Grounded by Gravity:

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1 Grounded by Gravty: A Well-Behaved Trade Model wth Industry-Level Economes of Scale Konstantn Kucheryavyy U Tokyo Gary Lyn UMass Lowell Andrés Rodríguez-Clare UC Berkeley and NBER October 7, 2016 Abstract Although economsts have long been nterested n the mplcatons of Marshallan externaltes (.e., ndustry-level external economes of scale) for tradng economes, the large number of equlbra that they typcally mply has kept such externaltes out of the recent quanttatve trade lterature. Ths paper presents a multndustry trade model wth ndustry-level economes of scale that nests a Rcardan model wth Marshallan externaltes as well as mult-ndustry versons of Krugman (1980) and Meltz (2003). The behavor of the model depends on two ndustry-level elastctes: the trade elastcty and the scale elastcty. We show that there s a unque equlbrum f the product of the trade and scale elastctes s weakly lower than one n all ndustres. The welfare analyss reveals that f ths condton s satsfed then all countres gan from trade, even when the scale elastcty vares across ndustres. The presence of scale economes tends to lower the gans from trade except f the country specalzes n ndustres wth relatvely hgh scale elastctes. On the other hand, scale economes amplfy the gans from trade lberalzaton except f t leads to reallocaton towards ndustres wth relatvely low scale elastctes. We thank James Anderson, Costas Arkolaks, Gaurab Aryal, Lorenzo Calendo, Arnaud Costnot, Rchard Cottle, Svetlana Demdova, Dave Donaldson, Jonathan Eaton, Gene Grossman, Nal Kashaev, Tm Kehoe, Hdeo Konsh, Sam Kortum, Beresford Parlett, Donald Rchards, Steve Reddng, Mchael Tsatsomeros, Guang Yang, X Yang, and Stephen Yeaple for valuable dscussons, and Kala Krshna for pontng us back to perfect competton. We thank Maurco Ulate and Pyush Pangrah for valuable research assstance. All errors are, of course, our own.

2 GROUNDED BY GRAVITY 1 1. Introducton The feld of nternatonal trade has made great strdes n recent years by mappng theory to data n the new quanttatve trade models (or so-called gravty models ). Ths has led to mportant nsghts nto the consequences of globalzaton. But a fundamental ssue has been mssng from these models: the role of localzed and ndustry-specfc external economes of scale. These externaltes played a large role n the world economy at the tme of Marshall (1890, 1930), and recent anecdotal and emprcal evdence suggests that they play, f anythng, an even larger role n the global economy today. 1 There s good reason for ths omsson. Early models yelded some dscomfortng results, ncludng a bewlderng varety of [multple] equlbra (Krugman, 1995) so that trade patterns need not conform to comparatve advantage, along wth the paradoxcal mplcaton that trade motvated by the gans from concentratng producton need not beneft the partcpatng countres (Grossman and Ross-Hansberg, 2010). At the heart of these pathologes seemed to lay the compatblty assumpton of ncreasng returns and perfect competton (see Chpman, 1965), namely that frms take productvty as gven even though productvty depends on total ndustry output. Ths leads to a crcularty whereby the scale of an ndustry affects ts productvty, whle an ndustry s productvty affects ts scale through the mpact on the pattern of comparatve advantage and specalzaton. In the standard analyss, ths leads to multple equlbra. 2 Grossman and Ross-Hansberg (2010, henceforth GRH) recently proposed a twocountry Rcardan model wth natonal ndustry-level external economes of scale (or Marshallan externaltes), whch attacks ths compatblty assumpton head-on. Instead of perfect competton, GRH assume Bertrand competton so that frms n each ndustry understand the mplcatons of ther decsons on ndustry output and productvty, ensurng that n equlbrum we have the rght allocaton of ndustres across countres n a smlar fashon to that of the constant returns to scale framework of Dornbusch, Fscher and Samuelson (1977). Whle the framework successfully elmnates the pathologes n a world free of trade costs, Lyn and Rodríguez-Clare (2013a,b) llustrate crcumstances under whch multple equlbra arse n the presence of trade costs. Coupled wth the fact that the equlbrum has mxed strateges for some levels of trade 1 See Krugman (2011) for a nce exposton of recent anecdotal evdence. For emprcal evdence see, for nstance, Caballero and Lyons (1989, 1990, 1992), Chan, Chen and Cheung (1995), Segoura (1998), and Henrksen, Steen and Ulltvet-Moe (2001). 2 See early work explorng ths by Graham (1923), Ohln (1933), Matthews (1949), Kemp (1964), Melvn (1969), Markusen and Melvn (1981), and Ether (1982a,b).

3 2 KUCHERYAVYY-LYN-RODRíGUEZ-CLARE costs, the framework quckly becomes ntractable, wth lttle hope of extendng t to a mult-country settng wth trade frctons. In ths paper we present a Rcardan model wth Marshallan externaltes that admts a unque equlbrum under ntutve parameter restrctons. Unlke GRH, we leave the compatblty assumpton ntact and approach the problem from a dfferent angle by relaxng the mplct assumpton n the standard framework (and n GRH) that frms wthn each ndustry are producng a homogeneous good. In partcular, we allow for ntra-ndustry heterogenety as n Eaton and Kortum (2002, henceforth EK) and fnd that ths adds some curvature that helps n establshng unqueness of equlbrum as long as the strength of Marshallan externaltes s not too hgh. The framework yelds the standard gravty-type equaton and so provdes a platform to assess quanttatvely the mportance of these externaltes for the welfare effects of trade. 3 The system of equatons that characterzes the equlbrum of the Rcardan multndustry model wth Marshallan externaltes turns out to be somorphc to the equlbrum system of a more general verson of the mult-ndustry Krugman (1980) model of product dfferentaton wth nternal economes of scale. 4 The exstence and unqueness result that we prove for our Rcardan settng can then be seamlessly appled to the mult-ndustry Krugman model. As far as we know, we are the frst to establsh unqueness of equlbrum for ths general case. Not surprsngly, the somorphsm extends also to the mult-ndustry Meltz (2003) model f the productvty dstrbuton s Pareto as n Chaney (2008) and the fxed exportng costs are pad n unts of labor of the destnaton country. 5 The common mathematcal structure that characterzes the equlbrum n all these mult-ndustry gravty models s governed by two elastctes that can vary across ndustres: the trade elastcty and the elastcty of productvty wth respect to ndustry sze, whch we wll refer to as the scale elastcty. The condton for unqueness s that (n all ndustres) the product of these two elastctes s not hgher than one. In the Rcardan 3 Our analyss restrcts to the case of Marshallan externaltes, whch operate nsde each ndustry. An alternatve case s the one n whch some of the externaltes operate across ndustres. Yatsynovch (2014) has recently shown condtons under whch a model wth such cross-ndustry externaltes exhbts a unque equlbrum for the case wth frctonless trade. 4 Abdel-Rahman and Fujta (1990), Allen et al. (2015) and Reddng (2016) explore smlar somorphsms for spatal equlbrum models n the economc geography lterature. 5 Somale (2014) ntroduces sector-specfc nnovaton nto a mult-sector Eaton and Kortum (2002) model (va mechansms from Eaton and Kortum, 2001) to quantfy ts mplcatons for welfare. Interestngly, although the model n Somale (2014) s dynamc, the balanced growth path s also characterzed by the same system of equatons as all the models that we consder n ths paper, and so our results extend to ths case as well.

