Pablo D. Fajgelbaum and Amit K. Khandelwal · 2020. 9. 20. · Pablo D. Fajgelbaum and Amit K. Khandelwal Individuals that consume different baskets of goods are differentially af-fected

MEASURING THE UNEQUAL GAINS FROM TRADE*

Pablo D. Fajgelbaum and Amit K. Khandelwal

Individuals that consume different baskets of goods are differentially af-fected by relative price changes caused by international trade. We develop amethodology to measure the unequal gains from trade across consumers withincountries. The approach requires data on aggregate expenditures and parame-ters estimated from a nonhomothetic gravity equation. We find that tradetypically favors the poor, who concentrate spending in more traded sectors.JEL Codes: D63, F10, F60.

I. Introduction

Understanding the distributional impact of internationaltrade is one of the central tasks pursued by international econo-mists. A vast body of research has examined this questionthrough the effect of trade on the distribution of earningsacross workers (e.g., Stolper and Samuelson 1941). A secondchannel operates through the cost of living. It is well knownthat the consumption baskets of high- and low-income consumerslook very different (e.g., Deaton and Muellbauer 1980b).International trade therefore has a distributional impact when-ever it affects the relative price of goods that are consumed atdifferent intensities by rich and poor consumers. For example, atrade-induced increase in the price of food has a stronger negativeeffect on low-income consumers, who typically have larger foodexpenditure shares than do richer consumers. How important arethe distributional effects of international trade through this ex-penditure channel? How do they vary across countries? Do theytypically favor high- or low-income consumers?

In this article we develop a methodology to answer thesequestions. The approach is based on aggregate statistics andmodel parameters that can be estimated from readily availablebilateral trade and production data. It can therefore be imple-mented across many countries and over time. A recent literature

*We thank the editor and three anonymous referees for comments and sug-gestions. We also thank Andrew Atkeson, Joaquin Blaum, Ariel Burstein, ArnaudCostinot, Robert Feenstra, Juan Carlos Hallak, Esteban Rossi-Hansberg, NinaPavcnik, Maria Jose Prados, Jonathan Vogel, and various seminar participants.We acknowledge funding from the Jerome A. Chazen Institute of InternationalBusiness at Columbia Business School and the National Science Foundation (NSFGrant 1529095).

! The Author(s) 2016. Published by Oxford University Press, on behalf of the Presidentand Fellows of Harvard College. All rights reserved. For Permissions, please email:[email protected] Quarterly Journal of Economics (2016), 1113–1180. doi:10.1093/qje/qjw013.Advance Access publication on March 14, 2016.

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in international trade, including Arkolakis, Costinot, andRodrı́guez-Clare (2012), Melitz and Redding (2014), andFeenstra and Weinstein (2010), measures the aggregate welfaregains from trade by first estimating model parameters from agravity equation (typically, the elasticity of imports with respectto trade costs) and then combining these parameters with aggre-gate statistics to calculate the impact of trade on aggregate realincome. We estimate model parameters from a nonhomotheticgravity equation (the elasticity of imports with respect to bothtrade costs and income) to calculate the impact of trade on thereal income of consumers with different expenditures within theeconomy.

The premise of our analysis is that consumers at differentincome levels within an economy may have different expenditureshares in goods from different origins or in different sectors.Studying the distributional implications of trade in this contextrequires a nonhomothetic demand structure with good-specificEngel curves, where the elasticity of the expenditure share withrespect to individuals’ total expenditures is allowed to vary acrossgoods. The Almost-Ideal Demand System (AIDS) is a naturalchoice. As first pointed out by Deaton and Muellbauer (1980a),it is a first-order approximation to any demand system; importantfor our purposes, it is flexible enough to satisfy the key require-ment of good-specific income elasticities and has convenient ag-gregation properties that allow us to accommodate within-country inequality.

We start with a demand-side result: in the AIDS, the welfarechange through the expenditure channel experienced by con-sumers at each expenditure level as a result of changes inprices, can be recovered from demand parameters and aggregatestatistics. These aggregate statistics include the initial levels andchanges in aggregate expenditure shares across commodities,and moments from the distribution of expenditure levels acrossconsumers. The intuition for this result is that conditioning onmoments of the expenditure distribution, changes in aggregateexpenditure shares across goods can be mapped to changes in therelative prices of high- versus low-income elastic goods by invert-ing the aggregate demand. These relative price changes anddemand parameters, in turn, suffice to measure the variation inreal income of consumers at each expenditure level throughchanges in the cost of living.

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To study the distributional effects of trade through the ex-penditure channel, we embed this demand structure into a stan-dard model of international trade, the multisector Armingtonmodel. This simple supply side allows us to cleanly highlightthe methodological innovation on the demand side.1 The modelallows for cross-country differences in sectoral productivity andbilateral trade costs, and within each sector goods are differenti-ated by country of origin. We extend this supply-side structurewith two features. First, the endowment of the single factor ofproduction varies across consumers, generating within-countryinequality. Second, consumer preferences are given by theAIDS, allowing goods from each sector and country of origin toenter with different income elasticity into the demand of individ-ual consumers. As a result, aggregate trade patterns are drivenby both standard Ricardian forces (differences in productivitiesand trade costs across countries and sectors) and demand forces(cross-country differences in income distribution and differencesin the income elasticity of exports by sector and country).

In the model, differences between the income elasticities ofexported and imported goods shape the gains from trade-cost re-ductions of poor relative to rich consumers within each country.We show how to use demand-side parameters and changes inaggregate expenditure shares to measure welfare changes expe-rienced by consumers at different income levels in response toforeign shocks. For example, a tilt in the aggregate importbasket toward goods consumed mostly by the rich may reveal afall in the import prices of these goods, and a relative welfareimprovement for high-income consumers. In countries where ex-ports are high-income elastic relative to imports, the gains fromtrade are relatively biased to poorer consumers because openingto trade decreases the relative price of low-income elastic goods.Nonhomotheticity across sectors also shape the unequal gainsfrom trade across consumers because sectors vary in their trade-ability (e.g., food versus services) and in the substitutabilityacross goods supplied by different exporters.

1. For example, the model abstracts away from forces that would lead todistributional effects through changes in the earnings distribution, as well as dif-ferentiated exporters within sectors, firm heterogeneity, competitive effects, orinput-output linkages. Future work could consider embedding the AIDS intomodels with a richer supply side.

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To quantify the unequal gains from trade, we need estimatesof the elasticity of individual expenditure shares by sector andcountry of origin with respect to prices and income. A salientfeature of the model is that it delivers a sectoral nonhomotheticgravity equation to estimate these key parameters from readilyavailable data on production and trade flows. The estimationidentifies which countries produce high- or low-income elasticgoods by projecting budget shares within each sector on standardgravity forces (e.g., distance, border, and common language) anda summary statistic of the importer’s income distribution whoseelasticity can vary across exporters.2 Consistent with the existingempirical literature, such as Hallak and Schott (2011) andFeenstra and Romalis (2014), we find that richer countriesexport goods with higher income elasticities within sectors. Theestimation also identifies the sectors whose goods are relativelymore valued by rich consumers by projecting sectoral expenditureshares on a summary statistic of the importer’s income distribu-tion. Consistent with Hallak (2010), our results also suggestnonhomotheticities across origin countries and across sectors.

Using the estimated parameters, we apply the results fromthe theory to ask: who are the winners and losers of trade withincountries, how large are the distributional effects, and what coun-try characteristics are important to shape these effects? Toanswer these questions, we perform the counterfactual exerciseof increasing trade costs so that each country is brought from itscurrent trade shares to autarky, and compute the gains fromtrade corresponding to each percentile of the income distributionin each country (i.e., the real income that would be lost by eachpercentile because of a shut down of trade).

