Year 1998 2002 2006 2010 2014 Markup 1.235 1.226 1.215 1.215 1.215 Markup- Continuers 1.234 1.174 1.134 1.130 1.132 For robustness, we also report the markup results implied by the oligopoly specification from Table 9. While markups are slightly higher in the oligopoly case, the results in terms of how the average markup changes over time are unchanged from the baseline. This can be seen in Table 12 below. Table 12: Median Markup Over Time (Sales-Weighted): Oligopoly case Year 1998 2002 2006 2010 2014 Markup 1.256 1.251 1.237 1.235 1.235 Markup- Continuers 1.288 1.260 1.224 1.180 1.192 5.3 Aggregate Import Price Index With our parameter estimates in hand, we can calculate the aggregate import price index for the United States from 1998 to 2014. The exercise is similar to that of Broda and Weinstein (2006) but with the two key differences that prices (and thus varieties) are supplier specific and that the preferences used here are a more flexible, non-homothetic generalization of the CES preferences used in their study. We calculate the aggregate import price index for each year from 1999 to 2014, with 1998 as the reference year. 5.3.1 Baseline Results The results are shown in Figure 4. The value of the price index in each year is also reported in table form in theAppendix. Thedashedlinesinthefigurerepresenterrorbands,computedbyrecalculatingtheaggregate 43Forcomparison,FeenstraandWeinstein(2017)estimateamedianmarkupacrossHS4digitU.S.importsectorsin2005of 30%overmarginalcost. 44FeenstraandWeinstein(2017)alsoestimatethatmarkupshavedeclinedinU.S.importsectorsbetween1992and2005. 28

import price index using values of (cid:27) that represent the 95% thresholds for this parameter shown in Table 4. We find a U-shaped pattern: by 2006, import prices were nearly 12% lower than in 1998. However, by 2014, prices are about 8% higher than in 1998. The fall in the import price index in the early part of the period is consistent with the large increase in the number of foreign varieties observed over this time period as well as the decrease in median markups. Figure 4: U.S. Import Price Index, 1998-2014 1.1 1.05 1 0.95 0.9 0.85 0.8 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Aggregate Import Price Index Figure 5 again plots our estimated aggregate import price index from 1998 through 2014, with 1998 normalizedtoone. Forcomparison, thefigurealsoplotstheBLSAll-CommodityImportPriceIndex, which is a Laspeyres price index constructed from survey data gathered through the International Price Program. Interestingly,thedeclineinouraggregateimportpriceindexfrom1998-2006isincontrastwiththepublished import price index from the BLS, which rises over this time period. Of course, the BLS price index is not variety adjusted or widely quality adjusted and likely suffers from a substitution bias from the base period weights used in the Laspeyres formula45. 45OurimportpriceindexnestsaLaspeyresindexasalimitingspecialcasewhen˛v D0,'vt D'v,'st D's,(cid:27)s !0,and (cid:27) !0,asshowninReddingandWeinstein(2016). 29

