Intersectoral Markup Divergence

Transcription

1 March 2018 Intersectoral Markup Dvergence Krstan Behrens, Sergey Kchko, Phlp Ushchev

2 Impressum: CESfo Workng Papers ISSN (electronc verson) Publsher and dstrbutor: Munch Socety for the Promoton of Economc Research CESfo GmbH The nternatonal platform of Ludwgs Maxmlans Unversty s Center for Economc Studes and the fo Insttute Poschngerstr. 5, Munch, Germany Telephone +49 (0) , Telefax +49 (0) , emal offce@cesfo.de Edtors: Clemens Fuest, Olver Falck, Jasmn Gröschl group.org/wp An electronc verson of the paper may be downloaded from the SSRN webste: from the RePEc webste: from the CESfo webste: group.org/wp

3 CESfo Workng Paper No Category 8: Trade Polcy Intersectoral Markup Dvergence Abstract We develop a general equlbrum model of monopolstc competton wth a traded and a nontraded sector. Usng a broad class of homothetc preferences that generate varable markups, dsplay a smple behavor of ther elastcty of substtuton, and nest the ces as a lmtng case we show that trade lberalzaton: () reduces domestc markups and ncreases mported markups n the traded sector; () ncreases markups n the non-traded sector; and () ncreases frm szes n both sectors. Thus, whle domestc and export markups n the traded sector converge across countres, markups dverge across sectors wthn countres. The negatve welfare effects of hgher markups and less consumpton dversty n the non-traded sector dampen the postve welfare effects of lower markups and greater dversty n the traded sector. JEL-Codes: F120, F150. Keywords: monopolstc competton, varable markups, trade lberalzaton, non-traded goods, markup dvergence. Krstan Behrens Department of Economcs Unversty of Québec at Montréal ESG-UQAM Montréal / Québec / Canada behrens.krstan@uqam.ca Sergey Kchko Natonal Research Unversty Hgher School of Economcs Sant Petersburg / Russan Federaton skchko@hse.ru sergey.kchko@gmal.com Phlp Ushchev Natonal Research Unversty Hgher School of Economcs Sant Petersburg / Russan Federaton ph.ushchev@gmal.com March 20, 2018

4 1 Introducton Havng been out of fashon for a long tme, monopolstc competton models wth varable elastcty of substtuton (ves) have attracted more attenton n recent years. They have been used to study, among others, the selecton effects of freer trade (Meltz and Ottavano, 2008), the effcency of the market outcome (Dhngra and Morrow, 2018), the welfare consequences of trade lberalzaton (Behrens and Murata, 2012), the nterdependence between prces and ncomes (Smonovska, 2015), and how preferences affect the market outcome n general (Zhelobodko et al., 2012). One key reason for ths renewed nterest s that mcro-level evdence on markups and frm szes s by and large nconsstent wth the constant elastcty of substtuton (ces) model. Whle ves models help us to make sense of the mcro-level data, they have largely abstracted from ntersectoral ssues. There are ndeed few general equlbrum models wth multple ves sectors. 1 Our paper ams to fll that gap by developng a tractable multsector general equlbrum framework wth ves preferences that allows us to study the market outcome under trade lberalzaton. Monopolstc competton models wth ves and multple sectors are mportant for at least three reasons. Frst, emprcal evdence shows that markups are varable and change systematcally wth market structure and trade lberalzaton (Syverson, 2007; De Loecker et al., 2012). Second, ndustral polces and trade lberalzaton affect the dstrbuton of resources across heterogeneous sectors and frms. In the presence of varable markups, ths has consequences for the composton of welfare gans (Feenstra and Wensten, 2017) and the allocatve effcency of the market outcome (Holmes et al., 2014; Dhngra and Morrow, 2018). 2 Thrd, dstortons also arse between sectors (Epfan and Ganca, 2011; Behrens et al., 2016), so that takng nto account both the ntra- and the ntersectoral allocaton of resources s mportant. The absence of multsector general equlbrum ves models n the lterature may be explaned by the fact that those models are dffcult to handle. Frst, there are only few specfcatons that reman analytcally tractable, even wth a sngle sector. Second, when extended to multple sectors, the ntersectoral allocaton and, especally, the sectoral budget shares are dffcult 1 Whle multsector ces models abound n the lterature (see, e.g., Bernard et al., 2007; and Costnot and Rodrguez-Clare, 2014, for a state-of-the-art survey), multsector ves models are notably absent. The vast majorty of multsector monopolstc competton models use ether only one ves sector and a compettve outsde sector (Zhelobodko et al., 2012), or quas-lnear preferences (Meltz and Ottavano, 2008) thus gvng those models a partal equlbrum flavor. To our knowledge, Behrens et al. (2016) s one of the few papers that develops a full general equlbrum framework wth heterogeneous frms and multple ves sectors to analyze ntra- and ntersectoral dstortons n the economy. See also d Aspremont and Dos Santos Ferrera (2015) for a closed-economy multsector model wth olgopolstc competton. 2 Analyzng optmalty and changes n allocatve effcency n the presence of varable markups has become an mportant topc n theoretcal studes (Arkolaks et al., 2018; Dhngra and Morrow, 2018; Holmes et al., 2014; Behrens et al., 2016). On the emprcal sde, usng Chlean data for , Wenberger (2015) shows that productvty gans are not always accompaned by allocatve effcency gans. Edmond et al. (2015) draw on Tawanese frm-level data to show that trade lberalzaton leads to a reducton of dstortons drven by varable markups by approxmately 66%, provded that domnant frms are exposed to tougher competton when trade gets freer. 2