We find a propoor bias of trade in every country. On average,the real income loss from closing off trade is 63% at the 10thpercentile of the income distribution and 28% for the 90th per-centile. This bias in the gains from trade toward poor consumershinges on the fact that these consumers spend relatively more onsectors that are more traded, whereas high-income individualsconsume relatively more services, which are among the leasttraded sectors. Additionally, low-income consumers happen toconcentrate spending on sectors with a lower elasticity of

2. When nonhomotheticities are shut down, the gravity equation in our modelcorresponds to the translog gravity equation estimated by Novy (2012) andFeenstra and Weinstein (2010).


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substitution across source countries. Larger expenditures inmore tradeable sectors and a lower rate of substitution betweenimports and domestic goods lead to larger gains from trade for thepoor than the rich. Although this propoor bias of trade is presentin every country, there is heterogeneity in the difference betweenthe gains from trade of poor and rich consumers across countries.In countries with a lower income elasticity of exports, the gainsfrom trade tend to be less favorable for poor consumers becauseopening to trade causes an increase in the relative price of low-income elastic goods. Similar results appear in counterfactualsinvolving smaller changes in trade costs than a movement to au-tarky; for example, a small reduction in the cost of importing inthe food or manufacturing sectors also exhibits a propoor bias.However, trade-cost reductions affecting only the service sectors(which are relatively high-income elastic) benefits the rich rela-tively more.

As mentioned, our approach to measure welfare gains fromtrade using aggregate statistics is close to a recent literature thatstudies the aggregate welfare gains from trade summarized byCostinot and Rodrı́guez-Clare (2014). This literature confrontsthe challenge that price changes induced by trade costs are notcommonly available by inferring them through the model struc-ture from changes in trade shares.3 These approaches aredesigned to measure only aggregate gains rather than distribu-tional consequences.4 In our setting, we exploit properties of anonhomothetic demand system that also allows us to inferchanges in prices from trade shares and to trace out the welfareconsequences of these price changes across different consumerswithin countries. We are motivated by the belief that an approachthat is able to quantify the (potentially) unequal gains from tradethrough the expenditure channel for many countries is useful inassessing the implications of trade, particularly because much ofthe public opposition toward increased openness stems from thebelief that welfare changes are unevenly distributed.

3. For example, autarky prices are rarely observed in data but under standardassumptions on preferences the autarky expenditure shares are generally known.The difference between autarky and observed trade shares can then be used to backout the price changes caused by a counterfactual movement to autarky.

4. Two exceptions are Burstein and Vogel (2012) and Galle, Rodrı́guez-Clare,and Yi (2014), which use aggregate trade data to estimate the effects of trade on thedistribution of earnings.


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Of course, we are not the first to allow for differences inincome elasticities across goods in an international trade frame-work. Theoretical contributions to this literature including Mar-kusen (1986), Flam and Helpman (1987), and Matsuyama (2000)develop models where richer countries specialize in high-incomeelastic goods through supply-side forces, while Fajgelbaum,Grossman, and Helpman (2011) study cross-country patterns ofspecialization that result from home market effects in verticallydifferentiated products. Recent papers by Hallak (2006), Fieler(2011), Caron, Fally, and Markusen (2014), and Feenstra andRomalis (2014) find that richer countries export goods withhigher income elasticity.5 This role of nonhomothetic demandand cross-country differences in the income elasticity of exportsin explaining trade data is an important motivation for our focuson explaining the unequal gains from trade through the expendi-ture channel.

These theoretical and empirical studies use a variety ofdemand structures. To our knowledge, only a few studies haveused the AIDS in the international trade literature: Feenstra andReinsdorf (2000) show how prices and aggregate expendituresrelate to the Divisia index in the AIDS and suggest that thisdemand system could be useful for welfare evaluation in a tradecontext, Feenstra (2010) works with a symmetric AIDS expendi-ture function to study the entry of new goods, and Chaudhuri,Goldberg, and Gia (2006) use the AIDS to determine the welfareconsequences in India of enforcing the Agreement on Trade-Related Intellectual Property Rights.6 Neary (2004) and Feen-stra, Ma, and Rao (2009) use the AIDS for making aggregatereal income comparisons across countries and over time usingdata from the International Comparison Project. Aguiar andBils (2015) estimated an AIDS in the United States to measure

5. See also Schott (2004), Khandelwal (2010), and Hallak and Schott (2011)who provide evidence that richer countries export higher-quality goods, which typ-ically have high-income elasticity of demand. In this article we abstract from qual-ity differentiation within sectors, but note that our methodology could beimplemented using disaggregated trade data where differences in the income elas-ticity of demand may be driven by differences in quality.

6. If good-specific income elasticities are neutralized, the AIDS collapses to thehomothetic translog demand system studied in an international trade context byKee, Nicita, and Olarreaga (2008), Feenstra and Weinstein (2010), Arkolakis,Costinot, and Rodrı́guez-Clare (2010), and Novy (2012).


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inequality in total consumption expenditures from consumptionpatterns.

A few publications study the effect of trade on inequalitythrough the expenditure channel. Porto (2006) studies the effectof price changes implied by a tariff reform on the distribution ofwelfare using consumer survey data from Argentina, Faber(2013) exploits Mexico’s entry into NAFTA to study the effect ofinput tariff reductions on the price changes of final goods of dif-ferent quality, and Atkin, Faber, and Gonzalez-Navarro (2015)study the effect of foreign retailers on consumer prices in Mexico.While these papers use detailed microdata for specific countriesin the context of major reforms, our approach provides a frame-work to quantify the unequal gains from trade across consumersover a large set of countries using aggregate trade and productiondata. Within our framework we are able to show theoreticallyhow changes in trade costs map to the welfare changes of indi-viduals in each point of the expenditure distribution, how to com-pute these effects using model parameters and aggregatestatistics, and how to estimate the parameters from cross-countrytrade and production data.

There is of course a large literature that examines trade andinequality through the earnings channel. A dominant theme inthis literature, as summarized by Goldberg and Pavcnik (2007),has been the poor performance of Stolper-Samuelson effects,which predict that trade increases the relative wages of low-skill workers in countries where these workers are relativelyabundant, in rationalizing patterns from low-income countries.7

We complement these and other studies that focus on the earn-ings channel by examining the implications of trade through theexpenditure channel.

The remainder of the article is divided into five sections.Section II uses standard consumer theory to derive genericexpressions for the distribution of welfare changes across con-sumers and applies these results to the AIDS. Section IIIembeds these results in a standard trade framework, derivesthe nonhomothetic gravity equation, and provides the

7. Several studies, such as Feenstra and Hanson (1996), Helpman et al. (2012),Brambilla, Lederman, and Porto (2012), Frias, Kaplan, and Verhoogen (2012), andBurstein, Cravino, and Vogel (2013) study different channels through which tradeaffects the distribution of earnings such as outsourcing, labor market frictions,quality upgrading, or capital-skill complementarity.


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expressions to determine the gains from trade across consumers.Section IV estimates the key elasticities from the gravity equa-tion. Section V presents the results of counterfactuals that simu-late foreign-trade cost shocks. Section VI concludes.

II. Consumers

We start by deriving generic expressions for the distributionof welfare changes in response to price changes across consumersthat vary in their total expenditures. We only use properties ofdemand implied by standard demand theory. In Section III, welink these results to a standard model of trade in generalequilibrium.

