Generalized Separability and the Gains from Trade

Transcription

1 Generalzed Separablty and the Gans from Trade Thbault Fally UC Berkeley ARE September 2018 Abstract The vast majorty of models n nternatonal trade make strong assumptons on the structure of preferences, ether assumng homothetc, drectly-separable or ndrectly-separable preferences. These assumptons lead to sharp mplcatons for the quanttatve gans from trade, condtonal on estmated elastctes of trade to trade costs and observed mport penetraton. Ths note examnes the gans from trade under a more flexble demand system that encompasses the three cases of demand usually mposed, whle retanng the property that prces can be summarzed by a sngle aggregator. Combned wth monopolstc competton and Pareto dstrbutons of margnal costs on the supply sde (whch leads to gravty equatons), ths demand system yelds more flexble responses of prces to ncome and trade, and a wder range of predctons for the gans from trade. To estmate the gans from trade, ths note brefly descrbes an dentfcaton strategy explotng prce responses to changes n market access and trade costs. Keywords: Gans from trade, Gravty, Generalzed Separablty, Sngle aggregator, Non-homothetc preferences. JEL Classfcaton: F10, D11, D43, L11 UC Berkeley, Agrcultural and Resource Economcs Department, NBER and CEPR. Contact: fally@berkeley.edu. Many many thanks to Andrés Rodríguez-Clare for very helpful comments. 1

2 1 Introducton How large are the gans from nternatonal trade? Followng Arkolaks, Costnot and Rodrguez-Clare (2012), a large body of work has examned the gans from trade through the lens of varous types of trade models, usng mport penetraton as a suffcent statstc. The conclusons of Arkolaks et al. (2012) have been extended by examnng alternatve assumptons on both the demand sde and the supply sde. 1 Lookng nto the demand sde, recent work has shown that t s crucal to depart from homothetc constant-elastcty (CES) preferences to allow for varable markups, and non-trval prce responses. 2 However, exstng papers have focused on specfc types of preferences. In partcular, Arkolaks, Costnot, Donaldson and Rodrguez-Clare (2015, henceforth ACDR) focus on drectlyseparable as well as homothetc preferences, whereas Bertolett, Etro and Smonovska (2018) focus on ndrectly-separable preferences. Ths paper unfes and generalzes these approaches by consderng the generalzed Gorman-Pollak demand system 3 whch encompasses CES, drectly-separable, ndrectly separable, and specal cases of homothetc preferences. 4 Such demand system allows for more flexble prce and ncome effects than drectly or ndrectly-separable preferences, whle retanng the property that prces can be summarzed by a sngle aggregator (generalzed separablty), whch s partcularly convenent under monopolstc competton. Ths general form of demand has been seldom used perhaps for the lack of formal statements on ntegrablty condtons, whch are now provded n a companon paper (Fally, 2018). 5 On the supply sde, the model ncorporates standard assumptons of recent trade models (e.g. as n ACDR, Chaney, 2008, Meltz, 2003), monopolstc competton wth heterogeneous frms and Pareto dstrbutons of margnal costs. Wth generalzed Gorman-Pollak demand, the model generates a response of prce levels to percapta ncome, as n Bertolett et al (2018), whle retanng a dependence of prces to trade shocks as n ACDR. Wth varable markups and ncomplete pass-through of costs to prces, prces do not only depend on margnal costs but also on reservaton prces (choke prces) n the destnaton market, leadng to prcng to market. Generalzed separablty allows for a more flexble relatonshp between the choke prce, ncome levels, and the degree of competton n the destnaton market, whch tself depend on trade costs and proxmty to thrd markets. The gans from trade can stll be expressed as a smple functon of the changes n mport penetraton but the formula leads to a wder range of estmates, condtonal on markup and trade elastctes. Indrectly-addtve preferences provde a lower bound of the gans from trade and homothetc preferences provde a hgher bound. Wthn these bounds, all ntermedate outcomes are theoretcally possble. Rather than assumng a specfc shape of preferences, these results ndcate that more precse 1 See, e.g., Feenstra (2018a) and Costnot and Rodríguez-Clare (2018) for recent surveys. 2 Models extendng Meltz (2003) to ncorporate varable markups nclude Arkolaks et al. (2015), Bertolett et al. (2018), Smonovska (2015), Behrens et al. (2014), Feenstra (2018b), among others. 3 See Gorman (1972, 1995), Pollak (1972), Fally (2018). 4 Ths homothetc case corresponds to the homothetc sngle aggregator (HSA) demand system studed n Matsuyama and Ushchev (2017), whch yelds the same outcomes as the mplctly-addtve homothetc preferences n ACDR. 5 A recent excepton s Bertolett and Etro (2017). 2