Figure 5: U.S. Import Price Indexes, 1998-2014 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Aggregate Import Price Index BLS All‐Commodity Import Price Index Tosummarize, wefindthatouraggregateimportpriceindex, with1998normalizedto1, wasabout1.08 in2014. Forcomparison,theAll-CommodityImportPriceIndexfromtheBLS,whichwith1998normalized to 1, takes a value of about 1.48 in 2014. The ratio of the two, 1.48/1.08, implies an upward bias in the Laspeyres import price index over our time period of about 37%, or about 2 percentage points per year46. 5.3.2 Components of the Aggregate Import Price Index Recall from Equation 21 that we can break down our import price index into a few major components. In particular, the CES portion of the price index can be written as ln.Œ X ' s (cid:27) t (cid:0)1P st 1(cid:0)(cid:27)(cid:141)1(cid:0) 1 (cid:27)/D N 1 S X . N 1 v X lnp vt / s2S t s2S st v2Gst 1 X 1 X 1 X (cid:0) . ln' /(cid:0) ln' NS Nv vt NS st t s2S st v2Gst t s2S 1 1 X 1 (cid:0) lnNS (cid:0) lnNv (41) (cid:27) (cid:0)1 t NS (cid:27)S (cid:0)1 st t s2S (cid:0) 1 ln. 1 X . 3 P 's s t t/1(cid:0)(cid:27) /(cid:0) 1 X 1 ln. 1 X . 4 p 'v v t t/1(cid:0)(cid:27)S /: (cid:27) (cid:0)1 N t S s2S .P 's s t t/1(cid:0)(cid:27) N t s s2S (cid:27)S (cid:0)1 N s v t v2Gst .p 'v v t t/1(cid:0)(cid:27)S ThisequationillustrateshowwecanconsiderwhatfactorscontributedtotheU-shapedtrendweobserve in our price index47. For example, we can look at the geometric average of observed variety-level prices, 46Forcomparison,BrodaandWeinstein(2006)useaCESaggregateimportpriceindexandfindanupwardbiasinthenonvariety-adjusted import price index of 28 percent, or 1.2 percentage points per year over the 1972-2001 time period. Relative totheirexercise,ourcomparisontotheLaspeyrespriceindexincludesadditionalsourcesofbiassuchasthesubstitutionbias. 47Sinceourestimated˛v termstendtobeverysmall,theapproximationofourpriceindexbytheCES-stylecomponenton theleftsideofEquation41islikelytobeaverygoodone. 30

the first term of the price index breakdown in Equation 41. This excludes factors such as the changes in the (average) number of varieties Nv in the third line and changes in the dispersion of variety-level prices st .3pvt/1(cid:0)(cid:27)S 'vt in the fourth line. If the geometric average of prices is very different from the actual import price .pvt/1(cid:0)(cid:27)S 'vt index, this is a clue that these two factors matter for the difference. As another example, the third line of Equation 41 is key for understanding the role of new varieties: one can hold the (average) number of varieties per sector fixed at 1998 levels and trace out an alternative import price index consistent with such anexperiment,illustratinghowmuchthechangingsetofavailablevarietiesintheaveragesectorcontributed to these changes. Figure 6 compares the geometric average of observed variety-level prices (the blue line), the first term of the price index breakdown in Equation 41, with our aggregate import price index (the green line). This geometric average of prices declined much less in the first half of the period than the overall index, while rising at a faster rate relative to the overall index in later years. In general, a widening gap between the two indicatesthatchangesinthenumberofvarietiesorchangesindispersioninpricesiscausingthedifferences. Which mattered more? The orange line in Figure 6 illustrates the exercise of holding fixed the (average) number of varieties in a sector at 1998 levels. Comparing the three lines makes clear that the increasing number of varieties did indeed lead to price index declines in the early part of the period. However, these effects are mostly washed out by the end of the period, meaning that in the end, changes in the number of available varieties did not affect prices much. Instead, changes in the dispersion of prices seem to matter much more for driving prices back up in the later half of the period, as the gap between the blue and green lines grows even as the gap between the green and orange lines shrinks. Figure 6: U.S. Import Price Index, Variants 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 IPI IPI‐Fixed No. of Varieties IPI‐Prices Only Interestingly, the geometric average of prices comes very close to the published BLS import price index. 31

Figure 7 compares the geometric average price index (the blue line) with the price index from the BLS (the red line) over the 2000 to 2012 period, now setting 2000 to one for both series. Remarkably, the two price indexes behave very similarly in this subperiod, with the exception that our price index falls significantly less in 2009 than the BLS price index. Figure 7: U.S. Import Price Indexes, 2000-2012 1.5 1.4 1.3 1.2 1.1 1 0.9 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 IPI‐Prices Only BLS All‐Commodity Import Price Index 5.3.3 The Role of China Another interesting question is the extent to which the well-known increase in U.S. trade with China contributed to changes in the aggregate price index. Although our model does not permit a full general equilibrium accounting of such an exercise, we can plot how the prices of non-Chinese varieties moved over this time period.48 This comparison can be seen in Figure 8. Beginning in 2002, import price inflation for the groupofnon-Chinesevarietiesrosemorethantheoverallpriceindex,implyingadeflationaryeffectofChina for U.S. consumers. According to our baseline index, overall import inflation was 0.47% annualized between 1998 and 2014, while the non-China import price inflation was 0.73%49. 48Notaccountingforgeneralequilibriumeffectsimpliesthatpricesandsalesoftherestoftheworld’svarietieswouldevolve equallycomparedwiththecasewithChinesevarietiesincluded. 49When we look at the results of the same exercise focusing just on consumer products at the decile level, we will find a differenceinannualaverageinflationratesofabout0.5percentagepoint,whichistwiceaslargeasthisaggregateresult. 32