II.A. Definition of the Expenditure Channel

We study an economy with J goods for final consumption

with price vector p ¼ pj� �J

j¼1 taken as given by h ¼ 1; . . .;H con-sumers. Consumer h has indirect utility vh and total expendituresxh. We denote the indirect utility function by v xh;pð Þ. We letsj;h � sj xh;pð Þ be the share of good j in the total expenditures of

individual h, and Sj ¼P

hxhPh0 xh0

� �sj;h be the share of good j in

aggregate expenditures.Consider the change in the log of indirect utility of consumer

h due to infinitesimal changes in log prices p̂j� �J

j¼1 and in the log of

the expenditure level x̂h:8

v̂h ¼XJj¼1

@ ln v xh;pð Þ@ ln pj

p̂j þ@ ln v xh;pð Þ@ ln xh

x̂h:ð1Þ

The equivalent variation of consumer h associated with p̂j� �J

j¼1and x̂h is defined as the change in log expenditures, !̂h, thatleads to the indirect utility change v̂h at constant prices:

v̂h ¼@ ln v xh;pð Þ@ ln xh

!̂h:ð2Þ

8. Throughout the article, we use ẑ � d ln zð Þ to denote the infinitesimalchange in the log of variable z.


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Combining equations (1) and (2) and applying Roy’s identitygives a well-known formula for the equivalent variation:9

!̂h ¼XJj¼1�p̂j� �

sj;h þ x̂h:ð3Þ

The first term on the right-hand side of equation (3) is anexpenditure-share weighted average of price changes. It repre-sents what we refer to as the expenditure effect. It is the in-crease in the cost of living caused by a change in prices appliedto the the preshock expenditure basket. Henceforth, we refer to!̂h as the welfare change of individual h, acknowledging that bythis we mean the equivalent variation, expressed as share ofthe initial level of expenditures, associated with a change inprices or in the expenditure level of individual h.

To organize our discussion it is useful to rewrite equation (3)as follows:

!̂h ¼ Ŵ þ ̂h þ x̂h;ð4Þ

where

Ŵ �XJj¼1�p̂j� �

Sjð5Þ

is the aggregate expenditure effect, and

̂h �XJj¼1�p̂j� �

sj;h � Sj� �

ð6Þ

is the individual expenditure effect of consumer h.The term Ŵ is the welfare change through the expenditure

channel that corresponds to every consumer either in the absenceof within-country inequality or under homothetic preferences. Italso corresponds to the welfare change through the cost of expen-ditures for a hypothetical representative consumer. In turn, the

term ̂h captures that consumers may be differentially affectedby the same price changes due to differences in the compositionof their expenditure basket. It is different from zero for someconsumers only if there is variation across consumers in howthey allocate expenditure shares across goods. The focus of

9. See Theil (1975).


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this article is to study how international trade impacts the dis-

tribution f ̂hgH

h¼1.

II.B. Almost-Ideal Demand System

The Almost-Ideal Demand System (AIDS) introduced byDeaton and Muellbauer (1980a) belongs to the family of logprice-independent generalized preferences defined by Muell-bauer (1975). The latter are defined by the indirect utilityfunction

v xh;pð Þ ¼ Fxh

a pð Þ

� � 1b pð Þ

" #;ð7Þ

where a pð Þ and b pð Þ are price aggregators and F �½ � is a well-behaved increasing function. The AIDS is the special case thatsatisfies

a pð Þ ¼ exp � þXJj¼1

�j ln pj þ1

2

XJj¼1

XJk¼1

�jk ln pj ln pk

!;ð8Þ

b pð Þ ¼ expXJj¼1

�j lnpj

!;ð9Þ

where the parameters satisfy the restrictionsPJ

j¼1 �j ¼ 1,PJj¼1 �j ¼

PJj¼1 �jk ¼ 0, and �jk ¼ �kj for all j, k.

10

The first price aggregator, a pð Þ, has the form of a homothetictranslog price index. It is independent from nonhomotheticitiesand can be interpreted as the cost of a subsistence basket of goods.The second price aggregator, b pð Þ, captures the relative price ofhigh-income elastic goods. For our purposes, a key feature ofthese preferences is that the larger is the consumer’s expenditurelevel xh relative to a pð Þ, the larger is the welfare gain from areduction in the cost of high-income elastic goods, as capturedby a reduction in b pð Þ: We refer to a and b as the homotheticand nonhomothetic components of preferences, respectively.

10. These parameter restrictions correspond to the adding up, homogeneity,and symmetry constraints implied by individual rationality, and ensure that theAIDS is a well-defined demand system. No direct utility representation of the AIDSexists, but this poses no restriction for our purposes. See Deaton and Muellbauer(1980b).


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Applying Shephard’s lemma to the indirect utility functiondefined by equations (7) to (9) generates an expenditure share ingood j for individual h equal to:

sj p; xhð Þ ¼ �j þXJk¼1

�jk ln pk þ �j lnxh

a pð Þ

� �ð10Þ

for j ¼ 1; . . .;J. We assume that equation (10) predicts nonneg-ative expenditure shares for all goods and consumers, so thatthe nonnegativity restriction is not binding. Since expenditureshares add up to 1, this guarantees that expenditure shares arealso smaller than 1. We discuss how to incorporate this restric-tion in the empirical analysis in Section IV.

These expenditure shares have two features that suit our pur-poses. First, the elasticity with respect to the expenditure level isallowed to be good-specific.11 Goods for which �j > 0 have positive

income elasticity, while goods for which �j < 0 have negative

income elasticity.12 Second, they admit aggregation: market-levelbehavior can be represented by the behavior of a representativeconsumer. The aggregate market share of good j is Sj ¼ sj p; ~xð Þ,where ~x is an inequality-adjusted mean of the distribution of ex-penditures across consumers, ~x ¼ xe�, where x � E xh½ � is the meanand � � E xh�x ln

xh�x

� �� is the Theil index of the expenditure distribu-

tion.13 We can write the aggregate shares as

Sj ¼ �j þXJk¼1

�jk ln pk þ �jy;ð11Þ

11. We note that the AIDS restricts these elasticities to be constant, thus rulingout the possibility that demand peaks at intermediate levels of income. Severaldiscrete-choice models of trade with vertically differentiated products—suchFlam and Helpman (1987), Matsuyama (2000), or the multiquality extension inSection VII of Fajgelbaum, Grossman, and Helpman (2011)—feature nonmono-tonic income elasticities. Banks, Blundell, and Lewbel (1997) and Lewbel andPendakur (2009) develop extensions of the AIDS that allow for nonconstantincome elasticities.

12. Note that �’s and �’s are semi-elasticities since they relate expenditureshares to logs of prices and income, but we refer to them as elasticities to savenotation. Note also that although we define xh as the individual expenditurelevel, we follow standard terminology and refer to �j as the income elasticity ofthe expenditure share in good j.

13. The Theil index is a measure of inequality that takes the minimum �¼ 0 ifthe distribution is concentrated at a single point. In the case of a log-normal expen-diture distribution with variance �2; it is � ¼ 12 �2:


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where y ¼ ln ~xa pð Þ

�

. Henceforth, we follow Deaton and Muellbauer

(1980a) and refer to y as the adjusted ‘‘real’’ income.

II.C. The Individual Expenditure Effect with Almost-IdealDemand

From equations (10) and (11), the difference in the budgetshares of good j between a consumer with expenditure level xhand the representative consumer is

sj;h � Sj ¼ �j lnxh~x

�:ð12Þ

Consumers who are richer than the representative consumerhave larger expenditure shares than the representative con-sumer in positive �j goods and lower shares in negative �jgoods. Combining equation (12) with the individual expenditureeffect defined in equation (6), we obtain

̂h ¼ �XJj¼1

�jp̂j

!|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

¼b̂

� ln xh~x

�;ð13Þ

where b̂ is the change in the log of the nonhomothetic compo-

nent b pð Þ. Note that b̂J equals the covariance between the good-specific income elasticities and the price changes.14 A positive

(negative) value of b̂ reflects an increase in the relative prices ofhigh- (low-) income elastic goods, leading to a relative welfareloss for rich (poor) consumers.