3 estmates of demand systems are necessary to quantfy the gans from trade. In the last secton, I dscuss an emprcal strategy to estmate the gans from trade wth such a demand system. I consder a sem-parameterzaton of demand that encompasses drectly and ndrectly-addtve preferences, as well as homothetc demand wth a sngle prce aggregator. Assumng that econometrcans also have estmates of trade and markup elastctes, 6 the key addtonal parameter to estmate determnes how choke prces vary wth ncome and trade costs. To estmate ths addtonal parameter, a suggeston s to use frm-level data on export prces for a gven source country and regress these prces on changes n ncome and market access of destnaton countres (as measured for nstance n Head and Mayer, 2010, and Reddng and Venables, 2004). 2 A flexble demand system Ths secton descrbes the demand sde, and the next secton provdes results from the full generalequlbrum model. The notaton remans close to Arkolaks et al., 2015 (ACDR), whch we can refer to for more detals, especally on the supply sde. 2.1 Demand system Consumer ncome and total expendtures are denoted by w. For each good, we assume that demand s determned by ts prce p, consumer ncome w and an aggregator Λ: q(p, w, Λ) = Q(Λ)D(F (Λ)p/w) where Λ = Λ(p, w) s tself a scalar functon of all prces and ncome, homogeneous of degree zero n (w, p). Λ s mplctly defned by the budget constrant,.e. t s the mplct soluton of: p(ω)q(λ)d(f (Λ)p(ω)/w)dω = w (1) ω Ω where Ω denotes the set of goods avalable to the consumer and p(ω) s the prce of each varety ω Ω. Condtons for ntegrablty Assumng that all three functons D, F and Q are dfferentable, and assumng that D s suffcently downward sloppng and elastc,.e. ε D d log D d log p > 1, the followng two condtons are suffcent to ensure that such demand system s ntegrable (Fally 2018),.e. can be derved from ratonal consumer behavor and utlty maxmzaton: ) ε F (Λ)ε D (y) ε Q (Λ) has the same sgn for all Λ and y ) For any set of prces p(ω) and ncome w, equaton (1) admts a soluton n Λ. where ε Q and ε F denote the elastcty of Q and F wth respect to Λ. 6 For nstance, Smonovska and Waugh (2014) and Amt et al. (2014) respectvely provde estmates of the trade and markup elastctes that are consstent wth the framework that we consder here. 3