5 to pn down. Even wth the workhorse Cobb-Douglas upper-ter utlty functon, the budget shares vary n a non-trval way wth the whole equlbrum allocaton when the lower-ter utlty s non-homothetc. These dffcultes largely explan why the Cobb-Douglas wth nested ces specfcaton has been adopted by the vast majorty of the extant multsector general equlbrum lterature wth mperfect competton. In ths paper, we focus on homothetc preferences, whch mples that prce ndces are well defned. In addton, we want preferences to have varable elastcty of substtuton, wth the latter dsplayng a smple behavor. Such preferences are necessarly non-addtve, whch makes the analyss a pror more nvolved. We provde, however, a smple characterzaton of Kmball s (1995) preferences by clarfyng what are the fundamental propertes of the elastcty of substtuton for that type of preferences. We show, n partcular, that the elastcty of substtuton depends solely on the ndvdual consumpton level scaled by a quantty ndex, thus mplyng that those preferences mmc the behavor of addtve preferences. They also nest ces preferences as a specal case. 3 We can therefore choose nstances of those preferences that are arbtrarly close to the ces, whch allows us to derve many results by contnuty n a small neghborhood of the ces. We refer to those preferences as ε-ces preferences. They make the analyss of the nteractons between sectors farly tractable whle allowng for emprcally relevant features lke varable markups and varyng frm szes. In addton, they make t straghtforward to brdge our results wth the exstng fndngs obtaned wthn ces-based settngs. We apply ths approach to develop an analytcally tractable two-sector ves trade model where one sector produces traded manufactured goods whle the other produces non-traded servces. Both ndustres produce a contnuum of horzontally dfferentated varetes and have varable markups. Havng non-traded goods makes the analyss both more tractable and realstc. It s well-documented that the share of non-traded goods n modern economes s hgh. It vares substantally across countres but averages around 45% (Lombardo and Ravenna, 2012). Furthermore, n the nternatonal macro context, varatons n the relatve prces of non-traded goods across countres are one of the common explanatons for real exchange rate volatlty (Bursten et al., 2005 and 2006). Thus, the presence of a non-traded sector mght have a sgnfcant mpact on trade patterns. Observe further that the presence of a non-traded sector automatcally mples that trade lberalzaton affects sectors dfferentally markups change n sector-specfc ways. We thus beleve t s mportant to look more closely at models that feature both non-traded goods and varable markups. 4 3 Snce ces preferences are the workhorse n the nternatonal trade lterature (e.g., Krugman, 1980; Eaton and Kortum, 2002; Meltz, 2003), stayng close to those preferences does not requre us to go nto completely new modelng drectons. Moreover, departng from the ces wthn ths setup does not render ntensve-margn and compettve effects fragle, as found by Bertolett and Epfan (2014) for addtve preferences. 4 The lnk between the elastcty of substtuton and the varaton n purchasng power party across countres s also wdely studed. For nstance, non-proportonal prcng-to-market s one of the explanatons for the large devatons from relatve purchasng power party (Atkenson and Bursten, 2008). Corsetta and Dedola (2005) nvestgate the mpact of exchange rate shocks on frms prcng decsons for domestc and foregn markets by 3