Collecting terms, the welfare change of consumer h is

!̂h ¼ Ŵ � b̂ � lnxh~x

�þ x̂h:ð14Þ

Given a distribution of expenditure levels xh across consumers,this expression generates the distribution of welfare changes inthe economy through the expenditure channel.

A useful property of this structure is that the terms fŴ ; b̂gcan be expressed as a function of demand parameters and aggre-gate statistics. Intuitively, these terms are simply weighted

14. That is, COVðf�jg; p̂jÞ � 1JP

jð�j � 1JP

j0 �j0 Þðpj � 1JP

j0 pj0 Þ ¼PJ

j¼1 p̂j�j, wherethe last equality follows from the fact that the elasticities f�jg add up to 0.


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averages of price changes, which can be expressed as a function ofchanges in aggregate expenditure shares and in the change inadjusted real income y after inverting the aggregate demandsystem in equation (11).

Let fS; Ŝg be vectors with the levels and changes in aggregateexpenditure shares, Sj and Ŝj. We also collect the parameters �jand �j in the vectors fa; bg and define C as the matrix with element�jk in row j, column k. With this notation, the demand system ischaracterized by the parameters f�; a; b;Cg. We choose an arbi-trary good n as the numeraire and assume that expenditurelevels are expressed in units of this good. Excluding good nfrom the demand system, the aggregate expenditure shares inequation (11) are represented by

S�n ¼ a�n þ C�n ln p�n þ b�ny;ð15Þ

where S�n is a vector with all expenditure shares but the numer-aire and C�n denotes that the nth row and the nth column areexcluded from C: We write the change in aggregate expenditureshares from equation (15) as dS�n ¼ C�np̂�n þ b�ndy and ex-press the vector of relative price changes as

p̂�n ¼ C�1�nðdS�n � b�ndyÞ:ð16Þ

Combining with the definition of the aggregate and the individ-ual expenditure effects from equations (5) and (6) yields

Ŵ ¼ �S0�nC�1�nðdS�n � b�ndyÞ;ð17Þ

b̂ ¼ �b0�nC�1�nðdS�n � b�ndyÞ:ð18Þ

These expressions show Ŵ and b̂ as functions of levels andchanges in aggregate shares, the substitution parameters �jk,the income elasticity parameters �j, and the change in adjustedreal income, dy. In addition, using dy ¼ ~̂x � â and Shephard’slemma allows us to express dy as follows:15

dy ¼ ~̂x � ½S0�n � yb0�n�C�1�ndS�n

1� ½S0�n � yb0�n�C�1�nb�n:ð19Þ

15. To derive equation (19), we use that â � @ ln a@ ln p�n0 p̂�n ¼ ½S�n0 � yb�n0 �p̂�n,

where thesecond line follows fromShephard’s lemma. Replacing p̂�n fromequation

(16) into this expression, using dy ¼ ~̂x � â, and solving for dy yields equation (19).


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Equations (17) to (19) allow us to express the aggregate andindividual expenditure effects as function of the level andchanges in aggregate expenditure shares, the parameters�j� �

; �jk� �

, the initial level of adjusted real income, y, andthe change in income of the representative consumer, ~̂x. Theseformulas correspond to infinitesimal welfare changes and can beused to compute a first-order approximation to the exact welfarechange corresponding to a discrete set of price changes.16

In deriving this result, we have not specified the supply sideof the economy, and we have allowed for arbitrary changes in thedistribution of individual expenditure levels, x̂h

� �. These

demand-side expressions can be embedded in different supply-side structures to study the welfare changes associated with spe-cific counterfactuals. In the next section, we embed them in amodel of international trade to compute the welfare effectscaused by changes in trade costs as function of observed expen-diture shares.

III. International Trade Framework

We embed the results from the previous section in anArmington trade model. Section III.A develops a multisectorArmington model with almost-ideal preferences and within-country income heterogeneity. Section III.B derives the non-homothetic gravity equation implied by the framework. SectionIII.C presents expressions for the welfare changes across house-holds resulting from foreign shocks.

III.A. Multisector Model

The world economy consists of N countries, indexed by n asimporter and i as exporter. Each country specializes in the pro-duction of a different variety within each sector s ¼ 1; :::;S, sothat there are J ¼ N � S varieties, each defined by a sector-origin dyad. These varieties are demanded at different incomeelasticities. For example, expenditure shares on textiles fromIndia may decrease with individual income, while shares onU.S. textiles may increase with income. We let psni be the price

16. In assuming that the changes in prices are small, we havenot allowed for thepossibility that consumers drop varieties in response to the price changes. When wemeasure the welfare losses from moving to autarky in the international trade setup,we account for this possibility.


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in country n of the goods in sector s imported from country i, andpn be the price vector in country n. The iceberg trade cost of ex-porting from i to n in sector s is �sni. Perfect competition impliesthat psni ¼ �snipsii.

Labor is the only factor of production. Country n has constantlabor productivity Zsn in sector s. Assuming that every country haspositive production in every sector, the wage per effective unit oflabor in country n is wn ¼ psnnZsn for all s ¼ 1; :::;S, and an indi-vidual h in country i with zh effective units of labor receivesincome of xh ¼ zh �wn. Each country is characterized by amean zn and a Theil index �n of its distribution of effectiveunits of labor across the workforce. Therefore, the income distri-bution has mean xn ¼ wnzn and Theil index �i. Income equalsexpenditure at the individual level (we use these terms inter-changeably) and at the aggregate level due to balanced trade.

We assume almost-ideal demand and reformulate the aggre-gate expenditure share equation (11) in this context. Let Xsni bethe value of exports from exporter i to importer n in sector s, andlet Yn be the total income of the importer. The share of aggregateexpenditures in country n devoted to goods from country i insector s is

Ssni ¼XsniYn¼ �sni þ

XSs0¼1

XNi0¼1

�ss0

ii0 ln ps0

ni0 þ �si yn;ð20Þ

where an ¼ a pn� �

is the homothetic component of the price

index (equation (8)) in country n and yn ¼ ln xnan

�þ�n is the

adjusted real income of the economy. The income elasticity �siis allowed to vary across both sectors and exporters. The richerthe importing country (higher xn) or the more unequal it is(higher �n), the larger its expenditure share in varieties withpositive income elasticity, �si > 0: In turn, the parameter �

sin

may vary across exporters, sectors, and importers, and it cap-tures the overall taste in country n for the goods exported bycountry i in sector s independently from prices or income in the

importer. These coefficients must satisfyPN

i¼1PS

s¼1 �si ¼ 0 andPN

i¼1PS

s¼1 �sni ¼ 1 for all n ¼ 1; . . .;N.