4 Specal cases These preferences are more general than typcally assumed n the trade lterature. They correspond to: Drectly-separable preferences when Q(Λ) s constant and F (Λ) = Λ. Indrectly-separable preferences when F (Λ) s constant and Q(Λ) = Λ. Homothetc Sngle Aggregator (Matsuyama and Ushchev 2018) when F (Λ) = Q(Λ) = Λ. Iso-elastc shfters, a useful parameterzaton: Q(Λ) = αλ β and F (Λ) = Λ γ. In the case wth so-elastc shfters Q and F, t s wthout loss of generalty to assume that γ {0, 1} (where the case γ = 0 corresponds to ndrectly-addtve preferences). 7 If γ = 1, a suffcent condton for ntegrablty s that β < ε D. Snce we assume that ε D > 1, a suffcent condton s β < 1. Note that β can be negatve, whch also ensures ntegrablty. Indrectly-addtve preferences can be seen as the lmt case β, whle drectly-separable preferences correspond to β = Choke prces Assume that D( ) s equal to zero when prces are hgh enough (.e. above a reservaton prce, or choke prce). Wthout loss of generalty, assume that D(x) = 0 f and only f x 1. Let us denote by P the choke prce, the prce above whch demand s null. It s then equal to: P = w F (Λ) (2) A central observaton by Smonovska (2015) and Bertolett et al. (2018) s that prces are more strongly correlated across countres wth per capta ncome than wth market sze or populaton. Ths motvates the use of ndrectly-addtve preferences rather than drectly-addtve preferences wth whch t s dffcult to quanttatvely reproduce the role of ncome. Wth ndrectly-addtve preferences, however, F s constant and the choke prce s proportonal to ncome. In contrast, the preferences used here allow for more flexblty n the general case. To fully characterze how the choke prce vares wth ncome n the more general case, we need to account for how ncome affects Λ. Dfferentatng w.r.t w, we obtan: d log Λ d log w = ε F ω Ω λ(ω)ε D(p(ω)/P )dω 1 ω Ω λ(ω)ε D(p(ω)/P )dω ε Q (3) where λ(ω) denotes the share of varety ω n total expendtures. Hence: d log P d log w = 1 ε d log Λ F d log w = 7 If γ 0, consder the change n varable Λ = Λ γ. ε F ε Q ε F ω Ω λ(ω)ε D(p(ω)/P )dω ε Q 4

5 where ε D s the elastcty of 1/D. In the so-elastc case wth F (Λ) = Λ and Q(Λ) = αλ β, we have: d log P d log w = 1 β ω Ω λ(ω)ε D(p(ω)/P )dω β where β can take any value between and 1. As β s assumed to be smaller than both 1 and ε D, rcher consumers have hgher choke prces and consume a larger varety of goods an emprcallyrelevant property (see e.g. Hummels and Klenow, 2005). 2.3 Markups under monopolstc competton Under monopolstc competton, we assume that each frm has a small market share and takes the prce aggregator Λ as gven. The prce elastcty that a frm s facng s gven the elastcty of functon D, and the markup s equal to: m = ε D(p/P ) ε D (p/p ) 1 = ε D(m c/p ) ε D (m c/p ) 1 (4) Convenently, ths s the same formula as n ACDR, because markups do not depend on Q and F once we know the choke prce P. As descrbed n Fally (2018), functon D can be very flexble, leadng to flexble patterns of markups across frms. As n ACDR, we further assume that the prce elastcty ε D s weakly ncreasng n the prce p/p (a.k.a. second law of demand ). Ths addtonal assumpton ensures that equaton (4) has a unque soluton n markups for each P/c, and mplctly defnes markups as a functon of the choke-prce-to-margnal-cost rato: m = µ(p/c). (5) Ths assumpton also mples that markups are larger for low-cost frms, whch s n lne wth emprcal evdence (see e.g. De Loecker and Warzynsk, 2012). 3 Trade, prces and the gans from trade We now embed these preferences n a trade model wth monopolstc competton as n ACDR. We consder multple asymmetrc countres, ndexed by and j. There s a sngle factor, labor, that s nelastcally suppled, wth populaton L n each country. Wages and per capta ncome are denoted by w, and L s also the number of consumers n country. Demand s as descrbed above, wth a choke prce denoted by P j = w j /F (Λ j ) n each destnaton country j. We assume blateral ceberg trade costs τ j 1 between and j. Ths leads to a cost thresholds c j = P j/(w τ j ). Frms n country wth margnal cost above w c j face zero demand n country j f they also ncur blateral ceberg trade costs τ j. We further assume that the margnal cost shfter c for each frm s drawn from a Pareto dstrbuton 5