Figure 8: U.S. Import Price Index, With and Without Chinese Varieties 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Import Price Index Import Price Index‐ No China Varieties 5.4 Import Price Inflation Across Consumers 5.4.1 Baseline Results Figure 9 reports import price indexes for selected income deciles over the entirety of 1998 through 2014, namelythelowest,median,andsecond-highestdecilesofincomeintheUnitedStates.50 Theresultsforthese and other deciles are reported in table form in the appendix. The dashed lines in the figure represent error bands, computed by recalculating the income-decile specific price indexes using values of (cid:27) that represent the 95% thresholds for this parameter above.51 There are several features of Figure 9 to discuss. First, note that the price index for the ninth decile of income is below the other deciles in every year after 2002. Thus, the higher-income households experienced less cumulative import price inflation than other households. Second, with the exception of 2009, the price index for the lowest decile of income was above the other deciles in almost every year after 2002. Therefore, the lowest-income households experienced the most cumulative import price inflation over this time period. 50TheU.S.CensusBureaudoesnotdiscloseincomeamountsforthethird,seventh,ortenthdeciles. 51Error bands calculated using the 95% thresholds from the nonlinear estimates of the sector level parameters ˇE and ˇC s s arealsoextremelytight,andareavailableuponrequest. 33

Figure 9: U.S. Import Price Index, 1998-2014, Selected Income Deciles 1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 1st Income Decile Median Income Decile 9th Income Decile Figure9canbecomparedwithchangesinnominalincomefordifferentdecilesoverthesametimeperiod. As can be seen in Figure 10, U.S. Census data on income thresholds indicate that the highest income for a person in the ninth income decile has risen about 7.5% from 1998 to 2014. At the same time, the highest income of consumers in the first decile has dropped by about 12.5% 52. . Figure 10: Cumulative Percent Change in U.S. Decile Income Thresholds, 1998-2014 10.0 7.5 5.0 2.5 0.0 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 ‐2.5 ‐5.0 ‐7.5 ‐10.0 ‐12.5 ‐15.0 1st Income Decile Median Income Decile 9th Income Decile 52WereportthefullsetofdecilethresholdsinTable18intheAppendix. 34

Figure 11 shows the average annual change in income thresholds by decile over the 1998 to 2014 period. Note that decile five is the median decile of income, and we do not report results for deciles three, seven, or ten. Notably, there is a positive relationship between the decile of income and its average annual income threshold change over this time period. Figure 11: Average Annual Income Threshold Change by Decile, 1998-2014 0.60 0.40 0.20 0.00 1 2 4 5 6 8 9 -0.20 -0.40 -0.60 -0.80 -1.00 Income Decile's Change in Income Threshold (A.R.) Figure 12 plots the import price inflation rates experienced by different deciles over the 1998 to 2014 period. We can see that the lowest-income households experienced the most inflation, while import price inflationwassubstantiallylowerforhigher-incomedeciles. Therefore,wefindanegativerelationshipbetween the decile of income and the average annual import price inflation over this time period. Importantly, the sign of this relationship is the opposite of what we saw in terms of nominal income changes. Thus, changes in import prices appear to be exacerbating increases in nominal income inequality over this time period. 35

Figure 12: Average Annual Import Price Inflation by Decile, 1998-2014 1.50 1.40 1.30 1.20 1.10 1.00 0.90 0.80 0.70 0.60 0.50 1 2 4 5 6 8 9 Income Decile's Import Price Inflation (A.R.) 5.4.2 Within- vs. Across-Sector Non-Homotheticity Our model permits both within-sector (through ˛ ) and across-sector non-homotheticity (through ' ). In v hst this section, we consider the relative importance of each of these channels to the differences in decile-level import price inflation described above. We first set all ˛ terms equal to zero and recalculate the decile-level price indexes. As can be seen from v Figure 13, shutting down within-sector non-homotheticity changes the lines only slightly and does not alter the qualitative picture that consumers in the lowest income decile experienced the highest level of import price inflation. 36