4 GROUNDED BY GRAVITY 3 model the scale elastcty s gven drectly by the strength of Marshallan externaltes, so the condton for unqueness s that the strength of these externaltes s not hgher than the nverse of the trade elastcty. In the Krugman or Meltz-Pareto models the scale elastcty s gven by the nverse of the trade elastcty, hence we are always at the edge of the regon of unqueness. One can easly add flexblty to the Krugman and Meltz-Pareto models to break the tght lnk between the two elastctes. For example, f we allow the elastcty of substtuton across varetes from dfferent countres to dffer from the elastcty of substtuton across varetes from the same country (wth nested CES preferences) then the product of the scale and trade elastctes can be dfferent than one. 6 When formulatng the equlbrum condtons n our model, we explctly allow for corner equlbra n whch ndustres shut down n some countres. We show that f the product of trade and scale elastctes s less than one then every country s actve n all ndustres, whle f the product s one then the equlbrum may exhbt corners. In partcular, as s known n the lterature, the mult-ndustry Krugman model can have countres completely specalzed n some subset of ndustres as an equlbrum outcome. Remarkably, however, the exstng lterature lacks a proof of unqueness of equlbra n the mult-ndustry Krugman model whle approprately dealng wth the complementarly slackness condtons relevant for ths case. The two papers that address the ssues of exstence and unqueness n the context of the usual mult-ndustry Krugman model (n whch the product of trade and scale elastctes s exactly one) are Hanson and Xang (2004) and Behrens, Lamorgese, Ottavano and Tabuch (2009). Hanson and Xang (2004) show exstence and unqueness for the case of two countres and a contnuum of ndustres under the explct assumpton that both countres produce n all ndustres (.e., no corner allocatons). Behrens et al. (2009) consder the case of many countres and two ndustres one ndustry beng the usual outsde good ndustry that pns down wages and show exstence of equlbrum whle allowng the equlbrum to exhbt corner allocatons. Ther unqueness proof, however, reles on the assumpton that there are no corner allocatons. Note also that snce one ndustry s modeled as an outsde good, the framework can essentally be vewed as one wth exogenous wages, multple countres, and one ncreasng-returns 6 Alternatvely, one can allow for heterogenety n worker ablty across ndustres as n Galle, Rodríguez-Clare and Y (2015). Ths ntroduces nter-ndustry curvature nto the model and reduces the scale elastcty n all ndustres below the nverse of the trade elastcty, thereby helpng ensure unqueness.

5 4 KUCHERYAVYY-LYN-RODRíGUEZ-CLARE ndustry. In ths paper we show exstence and unqueness of equlbrum n a settng wth multple ndustres and two countres whle allowng for complete specalzaton and for endogenous wages. Our exstence result s vald for more than two countres, whle for now we have only been able to extend the unqueness results to more than two countres under frctonless trade or wth exogeneous wages. Whle numercal smulatons ndcate that the equlbrum s also unque n the context of endogenous wages and multple countres, the theoretcal dffculty les n the fact that the labor excess demand system does not satsfy the gross substtutes property a suffcent condton for unque wages whch s often satsfed n smlar envronments. Provng unqueness n ths settng may requre applyng more powerful technques that we are currently studyng. 7 In the fnal secton we use the unfed framework to study the mplcatons of scale economes for the welfare effects of trade. We frst establsh that as long as we are n the regon of unqueness then all countres gan from trade. Ths s so even f the scale elastcty dffers across ndustres, for example because of cross-ndustry varaton n the strength of Marshallan externaltes n the Rcardan model. Ths s noteworthy n lght of prevous results wth ths type of model where countres could lose from trade. We extend the suffcent statstcs approach to the quantfcaton of the gans from trade n Arkolaks et al. (2012) to mult-ndustry models wth scale economes. somorphsm across models stll apples n ths settng n the sense that, for the same ndustry-level trade and scale elastctes, the dfferent models we consder delver the same gans from trade and the same counterfactual mplcatons gven ndustry-level data on trade, expendture and revenue shares. We next show that, perhaps surprsngly, f the scale elastcty s the same across ndustres, the gans from trade are lower wth scale economes than wthout. In contrast, for a smple case that we can solve analytcally, the gans from trade lberalzaton are hgher wth scale economes than wthout. The opposte consequences of scale effects on the gans from trade and on the gans from trade lberalzaton come from the fact that when we compute gans from trade we take trade shares (from the data) as gven, whle when we compute gans from trade lberalzaton we allow trade shares to endogenously respond to the declne n trade costs. 8 In other words, gans from trade reveal the 7 The exstence and unqueness result for a mult-sector gravty model n Corollary 1 of Allen et al. (2014) s not helpful n our settng because the condtons they mpose on ther mult-sector gravty model rule out sector-level economes of scale. 8 Somethng smlar happens n the standard one-ndustry gravty model, where a hgher trade elastcty leads to lower gans from trade and hgher gans from trade lberalzaton. As explaned by Costnot The