The coefficient �ss0

ii0 is the semi-elasticity of the expenditureshare in good (i, s) with respect to the price of good i0; s0ð Þ. Weassume no cross-substitution between goods in different sectors


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(�ss0

ii0 ¼ 0 if i 6¼ i0) and, within each sector s, we assume the sameelasticity between goods from different sources (�ssii0 is the same forall i0 6¼ i for each s, but allowed to vary across s). Formally,

�ss0ii0 ¼

� 1� 1N

� ��s if s ¼ s0 and i ¼ i0;

�s

Nif s ¼ s0 and i 6¼ i0;

0 if s 6¼ s0:

8>>>>><>>>>>:

ð21Þ

This structure on the elasticities is convenient because itsimplifies the algebra, but it is not necessary to reach analyticresults.17 It allows us to cast a demand system that looks similarto a two-tier demand system (across sectors in the upper tier andacross origins within each sector in the lower tier) and to relate itto homothetic multisector gravity models.18

Using equation (21), the expenditure share in goods fromorigin country i in sector s can be simplified to

Ssni ¼ �sni � �s ln psni� �

� 1N

XNi0¼1

ln psni0

" #þ �si yn:ð22Þ

The corresponding expenditure share for consumer h in goodsfrom country n in sector s is

ssni;h ¼ �sni � �s ln psni� �

� 1N

XNi0¼1

ln psni0

" #þ �si ln

xh~xn

� �þ yn

� �:ð23Þ

Adding up equation (22) across exporters, the share of sector sin the total expenditures of country n is:

Ssn ¼XNi¼1

Ssni ¼ �sn þ �syn;ð24Þ

17. The normalization by N in equation (21) only serves the purpose of easingthe notation in following derivations.

18. This nesting is a standard approach to the demand structure in multisectortrade models. For example, see Feenstra and Romalis (2014) or Costinot andRodrı́guez-Clare (2014). Imposing symmetry within sectors also allows us to com-pare results to estimates of gravity equations derived under a translog demandsystem from the literature (see later discussion).


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where

�sn ¼XNi¼1

�sni;

�s ¼

XNi¼1

�si :

In turn, the share of sector s in total expenditures of consumerh is

ssn;h ¼XNi¼1

ssni;h ¼ �sn þ �s

yn þ lnxh~xn

� �� :ð25Þ

Equations (24) and (25) show that the expenditure sharesacross sectors have an ‘‘extended Cobb-Douglas’’ form, whichallows for nonhomotheticities across sectors through �

son top

of the fixed expenditure share �sn. We refer to �s

in equation(24) as the ‘‘sectoral betas.’’19

III.B. Nonhomothetic Gravity Equation

The model yields a sectoral nonhomothetic gravity equationthat depends on aggregate data and the demand parameters.These parameters are the elasticity of substitution �s across ex-porters in sector s and the income elasticity of the goods suppliedby each exporter in each sector, �sn

� �. Combining equation (22)

and the definition of yn gives

XsniYn¼ �sni � �s ln

�snipsii

� snps

� �þ �si ln

xna pn� �

!þ�n

" #;ð26Þ

where

�sn ¼ exp1

N

XNi¼1

ln �sni� � !

19. If �s ¼ 0 for all s (so that nonhomotheticities across sectors are shut down),

sectoral shares by importer are constant at Ssn ¼ �sn, as it would be the case withCobb-Douglas demand across sectors.


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and

ps ¼ exp 1N

XNi¼1

ln psii� � !

:

Income of each exporter i in sector s equals the sum of sales to

every country, Ysi ¼PN

n¼1 Xsni. Using this condition and equation

(26), we can solve for �s lnps

ii

ps

�. Replacing this term back into

equation (26), import shares in country n can be expressed inthe gravity form:

XsniYn¼ Asni þ

YsiYW� �sTsni þ �si �n;ð27Þ

where YW ¼PI

i¼1 Yi stands for world income, and where

Asni ¼ �sni �XNn0¼1

Yn0

YW

� ��sn0i;ð28Þ

Tsni ¼ ln�sni� sn

� ��XNn0¼1

Yn0

YW

� �ln

�sn0i�sn0

� �;ð29Þ

�n ¼ lnxnan

� �þ�n

��XNn0¼1

Yn0

YW

� �ln

xn0

an0

� �þ�n0

�:ð30Þ

The first term in equation (27), Asni, captures cross-countrydifferences in tastes across sectors or exporters; this termvanishes if �sni is constant across importers n. The second term,Ys

i

YW, captures relative size of the exporter because of, for example,

high productivity relative to other countries. The third term, Tsni,measures both bilateral trade costs and multilateral resistance(i.e., the cost of exporting to third countries).

The last term in equation (27), �si �n; is the nonhomotheticcomponent of the gravity equation. It includes the good-specificEngel curves needed to measure the unequal gains from tradeacross consumers. This term captures the ‘‘mismatch’’ betweenthe income elasticity of the exporter and the income distributionof the importer. The larger �n is, because either average incomeor inequality in the importing country n are high relative to therest of the world, the higher the share of expenditures devoted togoods in sector s from country i when i sells high income-elastic


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goods ð�si > 0Þ: If nonhomotheticities are shut down, this lastterms disappears and the gravity equation in (27) becomes thetranslog gravity equation.

III.C. Distributional Impact of a Foreign-Trade Shock

Using the results from Section III, we derive the welfare im-pacts of a foreign-trade shock across the expenditure distribution.Without loss of generality we normalize the wage in country n to1, wn¼ 1. Consider a foreign shock to this country consisting of aninfinitesimal change in foreign productivities, foreign endow-ments, or trade costs between any country pair. From the per-spective of an individual consumer h in country n, this shockaffects welfare through the price changes p̂sni

� �i;s

and theincome change x̂h. From equations (21) and (22), the change inthe price of imported relative to own varieties satisfies:

p̂sni � p̂snn ¼ �

dSsni � dSsnn�s

þ 1�s

�si � �sn� �

dyn:ð31Þ

Because only foreign shocks are present, the change in in-come x̂h is the same for all consumers and equal to the change inthe price of domestic commodities, x̂h ¼ ~̂x ¼ p̂snn ¼ 0 for all h incountry N and for all s ¼ 1; :::;S.20 Imposing these restrictions,we can rewrite equation (17) as

Ŵn � ŴH;n þ ŴNH;n;ð32Þ

where

ŴH;n ¼XSs¼1

XNi¼1

1

�sSsni dS

sni � dSsnn

� �;ð33Þ

ŴNH;n ¼XSs¼1

XNi¼1

1

�sSsni �

sn � �si

� �dyn:ð34Þ

Using these restrictions, we can also rewrite the slope of theindividual effect in equation (18) as

b̂n ¼XSs¼1

XNi¼1

�si�s

dSsnn � dSsni þ �si � �sn� �

dyn� �

;ð35Þ

20. Note that because of the Ricardian supply-side specification, there is nochange in the relative price across domestic goods or in relative incomes acrossconsumers.


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and the change in the adjusted real income from equation (19) as

dyn ¼

XSs¼1

XNi¼1

1

�sSsni � �sniyn� �

dSsni � dSsnn� �

1�XS

s¼1

XNi¼1

1

�sSsni � �si yn� �

�sn � �si� � :ð36Þ

Expressions (32) to (36) provide a closed-form characterizationof the welfare effects of a foreign-trade shock that includes threenovel margins. First, preferences are nonhomothetic with good-specific income elasticities. Second, the formulas accommodatewithin-country inequality through the Theil index of expendituredistribution �n, which enters through the level of yn. Third, andkey for our purposes, the expressions characterize the welfarechange experienced by individuals at each income level, so thatthe entire distribution of welfare changes across consumers h incountry n can be computed from equation (14) using:

!̂h ¼ Ŵn � b̂n � lnxh~x

�:ð37Þ

The aggregate expenditure effect, Ŵn, includes a homotheticpart ŴH;n independent from the �

sn’s. When nonhomotheticities

are shut down, this term corresponds to the aggregate gainsunder translog demand.21 The aggregate effect also includesand a nonhomothetic part, ŴNH;n, which adjusts for the country’spattern of specialization in high- or low-income elastic goods andfor the change in adjusted real income.