Figure 13: U.S. Import Price Index for ˛ D0, Selected Deciles v 1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 1st Income Decile Median Income Decile 9th Income Decile However, shutting down cross-sector non-homotheticity by setting all household-level sectoral demand shifters ' equal to the sectoral average ' leads to a very different picture. Figure 14 shows that by hst st shutting down variation in the sectoral demand shifters across households, import price inflation differences across deciles collapses to the point of being indistinguishable.53 Figure 14: U.S. Import Price Index for ' D' , Selected Deciles hst st 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 1st Income Decile Median Income Decile 9th Income Decile 53Individualyearlyobservationsoftheindexarenotexactlyequal;theytendtodifferbyabout0.0001to0.0002percentage point. 37

5.4.3 The Role of Different Products Although we have shown that the total share of income spent on imported products does not differ much acrossincomedeciles,itisstillthecasethattheshareofincomespentonparticularimportedproductsdiffers widely across deciles. One particularly useful decomposition is separating out food and energy products from the overall price index to generate a “core” import price index. As can be seen in Figure 15, this index preserves many of the same features of the baseline index: a U-shaped pattern for prices, important differences between deciles, and the ninth decile of income facing the lowest level of import price inflation. Figure 15: U.S. Import Price Index for “Core" Products, Selected Deciles 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 1998 2000 2002 2004 2006 2008 2010 2012 2014 1st Income Decile Median Income Decile 9th Income Decile However, the import price index for food and energy (i.e. “Non-Core”) products in Figure 16 looks very different. Althoughtherichestdecilestillhasthelowestlevelofinflation,pricesfortheseimportedproducts rose steadily over the time period. Additionally, there are less stark differences between the median and lowest-income deciles for non-core products. 38

Figure 16: U.S. Import Price Index for “Non-Core" Products, Various Deciles 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 1998 2000 2002 2004 2006 2008 2010 2012 2014 1st Income Decile Median Income Decile 9th Income Decile 5.4.4 Adjusting the Aggregate Elasticity of Substitution Recall from Section 4 that, using a properly instrumented version of the sectoral price index to trace out the demand curve, we estimate a value of the aggregate elasticity of substitution of 2.8. As a robustness check, we allow for different deciles to have different elasticities of substitution by estimating Equation 38 decile by decile. The implied (cid:27) coefficients are listed in Table 13. We find higher substitution elasticities h for lower-income deciles. Table 13: Decile Specific (cid:27) h Decile 1 2 4 5 6 8 9 Avg. (cid:27) 3.26 3.11 3.02 3.18 2.49 2.66 1.93 2.81 h When we apply these decile-specific elasticities to the price indexes (see Figure 17), we find some minor differences compared with our baseline results. The contour remains the same, but the differences between the lines are less pronounced. 39

Figure 17: U.S. Import Price Index, Decile-Specific (cid:27) h 1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 1998 2000 2002 2004 2006 2008 2010 2012 2014 1st Income Decile Median Income Decile 9th Income Decile Ourestimateof(cid:27) D2:8ishigherthanReddingandWeinstein(2017)’sestimateof1.36,partlybecauseour estimationsampleforthesecondstageincludesonlythosesectorsthataredirectlyconsumedbyhouseholds, as is clear from Equation 38. A typical parametrization of the aggregate elasticity of substitution in the literature would be (cid:27) D 1, which corresponds to Cobb-Douglas preferences. If we were to adopt a value of (cid:27) that is close to the estimate from Redding and Weinstein (2017) and much closer to Cobb-Douglas preferences, then our price index is changed significantly in magnitude relative to the baseline, as seen in Figure 18. Differences between income groups are far larger in this case. Figure 18: U.S. Import Price Index, (cid:27) D1:3 2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 0.5 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 1st Income Decile Median Income Decile 9th Income Decile 40