6 GROUNDED BY GRAVITY 5 welfare consequences of a counterfactual exercse n whch we observe changes n trade shares, whle gans from trade lberalzaton does the same for a counterfactual exercse n whch we observe changes n trade costs of course, the mpled changes n ether trade shares or trade costs are dfferent n these two cases. We next revst a classcal result due to Venables (1987) that, f wages are pnned down by an outsde good, countres lose from unlateral trade lberalzaton and from a foregn technologcal mprovement n a monopolstcally compettve sector modeled as n Krugman (1980). We show that ths result generalzes to any source of scale economes (e.g., Marshallan externaltes) as long as the product of the trade and scale elastctes s above a threshold value that s a functon of sector-level mport and export shares. We complement our exploraton of the effect of scale economes on the gans from trade by applyng our framework to data on 31 ndustres from the World Input Output Database (WIOD, Tmmer et al., 2015) n We start by focusng on the case wth common trade and scale elastctes across ndustres. As explaned above, the presence of scale economes leads to a declne n the gans from trade, but now we also see that ths declne s more pronounced n countres that have a hgher degree of specalzaton across ndustres. Thus, for example, for the country wth the hghest degree of ndustry specalzaton, Korea, the gans from trade decrease from 6.6% to 4.1%, whle they barely decrease for the country wth the lowest degree of ndustry specalzaton, Brazl. We then study how the gans from trade are affected by scale economes when the scale elastcty vares across ndustres. We consder two possbltes. The frst s that scale economes are present only n manufacturng ndustres a typcal case consdered n the lterature (see, for nstance, Ether, 1982a,b). Not surprsngly, relatve to the case wth no scale economes, gans from trade ncrease for countres that specalze n manufacturng and the opposte happens for countres that specalze away from manufacturng. For example, gans from trade ncrease from 3 to 3.5% for Chna, whle they decrease from 5.7 to 3.5% for Greece. The second possblty we consder s that scale elastctes are nversely proportonal to trade elastctes (as n the standard mult-ndustry Krugman or Meltz-Pareto models), wth trade elastctes varyng across ndustres and calbrated to those estmated by Calendo and Parro (2015). We fnd that countres that specalze n ndustres wth lower than average scale economes gan less and Rodríguez-Clare (2014), gans from trade are lower when the trade elastcty s hgher snce ths makes t easer to substtute foregn for domestc goods when we move to autarky. In contrast, gans from trade lberalzaton are hgher when the trade elastcty s hgher snce ths allows for a stronger response of trade shares to the declne n trade costs.

7 6 KUCHERYAVYY-LYN-RODRíGUEZ-CLARE from trade wth scale economes than wthout, but the opposte may happen for countres that specalze n ndustres wth hgher than average scale economes. Thus, for example, movng from a model wthout scale economes to one wth scale economes leads to a declne n the gans from trade n Greece from 14.5% to 5.5% but an ncrease n the gans from trade n Japan from 2.4% to 6.1%. We use the model to quantfy the welfare mplcatons of unlateral trade lberalzaton and foregn productvty gans n an envronment wth economes of scale, comparng to the results to those n an envronment wthout economes of scale. To lnk ths exercse to the theoretcal analyss nspred by Venables (1987) we assume agan that the manufacturng sector exhbts scale economes whle all other sectors do not. We fnd that gans from unlateral trade lberalzaton n manufacturng decrease as we allow for scale economes n that sector, but (n the regon of unqueness) the gans are always postve. We show that ths arses because of wage adjustments that are ruled out n the Venables (1987) type analyss. On the other hand, we fnd that most countres experence losses from an mprovement n Chnese manufacturng productvty. Fnally, we explore the role of economes of scale n explanng trade flows and ndustrylevel specalzaton n the data. We fnd that f scale economes are as strong as those n the Krugman model then most of the ndustry-level specalzaton that we see n the data s due to economes of scale rather than pure Rcardan comparatve advantage. Costnot and Rodríguez-Clare (2014) compute gans from trade and gans from the declne n trade costs for economes wth and wthout scale economes. Compared to that paper, we further establsh analytcally that all countres gan from trade as long as the condtons for equlbrum unqueness are satsfed, we connect a country s declne n the gans from trade to ts degree of ndustry specalzaton, and we analyze how varyng scale elastctes across ndustres nteract wth a country s nter-ndustry trade pattern to affect ts gans from trade. Somale (2014) also analyzes how economes of scale matter for ndustry-level specalzaton. The dfference s that whereas he focuses on the the way n whch economes of scale affect the varance of comparatve advantage, we compare measures of trade and specalzaton between the data and those that would arse n a counterfactual world where everythng s the same except that there are no economes of scale.

8 GROUNDED BY GRAVITY 7 2. A Mult-Industry Gravty Model wth Scale Economes We frst present the key equlbrum equatons of the model and then dscuss how these equatons arse n three dfferent settngs: () our mult-ndustry Rcardan model wth Marshallan externaltes; () the mult-ndustry Krugman (1980) model wth possbly dfferent elastctes of substtuton between varetes from the same and dfferent countres; and () the mult-ndustry Meltz (2003) model wth Pareto-dstrbuted productvty as n Chaney (2008) and also allowng for dfferent elastctes of substtuton between varetes from the same and dfferent countres. There are N countres ndexed by n, and l, and K ndustres or sectors ndexed by k. The only factor of producton s labor, whch s mmoble across countres and perfectly moble across ndustres wthn a country. We use L and w to denote the nelastc labor supply and the wage level n country, respectvely. Each country has a representatve consumer wth upper-ter Cobb-Douglas preferences wth ndustry-level expendture shares β,k (0, 1) for all (, k) wth K k=1 β,k = 1 for all. Trade costs are of the standard ceberg type, so that delverng a unt of any ndustry-k-good from country to country n requres shppng τ n,k 1 unts of the good, wth τ,k = 1 for all and all k and τ nl,k τ n,k τ l,k for all n, l, and k. Let X n,k denote country-n s total expendture on ndustry k and let λ n,k denote the share of ths expendture devoted to mports from country. Balanced trade mples X n,k = β n,k w n Ln. We focus on models that generate ndustry-level economes of scale and a log-lnear gravty equaton for ndustry-level trade shares. Below we show that our Rcardan model wth ndustry-level external economes of scale as well as Krugman (1980) and Meltz (2003) satsfy ths crtera. Economes of scale are captured by an ndustry-level productvty shfter that can vary wth total ndustry employment accordng to S,k L ψ k,k, where S,k s a constant, L,k denotes total employment n ndustry (, k), and ψ k s the scale elastcty n ndustry k, whch s assumed to be common across countres. Industrylevel trade shares are gven by λ n,k = ( w τ n,k / S,k L ψ k,k ( w l τ nl,k / S l,k L ψ k l,k l ) εk ) εk, where ε k s the trade elastcty n ndustry k, defned formally by ε k ln(λ n,k/λ nn,k ) ln τ n,k.