6 Our key results may be summarzed as follows. Frst, whle trade lberalzaton promotes the convergence of markups across countres, t leads to the dvergence of markups across sectors wthn countres. Our model hence allows us to ratonalze the observaton that markups for traded manufactured goods have decreased n the wake of the Sngle Market Program of the European Unon n the 1990s, whereas markups for non-traded servces have ncreased (see Badnger, 2007). Although ant-compettve practces have been put forward as a possble explanaton for those changes, our model suggests that smple market-drven general equlbrum effects may be enough to generate that outcome. Second, as markups fall and product dversty expands n the traded sector, markups rse and product dversty shrnks n the non-traded sector. The reason for these opposng changes s the ntersectoral reallocaton of resources trggered by the lberalzaton of trade n one sector. The negatve welfare effects of hgher markups and less consumpton dversty n the non-traded sector dampen the postve welfare effects of lower markups and greater dversty n the traded sector. To gauge the overall welfare effect, we provde comparsons of welfare gans for our settng wth varable markups and the benchmark case of ces preferences. We show that the overall welfare gans are slghtly large n the ε-ces case than under the ces preferences,.e., varable markups are an addtonal source of gans. It s mportant to stress that our welfare results are fully drven by general equlbrum effects. Whle trade lberalzaton has a drect mpact on the traded sector, freer trade affects the non-traded sector only ndrectly through changes n the traded sector. Ths hghlghts the mportance of workng wth multsector models to nvestgate the postve and normatve effects of trade lberalzaton under monopolstc competton. The remander of the paper s organzed as follows. Secton 2 ntroduces and characterzes our preference structure and descrbes the trade model. Sectons 3 and 4 study the market outcomes n the non-traded and n the traded sectors, respectvely. Secton 4 ntroduces our concept of ε-ces preferences. Secton 5 explores the consequences of trade lberalzaton for sectoral markups, frm szes, and the ntersectoral allocaton. Secton 6 establshes our welfare results and provdes numercal llustratons to show how far beyond the ces we can go for our key qualtatve results to reman unchanged. Fnally, Secton 7 concludes. 2 The model We develop a two-sector general equlbrum model of monopolstc competton wth varable elastcty of substtuton and costly trade between two countres: home, H, and foregn, F. For smplcty, countres are assumed to have dentcal technologes and the same populaton, L. 5 employng a model wth non-tradables and country-specfc prce elastctes. We devate from ths lterature by studyng prce formaton n traded and non-traded sectors under the presence of an external shock n the traded sector,.e., trade lberalzaton. 5 We relax the assumpton of equal populaton sze n the Appendx D and show that our key results extend to the case of asymmetrc populaton szes. Introducng technologcal dfferences between sectors does not add 4