Essentially, higher levels of (cid:27) imply easier substitution across products for consumers and thus smaller levels of import price inflation. Combining this finding with our result that lower decile-specific (cid:27) values h tend to be found for the highest-decile consumers means that differences in import price inflation could be partly explained by assuming the elasticity of substitution to be equal across consumers. 5.4.5 Import Price Inflation by Decile Table 14 provides the annual average import price inflation rates experienced by each decile for the various exercises previously discussed. The results show a clear pattern, with the only exceptions being the case of non-coreproductsandwhenweshutdownacross-sectornon-homotheticity(' D' ), thathigher-income hst st households have experienced the lowest import price inflation and lower-income households the highest import price inflation over this time period. Table 14: Annual Average Import Price Inflation by Decile, 1998-2014 Decile 1 2 4 5 6 8 9 Baseline 1.33 1.17 1.16 1.24 0.70 1.00 0.90 ˛ D0 1.39 1.22 1.22 1.30 0.74 1.06 0.98 v ' D' 1.02 1.02 1.02 1.01 1.01 1.01 1.01 hst st Without China 1.85 1.68 1.67 1.76 1.21 1.51 1.42 Core Products 0.27 0.11 0.05 0.15 -0.49 -0.08 -0.11 Non-core Products 3.73 3.45 3.62 3.77 3.72 3.55 3.37 (cid:27) D1:3 3.83 2.80 2.75 3.27 0.01 1.79 1.23 Decile-specific (cid:27) 1.23 1.12 1.12 1.17 0.67 1.01 0.96 h 6 Conclusion We develop a new framework based on non-homothetic preferences and use detailed trade transaction data for the United States to estimate the import price index, both in the aggregate and for different income deciles. Ourframeworkallowschangesinaverageprices(consistingofmarginalcostmovementsandmarkup adjustment), changes in the dispersion of prices (i.e., changing opportunities for substitution), product quality changes, and an expansion (or contraction) in the set of available varieties to affect the import price index. Themodelpermitsbothcross-sectorandwithin-sectornon-homotheticity, thefirstofwhichcaptures differences in sectoral expenditure shares across consumers and the second which captures differences in product quality. Constructing the import price index requires estimating key parameters of the model. Using a richer framework and more detailed data than in prior work, we structurally estimate sectoral elasticities of substitution that are in line with the literature. However, we estimate that the overall aggregate elasticity of substitution (across consumer goods) is about 2.8, higher than the value of 1 typically assumed in the literature. This parameter is important quantitatively, and we show that our baseline results are conservative estimates given that we recover a higher aggregate elasticity of substitution than prior work. 41

We use our parameter estimates and the variety-level universe of goods trade data to construct the aggregate U.S. import price index. Relative to 1998, we find that aggregate import prices rose about 8% by 2014. For comparison, the Laspeyres-based BLS import price index, which does not capture substitution effects or changes in the set of imported varieties, rose by 48% over this time period. Therefore, we estimate anupwardbiasintheLaspeyresimportpriceindexoverourtimeperiodofabout37%,orabout2percentage points per year. WealsofindthatdevelopmentsinU.S.tradeoverthepasttwodecadeshavehadimportantdistributional consequences through the consumption channel. In particular, we find that lower-income households experiencedthemostimportpriceinflationfrom1998to2014,whilethehigher-incomehouseholdsexperiencedthe least import price inflation. In our baseline results, the 1st income decile experienced import price inflation of about 24% from 1998 to 2014, or about 1.33 percent per year. For comparison, the 9th income decile only experienced import price inflation of about 15% over that time period, or about 0.90 percent per year. Therefore, we do not find evidence that the consumption channel has mitigated the distributional effects of trade that have occurred through the nominal income channel in the United States over the past two decades. Instead, our results imply that import price changes have exacerbated the increase in nominal income inequality. Our results also indicate that cross-sector non-homotheticity is the key mechanism driving the differences in import inflation across import groups. References Ackerberg, Daniel and Marc Rysman, “UnobservedProductDifferentiationinDiscreteChoiceModels: Estimating Price Elasticities and Welfare Effects,” RAND Journal of Economics, 2005, 36 (4), 771–788. Aghion, Philippe, Antonin Bergeaud, Timo Boppart, Peter J. Klenow, and Huiyu Li, “Missing Growth from Creative Destruction,” NBER Working Paper 24023, 2017. Aguiar, Mark and Mark Bils, “Has Consumption Inequality Mirrored Income Inequality?,” American Economic Review, 2015, 105 (9), 2725–2756. Amiti, Mary, Mi Dai, Robert Feenstra, and John Romalis, “How Did China’s WTO Entry Benefit U.S. Consumers?,” NBER Working Paper 23487, 2017. Antràs, Pol, Alonso de Gortari, and Oleg Itskhoki, “Globalization, Inequality, and Welfare,” Journal of International Economics, 2017, 108, 387–412. Arkolakis, Costas, Arnaud Costinot, and Andres Rodríguez-Clare, “New Trade Models, Same Old Gains?,” American Economic Review, 2012, 102 (1), 94–130. , , David Donaldson, and Andres Rodríguez-Clare, “The Elusive Pro-Competitive Effects of Trade,” Review of Economic Studies, forthcoming. 42