9 8 KUCHERYAVYY-LYN-RODRíGUEZ-CLARE Lettng α k ε k ψ k and S,k S ε k,k, we rewrte trade shares more convenently as λ n,k (w, L k ) = S,kL α k,k (w τ n,k ) ε k l S l,kl α k l,k (w lτ nl,k ) ε, (1) k where w (w 1,..., w N ) s the vector of wages and L k (L 1,k,..., L N,k ) s the vector of labor allocatons to ndustry k across all countres. In turn, the prce ndex for ndustry k n country n s P n,k = µ n,k ( l S l,k L α k l,k (w lτ nl,k ) ε k) 1/εk, (2) and the aggregate prce ndex s P n = β K n k=1 P β n,k n,k, where µ n,k and β n are some constants. 9 We now ntroduce ndustry and labor market clearng condtons. In contrast to mult-ndustry gravty models wthout scale economes (e.g., Donaldson (2016), Costnot et al. (2012)), here we can have equlbra wth corner allocatons (.e., L,k = 0 for some k and for some, but not all, ), so we need to be careful when formulatng the market clearng condtons. Wth ths n mnd, we specfy the market clearng condton for any ndustry (, k) as a set of complementary slackness condtons, L,k 0, G,k (w, L k ) 0, L,k G,k (w, L k ) = 0, (3) where G,k (w, L k ) w 1 λ n,k (w, L k )β n,k w n Ln (4) L,k s the excess of the wage over revenue per worker n ndustry (, k). Note that for postve labor allocatons equaton (3) mples G,k (w, L k ) = 0, whch can be reformulated as w L,k = n λ n,kβ n,k w n Ln, a standard ndustry clearng condton. 10 n 9 The constant µ n,k wll be specfed below for each model, whle β n s the standard Cobb-Douglas term β n k β β n,k n,k. 10 A subtle ssue arses here wth the evaluaton of G,k (w, L k ) and L,k G,k (w, L k ) at ponts wth L,k = 0. If we thnk of the codoman of these functons as the set of real numbers then L k wth L,k = 0 (for at least some, but not all ) s not n ther doman. To avod ths, we defne the codoman as the extended real [ number lne R {, + } and we defne G,k (w, L k ) and L,k G,k (w, L k ) for L,k = 0 by lm x Lk w 1 ] [ n x λ n,k(w, x)β n,k w n Ln and lm x Lk x w 1 n x λ n,k(w, x)β n,k w n Ln ], re- spectvely. (Of course, for any pont wth L,k > 0 these alternatve defntons are perfectly consstent wth the ones n the text.) For each k, we stll leave the pont L k wth L,k = 0 for all outsde the doman. The formal defntons are n Appendx A.

10 GROUNDED BY GRAVITY 9 Fnally, the labor-market clearng condton for any country s smply L,k = L. (5) k Denote by L (L 1,..., L k ) the vector of labor allocatons across ndustres. The equlbrum of the economy s a wage vector and labor allocaton (w, L) R N ++ ( ) R NK + \ Z NK 0 such that (3) holds for all (, k) and (5) holds for all, where Z NK 0 { (x 1,..., x K ) R NK + : x k = 0 for some k } s the set of labor allocatons wth zero total labor (across countres) devoted to some ndustres A Rcardan Model wth Marshallan Externaltes We now show how the mult-ndustry Eaton and Kortum (2002, henceforth EK) model as developed by Costnot, Donaldson and Komunjer (2012, henceforth CDK), but extended to allow for Marshallan externaltes leads to the equlbrum condtons presented above. Each ndustry s composed of a contnuum of goods ω [0, 1]. Preferences are Cobb- Douglas across ndustres wth weghts β,k, and CES across goods wthn each ndustry k wth elastcty of substtuton σ k. The producton technology exhbts constant or ncreasng returns to scale due to natonal external economes of scale at the ndustry level (.e., Marshallan externaltes). In partcular, labor productvty for good ω n ndustry (, k) s z,k (ω)l φ k,k, where z,k (ω) s an exogenous productvty parameter, L,k s the total labor allocated to ndustry (, k), and φ k s the ndustry specfc parameter that governs the strength of Marshallan externaltes. We model z,k (ω) as n EK: z,k (ω) s ndependently drawn from a Fréchet dstrbuton wth shape parameter θ k and scale parameter T,k, and we assume that θ k > σ k 1. There s perfect competton, and the postve effect of ndustry sze on productvty, L φ k,k, s external to the frm. Thus, frms take as gven both prces and unt costs, whch are gven by c n,k (ω) = τ n,kw. Ths mples that p n,k (ω) = c n,k (ω). Snce z,k (ω)l φ k,k consumers can shop for the best deal around the world, prces must satsfy p n,k (ω) = mn 1 N {p n,k (ω)}. Followng the same procedure as n EK, trade shares can be shown

11 10 KUCHERYAVYY-LYN-RODRíGUEZ-CLARE to satsfy wth prce ndces gven by λ n,k = T,kL θ kφ k,k l T l,kl θ kφ k l,k (w τ n,k ) θ k (w l τ nl,k ) θ k P n,k = µ Rc k ( l T l,k L θ kφ k l,k (w l τ nl,k ) θ k) 1/θk, ( ) 1 where µ Rc 1 σ k Γ k +θ 1 σ k θ k, wth Γ beng the Gamma functon whch typcally arses n ths Rcardan settng. These two equatons collapse to the expressons for trade shares and ndustry prce ndexes n equatons (1) and (2) by settng wth S,k = T,k, ε k = θ k, ψ k = φ k and µ n,k = µ Rc k. See the frst row of Table 1. Fnally, the equlbrum condton (3) can be seen as capturng the standard complementary slackness condton n the Rcardan model requrng the prce to be weakly lower than the unt cost, wth equalty f there s postve producton n the ndustry. Multplyng both the prce and the unt cost by labor productvty (adjusted by trade costs), ths s the same as requrng that revenue per worker be weakly lower than the wage, wth equalty f there s postve employment n the ndustry. Table 1: Mappng to Dfferent Models Model Trade elastcty, ε k Scale elastcty, ψ k α k CDK wth ME θ k φ k θ k φ k 1 Mult-Sector Krugman σ k 1 σ k 1 1 Mult-Sector Meltz- 1 θ Pareto Model k θ k 1 Generalzed Mult-Sector Krugman Generalzed Mult-Sector Meltz-Pareto η k 1 1 σ k 1 η k 1 σ k 1 ( θ k ) 1 ( 1 ) 1+θ 1 k η k 1 1 θ k 1+θ 1 σ k 1 k η k 1 1 σ k A Krugman Model wth Two-Ter CES preferences Here we present a mult-ndustry Krugman model wth an added layer of product dfferentaton so that the elastcty of substtuton across varetes from dfferent countres s