6 wth coeffcent > 1, so that the cumulatve dstrbuton of the margnal cost s gven by: G (c) = b c where b s a parameter descrbng average productvty n country. Assumng that b s small enough, ths mples that the dstrbuton of margnal costs among frms n country exportng to destnaton j s a Pareto dstrbuton wth coeffcent and upper bound w c j. Fnally, we assume for smplcty that the mass of frms N s fxed Gravty As n ACDR, we obtan a gravty equaton thanks to the assumpton that margnal costs are drawn from a Pareto dstrbuton. Ths leads to aggregate trade between countres and j equal to: X j = χn b w τ j L j Q(Λ j )P 1+ j (6) where χ s a constant term (see Appendx). Trade X j s the product of a source-specfc term N b w, a dyadc trade cost term τj, and mportantly a destnaton-specfc term L jq(λ j )Pj 1+ whch depends on both the destnaton choke prce P j and the demand shfter Q(Λ j ). Note that the latter s constant when preferences are drectly separable, as often assumed wth gravty equatons. Let us remnd that Λ j s such that the budget constrant s satsfed, and that P j = w j /F (Λ j ). Summng equaton (6) for each destnaton j, ths s equvalent to mposng: X j /L j = χq(λ j )(w j /F (Λ j )) +1( N b (w τ j ) ) = w j (7) whch determnes Λ j, condtonal on all wages. Wth gravty equatons and Pareto dstrbutons, note also that aggregate profts are proportonal to aggregate trade: Π j = ζx j wth ζ < 1. As n ACDR, proportonalty holds even f we allow for varable markups. 3.2 Effect of trade costs on prces Let us normalze wages to unty w j = 1 for a specfc country j. Denote by ε Qj and ε F j the elastctes of Q and F evaluated at Λ j. By dfferentatng equaton (7) we obtan: d log P j = ε F j d log Λ j = ε F j ( + 1)ε F j ε Qj λ j d log(w τ j ) (8) where λ j refers to share of country n total mport of country j. Usng the nsght of Arkolaks et al. (2012), ths gravty framework wth elastcty mples that the changes n mport shares capture the changes n costs: λ jd log(w τ j ) = d log λ jj, and we obtan the followng proposton: 8 Note that the mass of frms remans constant under free entry wth sunk costs and Pareto-dstrbuted productvty. 6

7 Proposton 1 For small changes n trade costs, the change n the choke prce n country j s gven by: d log P j = ε F j ( + 1)ε F j ε Qj d log λ jj (9) Condtonal on changes n mport penetraton d log λ jj and the trade elastcty, the effect on prce levels can take any value between 0 (ndrectly-addtve) and 1 d log λ jj (homothetc). In the so-elastc case wth F (Λ) = Λ and Q(Λ) = αλ β we have: d log P j = β d log λ jj (10) We obtan the same formula as n ACDR but now wth a contnuum of β s nstead of mposng β to be equal to ether 0 or 1. Ths sem-parameterzaton captures more smply the added flexblty that we obtan thanks to generalzed separablty, and s vald as long as β remans smaller than the elastcty of D (assumed to be larger than unty). It ncludes all tractable cases studed recently and provdes a mcro-foundaton for ntermedate cases. The homothetc case (β = 1) yelds the largest effect on prces unless we assume a hgher lower bound on prce elastctes ε D. Relatve to homothetc preferences, the effect on prces s mtgated by a factor 1+ wth drectly-separable preferences (β = 0). The other extreme s obtaned wth ndrectly-separable preferences (lmt case β = ) where changes n trade costs do not affect the choke prce: d log P j = 0 (Bertolett et al 2018). Dstrbuton of prces and markups In ths framework, note that the choke prce serves as a suffcent statstc for prces. Relatve to the choke prce, the dstrbuton of prces remans nvarant to changes n trade costs, gven that the shape of the Pareto dstrbuton s nvarant to ts truncaton. Hence, Proposton 1 also descrbes how trade costs shft the entre dstrbuton of prces. In partcular, as n ACDR, the dstrbuton of markups remans nvarant when margnal costs are drawn from Pareto. Markups are determned by the rato of cost to the choke prce and the functon µ(p/c). Ths functon does not vary across countres, and the dstrbuton of P/c remans also nvarant (Pareto dstrbuton bounded at unty). Ths also mples that the average markup elastcty weghted by sales s also constant across countres and fully determned by functons µ, D and the Pareto parameter as n ACDR: ρ = 1 d log µ d log v µ(v)/vd(µ(v)/v)v 1 dv 1 µ(v )/v D(µ(v )/v )(v ) 1 dv (11) One can notce that the dstrbuton of markups (under monopolstc competton) does not depend on Q(Λ) and F (Λ), the demand and prce shfters. 7