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Appendix 6.1 Supplementary Tables Table 15: Aggregate Import Price Index, 1998-2014 Year Price Index 1998 1.0000 1999 0.9496 2000 0.8835 2001 0.8951 2002 0.8498 2003 0.8577 2004 0.8697 2005 0.8722 2006 0.8843 2007 0.9373 2008 1.0126 2009 1.0168 2010 0.9961 2011 1.0705 2012 1.0782 2013 1.0646 2014 1.0777 Table 16: Implied Expenditure by U.S. Decile of Income Year 1st Decile 2nd Decile 4th Decile 5th Decile 6th Decile 8th Decile 9th Decile 1998 22,691.64 26,500.19 39,771.70 47,594.73 59,164.12 77,489.22 80,510.08 1999 23,484.90 28,776.54 41,779.38 49,057.11 60,734.75 80,188.73 83,908.75 2000 25,042.57 29,985.88 43,442.33 48,546.20 60,319.89 80,070.49 84,368.44 2001 23,826.67 28,081.95 41,022.47 48,034.21 60,286.87 79,535.57 83,257.29 2002 23,527.28 27,756.49 41,300.90 47,736.58 59,840.07 78,754.93 82,282.56 2003 21,845.63 27,262.87 39,284.88 47,137.25 59,253.98 79,648.11 83,368.51 2004 21,265.66 25,953.35 37,191.67 44,776.65 55,781.26 78,565.46 83,005.80 2005 21,051.07 26,469.51 39,363.39 46,600.36 58,002.03 79,215.77 83,784.38 2006 23,141.08 27,433.93 39,338.86 47,844.29 59,555.79 81,156.70 85,570.55 2007 22,580.07 27,648.00 41,167.83 48,591.39 59,974.48 81,331.92 85,086.75 2008 22,304.27 27,779.90 38,852.76 47,057.79 58,678.04 78,509.91 83,324.13 2009 23,087.09 26,666.13 37,624.56 45,531.39 56,529.72 78,620.31 83,237.56 2010 20,341.74 25,329.12 37,464.52 43,315.35 54,060.81 77,366.00 82,566.92 2011 19,924.30 25,996.75 36,564.68 44,453.20 55,447.76 76,165.12 82,830.47 2012 20,094.04 26,401.61 38,077.93 44,352.56 56,145.12 76,452.87 82,485.29 2013 20,766.97 27,872.19 39,062.89 46,662.36 58,518.59 79,799.83 86,552.40 2014 21,796.67 27,533.58 40,352.67 46,514.05 59,131.45 79,960.80 86,283.93 49