7 There are two sectors n each country. The frst one produces a contnuum of horzontally dfferentated varetes of a traded good ( manufacturng ), whereas the second one produces a contnuum of varetes of a non-traded good ( servces ). Labor s the only producton factor. 2.1 Preferences Snce we are nterested n the behavor of markups, we cannot work wth standard ces preferences. Yet, we want to work wth preferences for whch we can defne prce ndces,.e., we would lke our preferences to be homothetc. As shown by Parent et al. (2017), no addtve preferences of the form U N 0 v(x )d satsfy jontly the two propertes of varable markups and homothetcty. Hence, we need to work wth non-addtve preferences. The dffculty wth non-addtve preferences s that the markups (or, alternatvely, the elastcty of substtuton) are no longer smple functons of solely the quantty x, as they are n the addtve case. 6 To deal wth that dffculty, we wll work wth a class of preferences that: () are homothetc; () have a smple behavor of ther elastctes of substtuton; and () dsplay varable markups. In other words, we seek for a class of homothetc preferences mmckng the behavor of addtve preferences (see equaton (1) below for a mathematcal formulaton). We show that preferences descrbed by Kmball s flexble aggregator (Kmball, 1995) satsfy these propertes. Snce they are closely related to addtve preferences, they dsplay a smlar behavor. In partcular, the elastcty of substtuton and hence the proft-maxmzng markups are lnked to the relatve rsk averson of Kmball s aggregator, whereas they are lnked to the relatve rsk averson of the lower-ter utlty functon v( ) n the addtve case. Gven ther conceptual smlarty, we can make use of the numerous tools developed prevously n the lterature to analyze the equlbrum propertes n that type of model. Ths sheds new lght on some fundamental propertes of these preferences and provdes also a new characterzaton for them. Let σ(x, x j, x) denote the elastcty of substtuton between varetes and j, where x denotes the whole consumpton profle. To combne the three desrable propertes mentoned above the exstence of a prce ndex, a smple behavor of the elastcty of substtuton, and varable markups we focus on homothetc preferences u for whch there exsts a functon s(x, u) such that σ(x, x, x) = s(x, u(x)) (1) anythng substantal to the analyss but makes the algebra more complcated. 6 As shown by Zhelobodko et al. (2012), under symmetrc addtve preferences the elastcty of substtuton between varetes and j, σ(x, x j, x), s ndependent of the remanng consumpton pattern x, gven that both varetes are consumed n equal quanttes. Put dfferently, f x = x j = x, we have σ(x, x, x) = σ(x). Thus, addtve preferences have a smple behavor wth respect to the elastcty of substtuton. Snce the substtutablty across varetes s a key feature of the demand sde n models of mperfect competton, the aforementoned property explans, at least partly, why addtve preferences are so popular. Examples nclude, among others: Krugman (1979, 1980); Eaton and Kortum (2002); Meltz (2003); Behrens and Murata (2007); Zhelobodko et al. (2012); and Smonovska (2015). 5

8 at a symmetrc consumpton pattern where x = x j. In expresson (1), u(x) s a consumpton ndex, whch s ncreasng, strctly quas-concave, and postve homogeneous of degree 1 n x. Moreover, observe that at a symmetrc consumpton pattern, gven by x = x 1 [0,N], where 1 [0,N] stands for the ndcator of [0, N] and where N s the mass of avalable varetes, we have u(x) = xν(n), so that s(x, u(x)) = s(x, xν(n)), (2) where ν(n) s defned as ν(n) u ( ) 1 [0,N]. As shown by Parent et al. (2017), when preferences are homothetc, the elastcty of substtuton σ depends solely on N at a symmetrc consumpton pattern. Combnng ths result wth (2) shows that s(x, u) s postve homogeneous of degree 0. Hence, expresson (1) bols down to σ(x, x, x) = σ(x/u(x)). (3) The ntuton behnd expresson (3) s as follows. If, for example, σ( ) s a decreasng functon, then a hgher consumpton level of both varetes makes them worse substtutes, whle an ncrease n the overall level of consumpton captured by u(x) does the opposte. However, a proportonal change n the consumpton of all varetes leaves the degree of substtutablty between any two of them unchanged. The followng proposton provdes an alternatve characterzaton of the preferences satsfyng (3). Proposton 1. (Kmball preferences) A symmetrc homothetc preference relatonshp satsfes (3) f and only f the utlty functon u s descrbed by N 0 θ ( ) x d = 1, (4) u(x) where θ( ) s an arbtrary non-negatve, ncreasng, concave, and twce contnuously dfferentable functon. Moreover, σ(x/u(x)) s gven by σ(x/u(x)) = where r θ ( ) s the relatve rsk averson of θ( ): Proof. See Appendx A.1. 1 r θ (x/u(x)), (5) r θ (x/u(x)) [x/u(x)]θ (x/u(x)). (6) θ (x/u(x)) To the best of our knowledge, Kmball (1995) was the frst to use (4) to represent a produc- 6