8 3.3 Gans from trade We can now examne the effect of trade costs on welfare, combnng the effect on prce levels, markups and ncome. If we mantan the same supply structure that s standard to ACDR and varous other frm-level trade models, the demand sde nfluences the gans from trade through two channels: ) the shape of the functon D determnes markups and the markup elastcty ρ; ) the demand shfters Q and F affects prce levels through the choke prce. Usng the expendture functon, we obtan the changes n welfare for country j: d log e j = (1 ρ) λ j d log(w τ j ) + ρ d log P j (12) The frst term s dentcal to ACDR and captures the drect effect of the changes n margnal costs and markups, whle the second term captures the ndrect effect on prce levels through the choke prce, whch dffers from ACDR as shown above. Wth λ j d log(w τ j ) = 1 d log λ jj (as n Arkolaks et al., 2012), the gans from trade dgt j = d log e j can be summarzed by the followng proposton: Proposton 2 For small changes n trade costs, the changes n welfare for country j are gven by: dgt j ( 1 ρ = + ρ ε F j ( + 1)ε F j ε Qj ) d log λ jj (13) Condtonal on changes n mport penetraton d log λ jj, trade elastcty and markup elastcty ρ, the range of admssble elastctes ε F j and ε Qj s such that the gans from trade can take any value between 1 ρ d log λ jj (ndrectly-addtve case) and 1 d log λ jj (homothetc case). Ths ncludes the followng three cases: dgt j = 1 ( d log λ jj ) wth homothetc preferences (ACR, ACDR) dgt j = 1 ρ + ρ +1 d log λ jj Drectly-separable preferences (ACDR) dgt j = 1 ρ d log λ jj Indrectly-separable preferences (Berlott et al, 2018) By shuttng down the effect on the choke prce, Bertolett et al (2018) obtan the lowest bound. Agan, t s practcal to assume F (Λ) = Λ and Q(Λ) = αλ β. Gans from trade then satsfy the same formula as n ACDR: dgt j ( 1 ρ = + ) ρ d log λ jj (14) + 1 β but now wth a contnuous representaton n β, where β can take any value between and 1 (and potentally hgher f t remans smaller than the prce elastcty ε D ), mplyng agan that lower and upper bounds correspond to the ndrectly-addtve and homothetc cases. 8

9 4 Emprcal mplementaton Wthout any pror on the shape of preferences, ths generalzaton leaves us wth at least an addtonal parameter to estmate f we want to quantfy the gans from trade, focusng on the the so-elastc case descrbed n equaton (14). Estmates of other parameters, the trade elastcty and the markup elastcty ρ, can be borrowed from prevous estmaton approaches gven that most of them are consstent wth the current framework. For nstance, Smonovska and Waugh (2014) provde an estmate of the trade elastcty ; Amt et al. (2014) and Arkolaks et al (2018), among many others, dscuss how to dentfy the markup elastcty ρ, whch determnes the pass-through of costs to prces. The new parameter β nfluences the gans from trade through the choke prce n partcular, and determnes how prce levels depends on ncome and trade costs. Hence, to estmate β, a natural approach s to examne how prces set by exportng frms depend on margnal costs as well as characterstcs of the destnaton country such as ncome and trade costs. As descrbed prevously, the prce of a product s determned by ts margnal cost and ts markup, whch can tself be expressed as a functon of the margnal cost and the choke prce. A frm wth a margnal cost draw c would charge a prce p jt (c) = τ jt w t c µ(p jt /(τ jt w t c)). Dfferentatng,.e. consderng small changes n costs and choke prces over tme, we obtan: log p jt (c) = (1 ρ j (c)) log(τ jt w t ) + ρ j (c) log P jt (15) The choke prce P j s not drectly observed, but ts relaton to trade costs can be ndrectly nferred through the gravty equaton. Dfferentatng equaton (7), we get: log P jt = 1 β 1 + β log w jt β log MA jt (16) where MA jt refers to market access for destnaton j: MA jt N b w τj. Ths measure of market access has been often used n the lterature and can be structurally estmated usng gravty equatons, takng the sum of estmated exporter fxed effects adjusted by trade costs. 9 averages across frms n country and combnng equatons (15) and (16) leads to: Next, takng log p jt = (1 ρ) log(τ jt w t ) + ρ 1 + β log MA jt + ρ (1 β) 1 + β log w jt (17) Ths relatonshp that can be used to dentfy both ρ and β, assumng that ncome s not perfectly correlated wth market access and that the econometrcan observes exogenous varatons n market access (e.g. as dscussed n Feyrer, 2009). In ths regresson, the frst coeffcent descrbes the pass-through of costs to prces, whch can be used to dentty the pass-through elastcty ρ, as often done n practce. In turn, the second term can ρ be used to dentfy 1+ β. A smaller β s assocated wth a lower senstvty of prces to market access n the destnaton country, and a hgher senstvty to ncome levels. For nstance, n the lmt case 9 See e.g. Reddng and Venables (2004), Head and Mayer (2010), and Fally et al. (2010) among others. 9