The key term for measuring unequal welfare effects is thechange in the nonhomothetic component b̂n. As we have estab-lished, b̂n < 0 implies a decrease in the relative price of highincome-elastic goods, which favors high-income consumers. Todevelop an intuition for how observed trade shares and parame-ters map to b̂n, consider the single-sector version of the model.

21. Feenstra and Weinstein (2010) measures the aggregate gains from trade inthe United States under translog preferences in a context with competitive effects,and Arkolakis, Costinot, and Rodrı́guez-Clare (2010) study the aggregate gainsfrom trade with competitive effects under homothetic translog demand andPareto distribution of productivity. The AIDS nests the demand systems in thesepapers, but we abstract from competitive effects. With a single sector, the translogterm in equation (33) becomes ŴH;n ¼

PNi¼1

1�Sni dSni � dSnnð Þ. Under CES prefer-

ences with elasticity �, the equivalent term is 11�� Ŝnn, which depends on just theown trade share. See Arkolakis, Costinot, and Rodrı́guez-Clare (2012).


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Setting S¼ 1 and omitting the s superscript from every variable,equation (35) can be written as

b̂n ¼1

��2�dyn � d�n

�

;ð38Þ

where �2� ¼PN

i¼1 �2i , and where

�n ¼XNi¼1

�iSni:ð39Þ

The parameter �2� is proportional to the variance of the �n’s andcaptures the strength of nonhomotheticities across goods fromdifferent origins. The term �n is proportional to the covariancebetween the Sni’s and the �i’s, and measures the bias in thecomposition of aggregate expenditure shares of country i towardgoods from high-� exporters. The larger is �n, the relativelymore economy n spends in goods that are preferred by high-income consumers. Suppose that d�n > 0, that is, a movementof aggregate trade shares toward high-�i exporters; if � > 0 andthe aggregate real income of the economy stays constant(dyn¼ 0), this implies a reduction in the relative price of importsfrom high-�i exporters, and a positive welfare impact on con-sumers who are richer than the representative consumer.22

Equations (32) to (36) express changes in individual welfareas the equivalent variation of a consumer that corresponds to aninfinitesimal change in prices caused by foreign shocks. To obtainthe exact change in real income experienced by an individual withexpenditure level xh in country n between an initial scenariounder trade (tr) and a counterfactual scenario (cf) we integrateequation (37),23

!tr!cfn;h ¼WcfnWtrn

� �xh~xn

� �� ln bcfnbtrn

�;ð40Þ

where Wcfn

Wtrnand b

cfn

btrncorrespond to integrating equations (32) to (36)

between the expenditure shares in the initial and counterfactual

22. At the same time, keeping prices constant, dyn> 0 would imply a movementof aggregate shares to high-�i exporters (d�n > 0). Therefore, conditioning on d�n, alarger dyn implies an increase in the relative price of high-income elastic goods.

23. An expression similar to equation (40) appears in Feenstra, Ma, and Rao(2009).


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scenarios. If !tr!cfn;h < 1, individual h is willing to pay a fraction

1� !tr!cfn;h of her income in the initial trade scenario to avoid themovement to the counterfactual scenario.

In Section V we perform the counterfactual experiment ofbringing each country to autarky and simulate partial changesin the trade costs. In each case, we compute equation (40) usingthe changes in expenditure shares that take place between theobserved and counterfactual scenarios. For that, we need theincome elasticities �sn

� �and the substitution parameters �s

� �.

The next section explains the estimation of the gravity equationto obtain these parameters.

IV. Estimation of the Gravity Equation

In this section, we estimate the nonhomothetic gravity de-rived in Section III.24 Section IV.A describes the data, and SectionIV.B presents the estimation results.

IV.A. Data

To estimate the nonhomothetic gravity equation we use datacompiled by World Input-Output Database (WIOD). The data-base records bilateral trade flows and production data by sectorfor 40 countries (27 European countries and 13 other large coun-tries) across 35 sectors that cover food, manufacturing, and ser-vices (we take an average of flows between 2005–2007 to smoothout annual shocks). The data record total expenditures by sectorand country of origin, as well as final consumption; we use totalexpenditures as the baseline and report robustness checks thatrestrict attention to final consumption. We obtain bilateral dis-tance, common language, and border information from CEPII’sGravity database. Price levels, adjusted for cross-country qualityvariation, are obtained from Feenstra and Romalis (2014).Income per capita and population are from the Penn WorldTables, and we obtain Gini coefficients from the World Income

24. In principle, one could obtain the parameters from other data sources, suchas household surveys, that record consumption variation across households withincountries. We have chosen to use cross-country data because it is internally consis-tent within our framework, and it is a common approach taken in the literature. InSection V.D, we explore results that use parameters estimated from the U.S. con-sumption expenditure microdata.


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Inequality Database (version 2.0c, 2008) published by the WorldInstitute for Development Research.25

The left-hand side of equation (27),Xs

ni

Yi, can be directly mea-

sured using the data from sector s and exporter i’s share in coun-try n’s expenditures. Similarly, we use country i’s sales in sector s

to constructYs

i

YW:

The term Tsni in equation (27) captures bilateral trade costsbetween exporter i and importer n in sector s relative to theworld. Direct measures of bilateral trade costs across countriesare unavailable, so we proxy them with bilateral observables.

Specifically we assume �sni ¼ d�s

ni�jg��sjj;ni~e

sni, where dni stands for

distance, �s reflects the elasticity between distance and tradecosts in sector s, the g’s are other gravity variables (commonborder and common language),26 and ~esni is an unobserved compo-nent of the trade cost between i and n in sector s.27 This allows usto rewrite the gravity equation as

XsniYn¼ Asni þ

YsiYW� �s�sð ÞDni þ

Xj

�s�sj

�Gj;ni þ �si �n þ esni;ð41Þ

where, letting dn ¼ 1NXNi¼1

ln dnið Þ;

Dni ¼ lndni

dn

� ��XNn0¼1

Yn0

YW

� �ln

dn0i

dn0

� �:ð42Þ

and where Gj;ni is defined in the same way as equation (42) butwith gj;ni instead of dni:

28 As seen from equation (45), becausewe do not directly observe trade costs we cannot separately

25. The World Income Inequality Database provides Gini coefficients from bothexpenditure and income data. Ideally, we would use Ginis from only the expendi-ture data, but this is not always available for some countries during certain timeperiods. We construct a country’s average Gini using the available data between2001–2006.

26. Since bilateral distance is measured between the largest cities in each coun-try using population as weights, it is defined when i¼n; see Mayer and Zignago(2011). Note that we parameterize trade costs such that a positive effect of commonlanguage and common border on trade is reflected in �sj > 0.

27. Waugh (2010) includes exporter effects in the trade-cost specification. Thegravity equation (27) would be unchanged in this case because the exporter effectwould wash out from Tsni in equation (29).

28. From the structure of trade costs it follows that the error term is esni ¼ ��s

ln~e sni~e sn

� ��PN

n0¼1Yn0YW

�ln

~e sn0 i

~e sn0

� �� where ~esn ¼ exp 1N

PNn0¼1 ln ~e

sn0i

� � �.