9 ton functon for a fxed range of varetes. 7 What s hence new n Proposton 1? Frstly, t provdes a new mathematcal characterzaton of the class of preferences descrbed by Kmball s flexble aggregator. Secondly, t shows that Kmball-type preferences are very close to addtve preferences, n a sense of a smple behavor of ther elastcty of substtuton. For these preferences, the whole relevant nformaton on the demand sde to understand the market outcome s captured by 1/r θ ( ),.e., the elastcty of substtuton, whch depends on x/u(x) only. In other words, Kmball preferences (4) mmc the property of addtve preferences where the whole relevant nformaton s captured by the relatve rsk averson of the lower-ter utlty functon r v (x) xv (x)/v (x). Ths leads to a substantal smplfcaton of the analyss and allows us to derve analytcal results on the mpacts of trade lberalzaton. Last, observe that the ces utlty wth elastcty of substtuton σ can be obtaned as a specal case of (4) by settng θ(z) z (σ 1)/σ. Ths wll prove useful later snce we can work wth preferences satsfyng (4), whch are arbtrarly close to ces preferences yet are homothetc and have varable markups. These ε-ces preferences allow us to derve many results nvokng contnuty arguments wthn the class of Kmball preferences. We return to ths pont n detal n Secton Consumers Each country hosts L consumers, who have dentcal preferences gven by U k = U(u(x kk, x lk ), v(y k )), k, l {H, F }, k l, (7) where U s an upper-ter utlty; u and v are lower-ter utltes for the consumpton of, respectvely, traded and non-traded goods; and x kk, x lk, and y k are vectors of ndvdual consumpton of, respectvely, locally produced varetes of the traded good n country k, mported varetes of the traded good n country k, and varetes of the non-traded good produced and consumed n k. We assume that U, u, and v are strctly ncreasng, strctly quas-concave, and homothetc. We also assume that the lower-ter utltes are Kmball type, as defned n Secton 2.1,.e., there exst ncreasng and concave functons θ and ψ, such that for any x kk, x lk, and y k, the lower-ter 7 Dotsey and Kng (2005) use the same approach to model preferences n a macroeconomc settng, havng agan a fxed range of varetes, whle Barde (2008) does the same n the settng of a economc geography model wth a varable range of varetes. Arkolaks et al. (2018) use a more general demand system, whch encompasses nonhomothetc addtve preferences and homothetc preferences whose deal prce ndex s descrbed by Kmball s (1995) flexble aggregator. They focus manly on measurng gans from trade under pro-compettve effects mostly usng emprcal estmates and numercal smulatons. We, nstead, work wth a narrower class of demand systems, whch allows us to provde an analytcal characterzaton of the equlbrum and to derve a number of comparatve statc results. 7

10 utltes u and v satsfy N k 0 θ ( ) x kk d + u k N l 0 θ ( ) x lk j dj = 1, u k M k 0 ψ ( ) y k d = 1, k, l {H, F }, (8) v k where u k u(x kk, x lk ), v k v(y k ); N k and N l are the masses of frms n the traded sector n countres k and l, respectvely; and M k s the mass of frms n the non-traded sector n country k. Note that N k, N l, and M k, whle treated parametrcally by consumers, wll be endogenously determned n equlbrum. Normalzng the wage to one by choce of numérare and recallng that countres are symmetrc and thus have the same equlbrum wage consumers maxmze utlty (7) subject ot ther budget constrant N k 0 pkk x kk d + N l 0 plk x lk d + M k 0 k y k d = 1, where p kk and p lk are the prces of domestc and mported traded varetes n country k; and k are the prces of nontraded varetes. Let P k and P k denote the prce ndces n the traded and n the non-traded sectors n country k, respectvely. Let also α(p k /P k ) denote the share of expendture for traded goods, whch depends solely on the rato of the traded and the non-traded sectoral prce ndces snce preferences are homothetc. The functonal form of α( ) s determned by the upper-ter utlty. The value of α(p k /P k ) s taken as gven by the consumers, although t changes wth the prce aggregates n the two sectors. The nverse demands can be derved as follows. Applyng standard two-stage budgetng technques, note frst that the consumers subproblem for the consumpton of the traded good s gven by: max u k, x kk, x lk s.t. u k N k 0 N k 0 N l p kk x kk d + θ ( ) x kk d + u k 0 N l 0 p lk x lk θ ( P d = α P ) ( x lk j u k ) dj = 1, (9) where we make use of the representaton of u. Settng z kk x kk /u k and z lk x lk /u k for notatonal convenence, we reformulate the constrants as N k 0 N l p kk z kk d + p lk z lk d = α(p/p), and 0 u k N k 0 θ ( ) N l z kk d + 0 θ ( ) z lk dj = 1. Snce maxmzng u k s equvalent to mnmzng 1/u k, we obtan the followng equvalent re- 8