10 where preferences are ndrectly-addtve and β =, the coeffcent for market access s null. In the ρ general case, ths coeffcent can be used to estmate the second term n equaton (14), +1 β log λ jjt, and thus estmate the gans from trade wth generalzed separablty. 5 Concluson Ths paper examnes the mplcatons of trade for consumer welfare by consderng a flexble yet tractable demand system (Gorman, 1972, 1995, Pollak, 1972, Fally, 2018) encompassng drectlyseparable, ndrectly-separable and homothetc preferences wth a sngle aggregator. Embedded n a general-equlbrum trade model featurng gravty, such demand system allows for more flexble effects of trade and ncome on prces. An upper bound of the gans from trade s provded by the homothetc case (as n Arkolaks et al., 2012), whle a lower bound corresponds to the case of ndrectly-separable preferences (as n Bertolett et al., 2018), allowng for a contnuum of ntermedate cases. In partcular, the added flexblty s reflected n how trade affects choke prces (reservaton prces). To estmate the gans from trade and dentfy addtonal parameters that characterze the demand system, a suggeston s to explot how changes n trade costs affect markups usng frm-level data on exports from a specfc source country. References Amt, M., O. Itskhok, and J. Konngs (2014). Importers, exporters, and exchange rate dsconnect. Amercan Economc Revew 104 (7), Arkolaks, C., A. Costnot, D. Donaldson, and A. Rodríguez-Clare (2015). The elusve pro-compettve effects of trade. NBER Workng Paper Arkolaks, C., A. Costnot, and A. Rodríguez-Clare (2012). New trade models, same old gans? Economc Revew 102 (1), Amercan Behrens, K., G. Mon, Y. Murata, and J. Südekum (2014). Trade, wages, and productvty. Internatonal Economc Revew 55 (4), Bertolett, P. and F. Etro (2017). Monopolstc competton, as you lke t. Dept. of Economcs Research Paper Seres, 08/2017. Unversty Ca Foscar of Vence, Bertolett, P., F. Etro, and I. Smonovska (2018). Internatonal trade wth ndrect addtvty. Amercan Economc Journal: Mcroeconomcs 10 (2), Chaney, T. (2008). Dstorted gravty: the ntensve and extensve margns of nternatonal trade. Amercan Economc Revew 98 (4), Costnot, A. and A. Rodríguez-Clare (2018). The US gans from trade: Valuaton usng the demand for foregn factor servces. Journal of Economc Perspectves 32 (2), De Loecker, J. and F. Warzynsk (2012). Markups and frm-level export status. Amercan economc revew 102 (6), Fally, T. (2018). Integrablty and generalzed separablty. NBER Workng Paper