12 GROUNDED BY GRAVITY 11 allowed to dffer from the elastcty of substtuton across varetes from the same country (wth nested CES preferences). We agan show that ths model leads to the equlbrum condtons n equatons (3) and (5). There s a contnuum of dfferentated varetes wthn each ndustry. Preferences are mult-tered: Cobb-Douglas across ndustres wth weghts β,k, CES across country bundles wthn an ndustry wth elastcty η k, and CES across varetes wthn a country bundle wth elastcty of substtuton σ k > 1. Let A,k be the exogenous productvty n (, k) whch s common across frms n that ndustry, let F,k denote the fxed cost (n terms of labor) assocated wth the producton of any varety n (, k), and let M,k the measure of goods produced n (, k). There s monopolstc competton and trade shares are λ n,k = (P n,k /P n,k ) 1 η k, where P n,k = M 1/(1 σ k),k ( σ k w τ n /A,k ) s the prce ndex n country n of country varetes of ndustry k, σ k σ k / (σ k 1) s the mark-up, and P n,k = ( P ) 1 η 1/(1 ηk ) k n,k. We now solve for equlbrum varety M,k as a functon of ndustry employment L,k and then use the result to derve an expresson for trade shares for ths model. Varable profts n (, k) are smply total ndustry revenues dvded by σ k. Lettng Π,k be total profts net of fxed costs n ndustry (, k), we then have Π,k = n λ n,kx n,k /σ k w M,k F,k. If L,k > 0 then free entry mples zero profts so total revenues must equal total wage payments n ndustry (, k), n λ n,kx n,k = w L,k. Combned wth Π,k = 0 we then have M,k = L,k /σ k F,k. Trade shares are then λ n,k = Aη k 1,k l Aη k 1 l,k F η k 1 σ k 1,k F η k 1 σ k 1 l,k η k 1 σ L k 1,k L η k 1 σ k 1 l,k (w τ n,k ) (η k 1) (w l τ nl,k ) (η k 1) wth prce ndces gven by where µ Krug k P n,k = µ Krug k = σ 1 σ k 1 k ( l A η k 1 l,k F η k 1 σ k 1 l,k ) 1/(ηk 1) η k 1 σ L k 1 l,k (w l τ nl,k ) (η k 1), σ k. These two equatons collapse to the expressons for trade shares F η k 1 and ndustry prce ndexes n equatons (1) and (2) by settng S,k = A η k 1 σ k 1,k,k, ψ k = (σ k 1) 1, ε k = (η k 1) and µ n,k = µ Krug k. Note also that f we set σ k = η k for all k then ths s just the standard mult-ndustry Krugman model, whle f σ k, then (η k 1)/(σ k 1) 0 and we obtan the mult-ndustry Armngton model. See rows 2

13 12 KUCHERYAVYY-LYN-RODRíGUEZ-CLARE and 4 of Table To deal wth the possblty of corner labor allocatons under monopolstc competton, we requre that profts per frm n ndustry (, k) be weakly lower than zero, wth strct equalty f L,k > 0, exactly as captured by the complementary slackness condtons n (3) A Meltz-Pareto Model wth Two-Ter Preferences We now brefly present a model à la Meltz (2003) wth Pareto dstrbuted productvty and the same preferences as n the Krugman model above and show that t leads to the same equlbrum condtons (3) and (5). 12 After payng a fxed entry cost F,k n unts of labor n country, frms are able to produce a varety n ndustry (, k) wth labor productvty drawn from a Pareto dstrbuton wth shape parameter θ k > σ k 1 and locaton parameter b,k. Frms from can then pay a fxed marketng cost f n,k n unts of labor of n to serve that market. 13,14 In Appendx B we show that ths leads to trade shares λ n,k = bθ kξ k,k l bθ kξ k l,k F ξ k,k Lξ k,k (w τ n,k ) θ kξ k F ξ k l,k Lξ k l,k (w l τ nl,k ) θ kξ k and prce ndces P n,k = µ Mel n,k ( l b θ kξ k l,k ) 1/θkξk F ξ k l,k Lξ k l,k (w l τ nl,k ) θ kξ k 1 where ξ k ( 1+θ 1 ), µ Mel k η k 1 1 n,k σ k 1 µ Mel k ( ) ( fn,k β n,k L n ) 1 σ k 1 1 θ k and µ Mel k s some constant 11 Is straghtforward to ncorporate Marshallan externaltes nto the mult-ndustry Krugman model presented above. For nstance, lettng A,k Ã,kL φ k,k yelds a settng wth scale and trade elastctes ψ k = (σ k 1) 1 + φ k and ε k = η k 1, respectvely. 12 Feenstra et al. (2014) also consder a mult-ndustry Meltz-Pareto model wth possbly dfferent elastctes of substtuton across varetes from dfferent countres and across varetes from the same country. 13 To smplfy the analyss, we assume that the fxed marketng cost to serve destnaton n does not vary across orgns. Allowng these fxed costs to vary across country pars would mply that nstead of a term S,k we would have a term S n,k that vares across country pars, but ths would not change any of our man conclusons below. 14 The assumpton that fxed marketng costs are pad n unts of labor of the destnaton country s crtcal for the result that ths model collapses to the general structure ntroduced above. Ths s related to the dscusson n ACR about how ther macro-level restrcton R3 obtans n the Meltz-Pareto model f and only f the fxed cost s pad n unts of labor of the destnaton country.

14 GROUNDED BY GRAVITY 13 defned n Appendx B. These two equatons collapse to the expressons for trade shares and ndustry prce ndexes n equatons (1) and (2), respectvely, by settng S,k = b θ kξ k,k F ξ k,k, ψ k = 1/θ k, ε k = θ k ξ k and µ n,k = µ Mel n,k. Note also that f we set σ k = η k for all k then ξ k = 1 and ths model s just a mult-ndustry verson of the Meltz-Pareto model n Arkolaks et al. (2008). See rows 3 and 5 n Table Characterzng Equlbrum To characterze the equlbrum we proceed n two steps: we frst characterze the equlbrum labor allocatons gven wages, and then we characterze wages that satsfy labor market clearng gven the correspondng equlbrum labor allocatons. Two-Step Equlbrum Defnton. The equlbrum labor allocatons for some wage vector w R N ++ are gven by L R NK + \ Z NK 0 that satsfy (3) for all (, k). Let L(w) be the set of such equlbrum allocatons. A wage vector w R N ++ s an equlbrum wage vector f there exsts an element L L(w) such that L also satsfes (5) for all. Note that gven wages, for each ndustry k we have a system of N nonlnear complementary slackness condtons n L,k for = 1,..., N specfed by (3). For the frst step we explot the fact that ths system s ndependent across k. We now ntroduce some addtonal notaton and defntons. Interor, Corner and Complete Specalzaton Allocatons. An allocaton L k s an nteror allocaton f L,k > 0 for all ; an allocaton L k s a corner allocaton f L,k = 0 for at least one ; and an allocaton L k s a complete specalzaton allocaton f there s a unque (k) such that L,k = 0 for all (k). 15 Industry-Level Equlbrum Labor Allocatons. Gven wage w, L k (w) denotes the set of equlbrum labor allocatons n ndustry k,.e., for any L k L k (w), L k satsfes complementary slackness condtons (3) for ndustry k Step 1: Equlbrum Labor Allocatons Before we proceed, let us ntroduce an addtonal assumpton on the matrx of trade costs whch we employ to prove our results n the case of α k = 1 for some k: 15 Note that there are many complete specalzaton allocatons. For nstance, t could be the case that producton n ndustry 1 s concentrated solely n country 1 ( (1) = 1) or n country 2 ( (1) = 2), and so on.