Table 17: Baseline Import Price Index by U.S. Decile of Income, 1998-2014 Year 1st Decile 2nd Decile 4th Decile 5th Decile 6th Decile 8th Decile 9th Decile 1998 1 1 1 1 1 1 1 1999 0.9991 0.9691 0.9902 1.0031 1.0080 0.9992 0.9962 2000 0.9161 0.8999 0.9298 0.9703 0.9759 0.9651 0.9589 2001 0.9563 0.9461 0.9661 0.9789 0.9775 0.9653 0.9620 2002 0.9161 0.9077 0.9123 0.9311 0.9288 0.9195 0.9164 2003 0.9383 0.8996 0.9210 0.9222 0.9183 0.8991 0.8950 2004 0.9478 0.9191 0.9411 0.9452 0.9473 0.9012 0.8972 2005 0.9430 0.9010 0.9006 0.9099 0.9142 0.8836 0.8787 2006 0.9532 0.9431 0.9605 0.9552 0.9588 0.9274 0.9237 2007 1.0184 0.9886 0.9877 0.9969 1.0041 0.9742 0.9760 2008 1.1110 1.0763 1.1090 1.1025 1.1060 1.0810 1.0762 2009 1.0693 1.0840 1.1128 1.1060 1.1131 1.0661 1.0628 2010 1.1367 1.1023 1.1036 1.1276 1.1164 1.0647 1.0570 2011 1.2477 1.1748 1.2152 1.2092 1.2035 1.1662 1.1470 2012 1.2665 1.1916 1.2128 1.2332 1.2027 1.1833 1.1673 2013 1.2622 1.1803 1.2145 1.2136 1.1770 1.1692 1.1500 2014 1.2363 1.2037 1.2019 1.2183 1.1177 1.1719 1.1547 Table 18: Income Thresholds by U.S. Decile of Income Year 1st Decile 2nd Decile 4th Decile 5th Decile 6th Decile 8th Decile 9th Decile 1998 14,281 23,727 44,768 57,248 71,163 110,418 149,137 1999 14,914 24,702 46,014 58,665 72,630 114,216 155,366 2000 14,754 24,985 46,009 58,544 72,742 114,000 156,153 2001 14,486 24,361 45,162 57,246 71,849 113,195 154,038 2002 14,173 23,911 44,545 56,599 70,950 112,127 152,293 2003 13,749 23,468 44,369 56,528 71,059 113,358 154,246 2004 13,857 23,489 44,059 56,332 70,177 111,818 153,576 2005 13,873 23,570 44,244 56,935 70,864 112,705 154,965 2006 14,285 23,850 44,967 57,379 71,425 115,508 158,325 2007 14,079 23,489 45,262 58,149 71,770 115,758 157,431 2008 13,557 23,089 43,476 56,076 69,924 111,744 154,172 2009 13,558 22,880 43,124 55,683 69,134 111,865 153,963 2010 13,057 22,017 41,832 54,245 67,702 110,116 152,772 2011 12,802 21,617 41,096 53,401 66,609 108,375 153,214 2012 12,791 21,533 41,568 53,331 67,511 108,818 152,623 2013 12,570 21,638 42,282 55,214 69,242 113,582 160,150 2014 12,445 21,728 41,754 54,398 69,153 113,811 159,652 50

6.2 Parameter Restriction for Interior Solution to UMP As in Barnett (1977), we require for regular interior solutions to the utility maximization problem that the restrictions in equation 3 hold. The final restriction given in that list is k <˛ <q ; (42) hv v hvt where k D(cid:0). p vt (cid:0)(cid:27)s' v (cid:27) t s(cid:0)1 /.Y (cid:0) X ˛ p /: (43) hv P .pkt/1(cid:0)(cid:27)s hst k kt k2Gs 'kt k2Gs Further, the regularity region is defined by the set of prices and income that satisfy X Y > ˛ p : (44) hst v vt v2Gs 6.3 Equilibrium Existence of partial equilibrium under monopolistic competition requires that incomes, equilibrium prices, and equilibrium quantities are such that consumer demands have a regular interior solution (the restrictions in section 6.2 hold for each household), each firm’s first order condition is satisfied (when " > 1, see vt equation 23), and each firm’s second-order condition is satisfied (when (cid:16) < 2, see equation 24). While we vt will ensure that these conditions hold for all of the observations in our data, it is clear that these conditions will not in general hold globally. Forfurtherdetails,seetherelatedtheoreticalliteratureonmonopolisticcompetitionwithgeneraladditive preferences, a class of models which includes our baseline model. Zhelobodko et al. (2012) characterize the free-entry equilibrium under monopolistic competition with additive consumer preferences. Mrázová and Neary (2017) characterize comparative statics of a general equilibrium model of international trade using generalized CES preferences (they call them Pollak preferences). Dhingra and Morrow (forthcoming) characterize the allocational efficiency of general equilibrium with monopolistic competition and additive preferences. 51