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identify �s and �s. Following Novy (2012) we set �s ¼ � ¼ 0:177for all s.29

The term �n in equation (41) captures importer n’s inequal-ity-adjusted real income relative to the world. To construct thisvariable, we assume that the distribution of efficiency units ineach country n is log-normal, ln zh � N n; �2n

� �. This implies a

log-normal distribution of expenditures with Theil index equal to�2n2 where �

2n ¼ 2 ��1

gininþ12

�h i2. We construct xn from total expen-

diture and total population of country n. We follow Deaton andMuellbauer (1980a), and more recently Atkin (2013), to proxy thehomothetic component an with a Stone index, for which we usean ¼

Pi Sni ln pnnd

�ni

� �, where pnn are the quality-adjusted prices

estimated by Feenstra and Romalis (2014). The obvious advan-tage of this approach is that it avoids the estimation of the �sni,which enter the gravity specification nonlinearly and are not re-quired for our welfare calculations. The measure of real spending

per capita divided by the Stone price index, xiai, is strongly corre-

lated with countries’ real income per capita; this suggests that �iindeed captures the relative difference in real income acrosscountries.

To measure Asni, we decompose �sni into an exporter effect �i, a

sector-specific effect �s, and an importer-specific taste for eachsector "sn:

�sni ¼ �i �s þ "sn� �

:ð43Þ

We further impose the restrictionPN

i¼1 �i ¼ 1. Under the as-sumption of equation (43), the sectoral expenditure sharesfrom the upper-tier equation (24) becomes:

Ssn ¼ �s þ �syn þ "sn:ð44Þ

This equation is an Engel curve that projects expenditureshares on the adjusted real income.30 Specifically, it regressessector expenditure shares on sector dummies and the importer’s

29. Later we explore the sensitivity of the results to alternative values of thisparameter.

30. Note that sectoral shares in value added and efficiency units are allowed tovary independently from expenditure shares depending on the distribution of sec-toral productivities Zsn and trade patterns. The sectoral productivities are not esti-mated and are not needed to perform the counterfactuals.


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adjusted real income interacted with sector dummies. The in-teraction coefficients will have the structural interpretation asthe sectoral betas �

s.31 Using equations (28), (43), and (44), we

can write Asni ¼ �i Ssn � SsW � �s�nÞ

�, where SsW is the share of

sector s in world expenditures. Combining this with the gravityequation (41), we reach the following estimating equation:

XsniYn¼ Y

si

YWþ�i Ssn�SsW

� �� s�sð ÞDniþ

Xj

�s�sj

�Gj;niþ �si ��i�

sÞ�nþ esni�

ð45Þ

The gravity equation (45) identifies �si � �i�sÞ

�using the var-

iation in �n across importers for each exporter. Using the �s

estimated from the sectoral Engel curve in equation (44) andthe �i estimated from equation (45) we can recover the �

si

(which is needed to perform the counterfactuals). Since themarket shares sum to 1 for each importer, it is guaranteed thatP

i

Ps �

si ¼ 0 in the estimation, as the theory requires. We cluster

the estimation at the importer level to allow for correlation in theerrors across exporters.

The sectoral gravity equation aggregates to the gravity equa-tion of a single-sector model. Summing equation (45) across sec-tors s gives the total expenditure share dedicated to goods from iin the importing country n,

XniYn¼ Yi

YW� ��ð ÞDni þ

Xj

��j� �

Gj;ni þ �i�n þ eni;ð46Þ

where �� PS

s¼1 �s�s; �i �

PSs¼1 �

si , and eni �

PSs¼1 e

sni. We can

readily identify equation (46) as the gravity equation thatwould arise in a single-sector model (S¼ 1). Thus, summingour estimates on the gravity terms from equation (45) willmatch the gravity coefficients from a single-sector model.Likewise, the sum of the sector-specific income elasticities byexporter

Ps �

si estimated from equation (45) matches the

income elasticity �i estimated from equation (46).

31. The term "si captures cross-country differences in tastes across sectors thatare not explained by differences in income or inequality levels. As in Costinot,Donaldson, and Komunjer (2012) or Caliendo and Parro (2012), this flexibility isneeded for the model to match sectoral shares by importer. This approach to mea-suring taste differences is also in the spirit of Atkin (2013), who attributes regionaldifferences in tastes to variation in demand that is not captured by observables.


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IV.B. Estimation Results

We begin by estimating the single-sector gravity model inequation (46). This regression aggregates across the sectors inthe data, and as illustrated in equation (46), the baselinemultisector gravity equation aggregates exactly to this single-sector gravity equation. The results are reported in Table I.Consistent with the literature, we find that bilateral distance re-duces trade flows between countries, which is captured by thestatistically significant coefficient on Dni. Under the assumptionthat � ¼ 0:177, the estimate implies � ¼ 0:24 (¼ :043:177).

32 The addi-tional trade costs—common language and a contiguous borderterm—also have the intuitive signs.

The table also reports estimates of the 40 �i parameters, onecorresponding to each exporter, in the subsequent rows. The ex-porters with the highest �’s are the United States and Japan,whereas Indonesia and India have the lowest �’s. This meansthat the United States and Japan export goods that are preferredby richer consumers, and the latter export goods preferred bypoorer consumers. To visualize the �’s, we plot them against theper capita income in Figure I. The relationship is strongly posi-tive and statistically significant. We emphasize that this relation-ship is not imposed by the estimation. Rather, these coefficientsreflect that richer countries are more likely to spend on productsfrom richer countries, conditional on trade costs. We also notethat the �’s are fully flexible, which is why the coefficients areoften not statistically significant, but the null hypothesis that allincome elasticities are zero is rejected.33 Moreover, the finding

32. This estimate is close to the translog gravity equation estimate of � ¼ 0:167estimated by Novy (2012). Feenstra and Weinstein (2010) report a median � of 0.19using a different data, level of aggregation and estimation procedure, so our esti-mate is in line with the few publications that have run gravity regressions with thetranslog specification.

33. If we reduce the number of estimated parameters by imposing a relationshipbetween income elasticities and exporter income, we find a positive and statisticallysignificant relationship between the two variables. Specifically, we can impose that�i ¼ B0 þ B1yi, which is similar to how Feenstra and Romalis (2014) allow fornonhomotheticities. The theoretical restriction

Pi �i ¼ 0 implies that

B0 ¼ �B1 1NP

i yi, transforming this linear relationship to �i ¼ B1 yi � 1NP

i0 yi0� �

and reducing the number of income elasticity parameters to be estimated from 40to 1. If we impose this to estimate the gravity equation, we find B1 ¼ 0:0057 (stan-dard error of 0.0026). This estimate is very close to regressing our estimated �i’s

reported in Table I on yi � 1NP

i0 yi0� �

, which yields a coefficient of 0.008 (standard

error of 0.0035).


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TABLE I

GRAVITY ESTIMATES: SINGLE SECTOR

Variables (1A) (1B)

�Distanceni 0.043***(0.005)

Languageni 0.131***(0.021)

Borderni 0.135***(0.023)

�n X �n X�-USA 0.052** �-POL �0.001

(0.022) (0.011)�-JPN 0.028*** �-IDN �0.023

(0.008) (0.032)�-CHN 0.008 �-AUT �0.001

(0.031) (0.009)�-DEU �0.015 �-DNK 0.003

(0.013) (0.009)�-GBR 0.005 �-GRC 0.018*

(0.013) (0.009)�-FRA �0.013 �-IRL �0.009

(0.011) (0.013)�-ITA 0.006 �-FIN 0.013

(0.006) (0.010)�-ESP �0.004 �-PRT �0.001

(0.006) (0.005)�-CAN �0.017 �-CZE �0.003

(0.015) (0.006)�-KOR 0.006 �-ROM 0.003

(0.012) (0.015)�-IND �0.048 �-HUN 0.008

(0.042) (0.012)�-BRA �0.010 �-SVK 0.005

(0.017) (0.010)�-RUS �0.003 �-LUX �0.012*

(0.022) (0.007)�-MEX �0.029* �-SVN �0.002

(0.017) (0.005)�-AUS 0.011 �-BGR 0.004

(0.012) (0.016)�-NLD �0.008 �-LTU 0.004

(0.009) (0.010)�-TUR 0.006 �-LVA 0.006

(0.016) (0.009)�-BEL �0.025** �-EST 0.007

(0.011) (0.007)�-TWN 0.017 �-CYP 0.016**

(0.011) (0.008)�-SWE 0.006 �-MLT �0.006

(0.008) (0.010)

Joint F-test p-value forincome elasticities

0.00

R2 0.47Observations 1,600Implied � 0.24

Notes. Table reports the estimates of the single-sector gravity equation thataggregates the data across the 35 sectors. There are 40 income elasticity param-

eters �i. We assume that �¼ 0.177, and the implied � ¼ fcoefficient on � Distanceni g� isnoted at the bottom of the table. Standard errors are clustered by importer.Significance * .10, ** .05, *** .01.