15 14 KUCHERYAVYY-LYN-RODRíGUEZ-CLARE Assumpton 1. Matrx s non-sngular. τ ε k 11,k... τ ε k 1N,k.. τ ε k N1,k... τ ε k NN,k We wll be explct about where ths assumpton s used n the results below. For now, note that ths assumpton s volated f trade s free (.e., τ n,k = 1 for all n and ). Gven the prevous defntons, we are now ready to state our frst Proposton. Proposton 1. If ether (a) 0 α k < 1, or (b) α k = 1 and Assumpton 1 holds, then the set L k (w) s a sngleton; f α k > 1, then the set L k (w) contans multple allocatons, ncludng (but not necessarly lmted to) one for each complete specalzaton allocaton. Moreover, the unque allocaton n L k (w) s an nteror allocaton f 0 α k < 1, whle t may be an nteror or a corner allocaton f α k = 1. Ths proposton states condtons under whch, gven any vector of postve wages and any ndustry k, the system (3) of N non-lnear complementary slackness condtons n L,k for = 1,..., N has a unque soluton, wth L,k > 0 for all f 0 α k < 1. The case wth α k = 0 s trval: gven wages, labor allocatons are explctly obtaned from the condtons L,k G,k (w, L k ) = 0. Below we focus on the case wth α k > 0. Before provng the proposton, we smplfy notaton by suppressng the sub-ndex k and by transformng varables wth x w L, x (x 1,..., x N ), a n S (w τ n ) ε w α, and b n β n w n Ln. For future purposes, note that Assumpton 1 guarantees that matrx A = (a n ) s non-sngular. Wth a slght abuse of notaton, we now rewrte the functon G,k (w, L k ) as G (x) 1 n a n x α 1 l a nlx α l We are dvdng the orgnal G,k functon by w and then suppressng w as a separate argument none of ths matters here snce we are treatng wages as gven for now. 16 The system n (3) can now be wrtten as a non-lnear complementarty problem (NCP) n x: x 0, G (x) 0, x G (x) = 0, = 1,..., N. (6) 16 Analogously to our treatment of the orgnal functons G,k (w, L k ) and L,k G (wl k ) at L,k = 0 (see footnote 10), we defne values of G (x) and x G (x) at x = 0 by ther lmts. b n.

16 GROUNDED BY GRAVITY 15 Note that f x solves (6) then x G (x) = 0 and hence x = b. Ths mples that the soluton to (6) satsfes x Γ { x R N x 0, = 1,..., N; x = b }. To prove Proposton 1 we follow a popular approach n the economcs lterature that conssts of characterzng equlbra of general equlbrum models as solutons to optmzaton problems. 17 Dong ths s possble f, for example, the functon G(x) (G 1 (x),..., G N (x)) has a Jacoban that s symmetrc at all ponts n ts doman, snce n ths case the functon G s the gradent of some other functon F that we can use n the optmzaton problem. 18 Fortunately, our functon G satsfes ths symmetry condton. In fact, t s easy to see that G s the gradent of functon F : R N + \ {0} R defned by F (x) α n x n n ( ) b n ln a n x α. (7) As we establsh formally below, ths makes t possble to solve the NCP n (6) by way of solvng arg mn x Γ F (x). 19 We now focus on the characterzaton of the optmzaton problem arg mn x Γ F (x) and then establsh formally the connecton between ths problem and the NCP n (6). Exstence of a soluton to arg mn x Γ F (x) follows mmedately from the fact that Γ s a compact set and F ( ) s contnuous on Γ. To establsh unqueness, we show n Appendx C that under the condtons of Proposton 1 functon F ( ) s strctly convex on Γ. Thus, snce Γ s a convex set, F ( ) has at most one global mnmum on Γ. Ths establshes the followng result: Lemma 1. If ether (a) 0 < α < 1, or (b) α = 1 and Assumpton 1 holds, then F ( ) has a unque global mnmum on Γ. Let us denote the unque global mnmum of F ( ) on Γ by x. In Appendx C we prove the followng result: Lemma 2. If 0 < α < 1 then x > 0 for all = 1,..., N. Fnally, we prove the part of Proposton 1 concernng the case of α 1 by combnng the two prevous lemmas wth the followng equvalence result: 17 Negsh (1960) s probably the most well-known example of ths approach n whch market equlbra are characterzed as solutons to a socal planner s problem. Kehoe, Levne and Romer (1992) descrbe a more general framework n whch the optmzaton problem does not necessarly have an economc nterpretaton. Our case fts nto ther general framework. 18 A classcal result n mathematcs states that a vector functon s a gradent map f and only f ts Jacoban s symmetrc n the doman of the functon (see, for example, Theorem on page 95 n Ortega and Rhenboldt, 2000). 19 We thank Anca Curte and Ioan Rasa for pontng us n ths drecton.