11 Fally, T., R. Pallacar, and C. Terra (2010). Economc geography and wages n brazl: Evdence from mcro-data. Journal of Development Economcs 91 (1), Feenstra, R. C. (2018a). Alternatve sources of the gans from nternatonal trade: Varety, creatve destructon, and markups. Journal of Economc Perspectves 32 (2), Feenstra, R. C. (2018b). Restorng the product varety and pro-compettve gans from trade wth heterogeneous frms and bounded productvty. Journal of Internatonal Economcs 110, Feyrer, J. (2009). Trade and ncome explotng tme seres n geography. Techncal report. NBER Workng Paper Gorman, W. M. (1972). Condtons for generalzed addtve separablty. Unpublshed transcrpt, now prnted n Gorman (1995). Gorman, W. M. (1995). Collected Works of WM Gorman: Separablty and Aggregaton, Volume 1. Oxford Unversty Press. Head, K. and T. Mayer (2010). Gravty, market potental and economc development. Journal of Economc Geography 11 (2), Hummels, D. and P. J. Klenow (2005). The varety and qualty of a naton s exports. Amercan Economc Revew 95 (3), Matsuyama, K. and P. Ushchev (2017). Beyond CES: Three alternatve classes of flexble homothetc demand systems. Workng paper. Meltz, M. J. (2003). The mpact of trade on ntra-ndustry reallocatons and aggregate ndustry productvty. Econometrca 71 (6), Pollak, R. A. (1972). Generalzed separablty. Econometrca: Journal of the Econometrc Socety, Reddng, S. and A. J. Venables (2004). Economc geography and nternatonal nequalty. Journal of nternatonal Economcs 62 (1), Smonovska, I. (2015). Income dfferences and prces of tradables: Insghts from an onlne retaler. The Revew of Economc Studes 82 (4), Smonovska, I. and M. E. Waugh (2014). The elastcty of trade: Estmates and evdence. Journal of nternatonal Economcs 92 (1), Appendx A) Effect of ncome on the prce aggregator Dfferentatng equaton (1) yelds: ω Ω [ ] d log Λ λ(ω) ε Q d log w ε d log Λ Dε F j d log w + ε D dω = 1 We obtan equaton (3) by solvng for d log Λ d log w. Note that ε Q and ε F product varetes and depend on F (Λ)p(ω)/w = p(ω)/p. depend only on Λ whle ε D vares across 11

12 B) Gravty equaton Margnal costs are gven by w c where c s drawn from a cumulatve dstrbuton G (c) = b c. Frms wth draws above c j = P j/(w τ j ) do not sell to market j. As n ACDR, t s then convenent to make a change n varable and defne v = c j /c wth a cumulatve dstrbuton 1 v wth the adjusted mass of frms N b Pj (w τ j ). For frms n country exportng to j wth margnal cost c = c j /v = P j/(w τ j v), ther prce s P j µ(v)/v and ther sales are gven by: x(v)q(λ j )P j = µ(v)/vd(µ(v)/v) Q(Λ j )P j where µ(v) s defned mplctly by equaton (5). Summng up across all frms n country, total exports to country j correspond to: X j = N b P j (w τ j ) + v=1 x(v) Q(Λ j )P j v 1 dv = χn b (w τ j ) Q(Λ j )P 1+ j wth wth constant term χ + v=1 µ(v)/vd(µ(v)/v)v 1 dc. Ths gves equaton (6) n the text. C) Proposton 1 Dfferentatng equaton (7) w.r.t Λ j yelds: (ε Qj ( + 1)ε F j ) d log Λ λ j d log(w τ j ) = 0 d log Λ j = λ j d log c j ( + 1)ε F j ε Qj and thus we obtan equaton (8) n the text for d log P j = ε F j d log Λ j. Next, we can apply the same trck as n Arkolaks et al. (2012) to obtan: λ j d log(w τ j ) = 1 λ j d log(w τ j ) = 1 λ j d log(λ jj /λ j ) = 1 d log λ jj 1 λ j d log λ j = 1 d log λ jj 1 d(λ j ) = 1 d log λ jj D) Proposton 2 Gven that we normalze ncome w j = 1 n country j, welfare changes are gven by the change n the expendture functon. Applyng Shephard s Lemma, ths s equal to the change n log prces weghted by expendture shares: d log e j = λ j (ω)d log p j (ω) ω Ω = (1 ρ) λ j d log(w τ j ) + ρ d log P j where ρ s the sales-weghted average of the markup elastcty defned n equaton (11). We obtan Proposton 2 as a corollary of Proposton 1 combned wth the welfare change above. 12