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that a subset are statistically significant is sufficient to rejecthomothetic preferences in the data and is consistent with theexisting literature that finds that richer countries export goodswith higher income elasticities.34

Next we report the results for the multisector estimation. Asnoted earlier, the analysis involves estimating the Engel curve inequation (44), which projects sectoral expenditure shares on ad-justed real income across countries. Table II reports the sectoralbetas, �

s. Compared to food and manufacturing sectors (listed in

column (1A)), service sectors (listed in column (1B)) tend to behigh-income elastic.35 This pattern can be visualized by plottingcountries’ expenditure shares in these three broad categories

USA

JPN

CHN

DEU

GBR

FRA

ITA

ESP

CAN

KOR

IND

BRA

RUS

MEX

AUS

NLD

TUR

BEL

TWN

SWE

POL

IDN

AUTDNK

GRC

IRL

FIN

PRTCZE

ROMHUN

SVK

LUX

SVN

BGR LTULVAEST

CYP

MLT

−.0

50

.05

β i

8 9 10 11 12Log GDP per Capita

FIGURE I

�i and GDP per Capita

Figure plots exporter income elasticity against its per capita GDP.

34. See Hallak (2006), Khandelwal (2010), Hallak and Schott (2011), andFeenstra and Romalis (2014).

35. To see this, we aggregate the �s

into three categories: food includes‘‘Agriculture’’ and ‘‘Food, Beverages, and Tobacco,’’ manufacturing includes theremaining sectors listed in column (1A) of Table II, and services is composed ofthe 19 sectors in column (1B). The corresponding elasticities for food, manufactur-ing, and services are�0.0343,�0.0410, and 0.0753, respectively. (Again, the sum ofthese three broad classifications is 0.)


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TABLE II

ENGEL CURVE ESTIMATION: BASELINE

Variables (1A) (1B)

Agriculture �0.0218*** Electricity, gas, and watersupply

�0.0033(0.002) (0.002)

Mining �0.0080*** Construction �0.0053(0.002) (0.003)

Food, beverages, and tobacco �0.0125*** Sale, repair of motor vehicles 0.0027***(0.003) (0.001)

Textiles �0.0063*** Wholesale trade and com-mission trade

0.0010(0.001) (0.003)

Leather and footwear �0.0009*** Retail trade �0.0020(0.000) (0.002)

Wood products �0.0008 Hotels and restaurants 0.0021(0.001) (0.001)

Printing and publishing 0.0014* Inland transport �0.0089***(0.001) (0.003)

Coke, refined petroleum, nu-clear fuel

�0.0056*** Water transport �0.0007(0.002) (0.001)

Chemicals and chemicalproducts

�0.0046*** Air transport 0.0007*(0.001) (0.000)

Rubber and plastics �0.0016* Other auxiliary transportactivities

0.0038***(0.001) (0.001)

Other nonmetallic minerals �0.0027*** Post and telecommunications 0.0012(0.001) (0.001)

Basic metals and fabricatedmetal

�0.0031 Financial intermediation 0.0280(0.004) (0.018)

Machinery �0.0028 Real estate activities 0.0095***(0.002) (0.003)

Electrical and opticalequipment

�0.0021 Renting of M&Eq 0.0243***(0.003) (0.003)

Transport equipment �0.0033* Public admin and defense 0.0038(0.002) (0.003)

Manufacturing, nec �0.0005 Education 0.0022**(0.001) (0.001)

Health and social work 0.0128***(0.003)

Other community and socialservices

0.0031(0.003)

Private households with em-ployed persons

0.0003**(0.000)

Sector FEs yesJoint F-test p-value for sectoral elasticities 0.00R2 0.67Observations 1,400

Notes. Table reports the sectoral income elasticities from the Engel curve equation. It is a regression ofimporters’ sectoral expenditures shares on the adjusted real income interacted with sector dummies. SectorsAgriculture and Food, beverages, and tobacco are the food sectors, and the remaining sectors in column (1A)are the manufacturing sectors; the service sectors are listed in column (1B). The regression also includessector fixed effects. Standard errors are clustered by importer. Significance * .10, ** .05, *** .01.


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against their income per capita in Figure II: the Engel curve forservices is positively sloped, whereas it is negatively sloped forfood and manufacturing.36 These sectoral elasticities are highlycorrelated with sectoral elasticities estimated using a differentnon-homothetic framework on different data by Caron, Fally,and Markusen (2014); see Appendix Figure A.1, which plots thetwo sets of elasticities against each other.37

The results of the sectoral gravity equation in equation (45) isreported in Table III. Columns (1A) and (1B) report the 35 sector-specific distance coefficients, ��s (where � ¼ 0:177 as before).Recall, these these coefficients sum to coefficient on distancefrom the single-sector model (see Table I). Likewise, the sector-specific language and border coefficients in columns (2) and (3) ofTable III sum exactly to the corresponding coefficients in thesingle-sector estimation.

0.2

.4.6

.8S

hare

of A

ggre

gate

Exp

endi

ture

s


Food Mfg Services

FIGURE II

Engel Curves by Broad Sector Groups

Figure displays Engel curves of expenditure shares in broad sectors againstper capita GDP. See the note in Table II for the list of sectors.

36. This is consistent with the literature on structural transformation; seeHerrendorf, Rogerson, and Valentinyi (2014).

37. Caron, Fally, and Markusen (2014) estimate sectoral income elasticities onGTAP data using constant relative income preferences. We match GTAP sectorclassifications with WIOD sector classifications to produce the scatter plot.


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We suppress the estimates of �sn� �

for readability purposes,but recall that

Ps �

si equals the exporter income elasticities �i

reported in the single-sector gravity equation. Also note that by

construction,P

n �sn equals the sectoral elasticities �

sdisplayed in

Table II. To see how �sn relate to exporter income per capita, weaggregate these coefficients to three broad classifications—food,manufacturing, and services—and report the plot in Figure III.Analogous to the single-sector estimates in Figure I, we find thatthe positive relationship between exporter income per capita andincome elasticities holds within sectors as well.

V. The Unequal Gains from Trade

This section conducts counterfactual analyses to measure thedistributional consequences of trade. Section V.A explains howwe numerically implement the expressions from Section III.C.Section V.B shows the results of autarky counterfactuals in thesingle-sector version of our model. Section V.C presents the mainresults: the autarky counterfactuals in the baseline multisector

−.0

20

.02

.04

Bet

a


Food Mfg Services

FIGURE III

� by Exporter and Broad Sector Group versus GDP per Capita

Figure plots income elasticities, summed across broad sectors for each coun-try, aga

Documents

Pablo D. Fajgelbaum and Amit K. Khandelwal · 2020. 9. 20. · Pablo D. Fajgelbaum and Amit K. Khandelwal Individuals that consume different baskets of goods are differentially af-fected