17 16 KUCHERYAVYY-LYN-RODRíGUEZ-CLARE Lemma 3. If ether (a) 0 < α < 1, or (b) α = 1 and Assumpton 1 holds, then x s a global mnmum of F ( ) on Γ f and only f x s a soluton to (6). The proof of ths result s almost trval, because the condtons n (6) are just the frst-order condtons for the mnmzaton of F ( ) on Γ. The only complcaton s that to nvoke the frst-order condtons, we need to have dfferentablty of F ( ) on Γ whch s understood as dfferentablty of F ( ) on some open set contanng Γ. In case of α < 1 any such open set necessarly ncludes ponts x wth x 0, at whch F ( ) s not dfferentable. We deal formally wth ths complcaton n Appendx C. One mght wonder f the equlbrum labor allocaton s contnuous n α as we approach α = 1 from below. Economcally speakng, one would expect ths to be the case, so that f at α = 1 we have a corner allocaton wth x = 0 for some country then x (α) > 0 for all α < 1 but lm α 1 x (α) 0. Mathematcally, however, ths result s not trval because the functon G s not jontly contnuous n x and α for α = 1 and ponts x wth x = 0 for some. Stll, thanks to the optmzaton approach followed n the prevous lemmas, we can establsh the left contnuty of x(α) by nvokng the Theorem of the Maxmum (see Theorem 3.6 n Stokey, Lucas and Prescott, 1989) (see Appendx C for detals). Lemma 4. If Assumpton 1 holds, then x(α) s contnuous as a functon of α for all α (0, 1]. In partcular, lm α 1 x(α) = x(1). Consder now the case wth α > 1. We can easly show that there are many solutons to (6). To see ths, choose some and set x = n b n and x = 0 for. It s easy to check that ths satsfes G (x) = 0 and G (x) = 1 0 for all, so that all condtons n (6) are satsfed. Of course, ths s only one example and there are many other possble equlbrum allocatons for ths case. However, characterzng the complete set of equlbra s not the focus of our analyss. We fnsh ths subsecton by commentng on the role of Assumpton 1 n Proposton 1. Whle ths assumpton plays no role n the proof of unqueness when α k < 1, we cannot rule out multplcty of equlbra f t s volated when α k = 1. As mentoned above, Assumpton 1 s volated f trade s frctonless. In fact, t s enough that trade be frctonless between any set of countres for multplcty to arse n the case wth α k = 1. Suppose, for nstance, that there were no trade costs between two countres and j. The trangular nequalty mples the trade costs between and j and all other countres are the same (.e., τ n,k = τ nj,k and τ n,k = τ jn,k for all n, j). It then follows that the and

18 GROUNDED BY GRAVITY 17 j row n the matrx of Assumpton 1 are the same, so the non-sngularty requrement s volated. Notce, however, that the multplcty that arses n ths case s that at most the overall labor allocaton L,k + L j,k s determned, but not L,k or L j,k. Ths type of nonunqueness s rrelevant for welfare: real wages are the same across any two equlbra n ths set. Moreover, wth any small trade costs between and j the non-unqueness dsappears, renderng these cases non-generc Step 2: Equlbrum Wages In what follows, we restrct the analyss to the case 0 α k 1. For ths case, Proposton 1 establshes that the soluton of the system of complementary slackness condtons (3) determnes a functon from wages to labor allocatons, L(w), for w R N ++. Lettng Z (w) L,k (w) L (8) k be the excess labor demand n country defned for all w R N ++ and lettng Z(w) (Z 1 (w),..., Z N (w)), the labor-market clearng condtons for all countres can be wrtten smply as Z(w) = 0. (9) To establsh exstence of a soluton to ths system of equatons, we wll nvoke contnuty of L k (w). We agan explot the equvalence between the system n (3) and a constraned optmzaton problem and nvoke Theorem of the Maxmum from Stokey, Lucas and Prescott (1989) to establsh that L k (w) s a contnuous functon for all w R N ++ (see Appendx C for detals). Lemma 5. If ether (a) 0 α k < 1, or (b) α k = 1 and Assumpton 1 holds, then the functon L k (w) s contnuous for all w R N ++. We now state our result for exstence of equlbrum. Proposton 2. Assume that for all k ether (a) 0 α k < 1, or (b) α k = 1 and Assumpton 1 holds. Then there exsts a vector of wages w R N ++ that satsfes (9). Proof. The case wth α k = 0 s a smple extenson of the exstence proof by Alvarez and Lucas (2007) to the case of multple ndustres. Here we focus on the case wth 0 < α k 1. To establsh exstence of a soluton to (9), t suffces to show that the followng propertes as outlned n Proposton 17.B.2 n Mas-Colell, Whnston and Green (1995,

19 18 KUCHERYAVYY-LYN-RODRíGUEZ-CLARE MWG) are satsfed: () Z(w) s contnuous; () Z(w) s homogeneous of degree zero; () w Z(w) = 0 for all w (Walras law); (v) there s an A > 0 such that Z (w) > A for all and w; (v) f w s w as s, where w 0 and w = 0 for some, then Max {Z 1 (w s ),..., Z N (w s )} as s. Property () follows from Lemma 5, whle propertes ()-(v) are mmedate. The proof of (v) s n Appendx C. In what follows, we frst provde suffcent condtons for a unque equlbrum n the case of two countres (N = 2), and n the case of multple countres wth free trade. After that we dscuss addtonal complextes that arse n the general settng wth multple countres and costly trade. Proposton 3. Assume that N = 2 and that for all k ether (a) 0 α k < 1, or (b) α k = 1 and Assumpton 1 holds. Then there exsts a unque (normalzed) vector of wages w R N ++ that satsfes (9). Proposton 4. Assume that 0 α k < 1 for all k and trade s frctonless n all ndustres,.e., that τ n,k = 1 for all n,, and k. Then there exsts a unque (normalzed) vector of wages w R N ++ that satsfes (9). We prove both Propostons 3 and 4 by showng that the labor excess demand functon Z(w) has the gross substtutes property under the assumptons of these propostons. Unqueness of soluton then follows from Proposton 17.F.3 from MWG. The prevous results establsh that f there are two countres, or f there are many countres but no trade costs, or f there are many countres and postve trade costs but wages are pnned down by an outsde good, then the equlbrum exsts and s unque. 20 Wth postve trade costs and more than two countres, our excess labor demand system does not, n general, satsfy the gross-substtutes property, and so ths property can no longer be nvoked for establshng a unque vector of wages for N > 2 and costly trade. Wth ndustry-level externaltes and trade costs one has to contend wth addtonal complcatons that arse when there are more than two countres. In partcular, 20 If we assume that there s a freely traded outsde good ndustry n whch producton exhbts constant returns to scale and assume that all countres produce a postve amount of ths good, as s typcally done n the lterature, then wages are exogenous and the proof from Proposton 1 whch s vald for any fnte N mples a unque allocaton of labor across ndustres. Note, however, that we need to assume that all countres produce the outsde good f some countres do not produce that good then wages are not pnned down and we don t have a proof of unqueness for more than two countres for ths case.

20 GROUNDED BY GRAVITY 19 whle these externaltes act to renforce the gross substtutes property when there are two countres, the same s not necessarly true for three or more countres. For nstance, a rse n the wage n one country, say country 1, may reduce the demand for labor there, whle at the same tme rasng the demand for labor n another country, say country 2, whch s so far consstent wth the gross substtutes property. The complextes arse from the fact that the ncreased labor demand n country 2 can generate productvty effects that can lead to ncreased exports to a thrd country, say country 3, whch can, n turn, result n a fall n the demand for labor there. In other words, a rse n wages n country 1 can result n a fall n the demand for labor n country 3, thereby, volatng the gross substtutes property. Whle we have not yet been able to prove our unqueness result n Proposton 3 for the case N > 2, extensve numercal smulatons ndcate that the determnant of the negatve of the Jacoban of the normalzed excess labor demand system (.e., the Jacoban of Z(w) after removng the last column and the last row) s always postve, so that one could nvoke the Index Theorem to show unqueness of equlbrum (Kehoe, 1980). The challenge here s that the Jacoban of the aggregate labor demand s the sum of the Jacobans of the labor demand comng from each sector (.e., DZ(w) = k DL k(w)), and establshng condtons on the determnant of a sum of matrces s extremely dffcult. The advantage of the gross-substtutes property s that t mples that the negatve of the Jacoban of the labor demand n each sector has all dagonal terms postve and all off-dagonal terms negatve, and hence a postve determnant. Snce the grosssubsttutes property survves under summaton, the determnant of the negatve of the Jacoban of the aggregate labor demand s postve as well. Wthout the gross-substtutes 21, 22 property, we need an alternatve approach. 21 In Appendx C we explore whether the technques developed n Allen, Arkolaks and L (2015) can be appled to establsh unqueness for our system. Unfortunately, we fnd that the suffcent condton for unqueness n ther Theorem 1 does not hold n our economy. 22 We have also explored the queston of unqueness numercally. Focusng on the case of N = 3 and K = 2, we smulated more than a mllon economes wth α = 0.9 and randomly chosen values for all other parameters and then computed the equlbrum for each economy, startng at 400 ntal ponts. We never found an nstance of multple equlbra. In contrast, usng the same code for α = 2 leads to multple equlbra for randomly generated parameters. For the case wth α = 0.9 we also computed the sgn of the determnant of the (negatve of the) excess labor demand. By the Index Theorem, a negatve value would mply multplcty, whle unqueness would mply a postve value. We always found ths sgn to be postve.

21 20 KUCHERYAVYY-LYN-RODRíGUEZ-CLARE 3.3. Computaton of Equlbrum The precedng analyss suggests two alternatve approaches to numercally compute the equlbrum. Frst, one can use an algorthm that properly deals wth the complementary slackness condtons n the system of Equatons (3) and (5) for (w, L). Ths requres an algorthm for non-lnear complementarty problems, such as the PATH solver (Ferrs and Munson, 1999). Second, one can follow the approach used above to prove exstence and unqueness of equlbrum and break the problem n two steps: frst, for each wage vector w fnd L k (w) for each k by solvng the optmzaton problem assocated wth (7), and second, fnd the wage vector such that the excess labor demand Z(w) k L k(w) L s zero usng the tatonnement teratve procedure proposed by Alvarez and Lucas (2007). It turns out, however, that a thrd approach does best. Consder the functon w(t ) that one would get smply by solvng for wages n the standard mult-sector model wth no scale economes and technology parameters T = {T,k }, and let L d,k (T, w) be labor demand as a functon of technology parameters and wages also n that model. Let T (L) be defned by T,k (L) = S,k L α k,k and let H(L) Ld (T (L), w(t (L))). By defnton of w(t ) we must have k Ld,k (T (L), w(t (L))) = L for all, and hence f L s a fxed pont of the mappng H(L) then (w, L ) = (w(t (L ), L ) s an equlbrum of our economy wth economes of scale. Note that H(L) s a contnuous mappng from the compact set Λ {L k L,k = L } to tself, and we know from Proposton 2 that an nteror soluton exsts f α k < 1 for all k. We can then use the teratve procedure gven by L t+1 = H(L t ) to compute the equlbrum ponts. In usng ths algorthm for the quanttatve analyss n the followng secton we fnd that t can easly handle corners and that t s very robust. 4. Scale Economes and the Welfare Effects of Trade In ths secton we explore the mplcatons of scale economes for the welfare effects of trade. We restrct the analyss to the case n whch 0 α k 1 for all k. We frst study how scale economes affect the gans from trade and the welfare effects from trade lberalzaton, and we conclude by quantfyng the dfferent effects usng counterfactual analyss when the model s made to be perfectly consstent wth the data.

22 GROUNDED BY GRAVITY Gans from Trade In prncple, countres that specalze n ndustres wth weak economes of scale could even lose from trade the premse of Frank Graham s argument for protecton. It turns out, however, that ths cannot happen f 0 α k 1 for all k. The formal proof s n the Appendx D.1, but the basc dea can be understood by the followng smple argument. Settng w = 1 by choce of numerare, ndustry level prce ndces can be wrtten as P ε k,k = µ ε k k S,k L α k,k /λ,k. (10) Wthout scale economes, gans from trade are assured by the fact that λ,k < 1 mples P,k < µ k S 1/ε k,k, where the RHS term s the prce ndex under autarky. Scale economes mply that L,k could fall wth trade, and so now from Equaton (10) we see that P,k could be hgher wth trade relatve to autarky. But note that n equlbrum we must have L,k > λ,k β,k L, snce the RHS s just the total labor cost assocated wth domestc ( ) sales. Combnng ths wth Equaton (10) yelds P ε k,k > µ ε k αk k S,k β,k L λ α k 1,k. Thus, f 0 α k 1 then P,k < µ k S 1/ε ( ) k αk /ε k,k β,k L, whch s the prce under autarky when there are scale economes. As we can also see, f α k > 1 then one could have hgher prces n some ndustres wth trade than wthout, leadng to the possblty of losses from trade. Ths argument establshes the followng Proposton. Proposton 5. If 0 α k 1 then all countres gan from trade. Proof. See Appendx D.1. Ths result can be seen as a generalzaton of Proposton 1 n Venables (1987), whch states that n a Krugman (1980) model wth an outsde good all countres gan from trade. Formally, the model n Venables (1987) s somorphc to ours when we consder two countres and two ndustres, one havng no trade costs, no scale economes, and an nfnte trade elastcty (the outsde good ), and the other havng trade costs, scale economes, and a fnte trade elastcty, wth α k = 1. Proposton 5 shows that ths generalzes to a case wthout an outsde good, wth multple sectors and arbtrary scale economes as long as α k 1 for all k. To further explore the mplcatons of scale economes for the magntude of the gans from trade, we assume that the equlbrum s nteror so that all trade shares and labor allocatons are strctly postve. Ths allows us to derve an expresson for the gans from