11 formulaton of (9) as an expendture mnmzaton problem: mn z N k 0 N l p kk z kk d + p lk z lk d 0 s.t. N k 0 θ ( ) N l z kk d + 0 θ ( ) z lk dj = 1. (10) The frst-order condtons of (10) are gven by θ (z kk ) µ k = p kk, θ (z lk j ) µ k = p kl j, (11) where µ k s a sectoral market aggregate that nvolves the Lagrange multpler of the budget constrant and the margnal utltes of traded varetes. Usng the subproblem for the nontraded sector, we analogously obtan the nverse demands for the non-traded goods as follows: ψ ( y k /v ) = k λ k, (12) where λ k s a sectoral market aggregate for the non-traded good. Note from (11) and (12) that the frst-order condtons for Kmball preferences are very smlar to those for addtve preferences. To see ths smlarty, t suffces to replace θ wth v and the scaled quanttes z wth normal quanttes x. 2.3 Frms Frms n both sectors ncur a constant fxed cost, f, and a constant margnal cost, c, both pad n terms of labor. The costs for shppng goods between countres n the traded sector are of the standard ceberg form: τ 1 unts of the good have to be dspatched for one unt to arrve. We now turn to the profts of frms n country k. The profts of frm n the traded sector and of frm j n the non-traded sector are gven by Π k = (p kk c)lx kk + (p kl cτ)lx kl f and Π k j = ( k j c)ly k j f, respectvely. Because countres are symmetrc, we naturally focus on a symmetrc outcome. In what follows, we use the notaton, where d denotes domestc and m denotes mport values of varables. 8 As before, we also defne z d x d /u d, z m x m /u m, and z n y/v. Because we work wth a contnuum of monopolstcally compettve frms, no sngle frm has any mpact on the market aggregates µ and λ. Combnng ths wth the defnton (6) of r θ and the nverse demands (11), we fnd that r θ s the nverse demand elastcty. Hence, applyng the standard monopoly 8 More formally, x d x HH = x F F, x m x F H = x HF, y y H = y F, p d p HH = p F F, p m p F H = p HF, H = F, λ λ H = λ F, µ µ H = µ F, N N H = N F, M M H = M F, u u H = u F, v v H = v F. 9

12 prcng rule yelds p d = c 1 r θ (z d ), pm = cτ 1 r θ (z m ), = c 1 r ψ (z n ), (13) where r θ (z d ) s the markup for domestcally produced varetes of the traded good; r θ (z m ) s the markup for mported varetes; and r ψ (z n ) s the markup for non-traded varetes, where we replace u by v n (6). 9 The second-order condtons are relegated to Appendx B. Expressons (13) mply that, unlke n the standard ces model of nternatonal trade, markups vary wth the mass of frms. The mpact of entry on the markups s channeled through changes n the relatve consumptons z d, z m, and z n. Intutvely, per varety relatve consumpton should decrease as the product range expands, because of love for varety. As consumers have more consumpton choces, they dversfy ther consumpton bundles so that the ndvdual consumpton levels of each varety decrease. Ths s not the end of the story, however. Love for varety also mples that an ncrease n the mass of varetes makes consumers better off,.e., ncreases the lower-ter utltes u and v, thereby magnfyng the reducton of relatve consumptons. Note that the former effect has been studed n Zhelobodko et al. (2012), whle the latter effect cannot be captured by a model wth addtve preferences. What happens to the markups as more frms enter? As mpled by (13), the answer to ths queston s fully determned by the behavor of the functons r θ ( ) and r ψ ( ). We wll focus mostly on the case where both functons are ncreasng: ths entals a pro-compettve effect,.e., entry of frms ncreases competton, thus leadng to a fall n the markups. Emprcal evdence ponts to the exstence of pro-compettve effects (e.g., De Loecker et al., 2012). 10 An alternatve justfcaton for ths assumpton can be derved from the conventonal wsdom prevalng n ndustral organzaton: the more frms operate n the market, the less dfferentated ther products are. Because σ = 1/r θ by Proposton 1, r θ ( ) and r ψ ( ) can be vewed as measures of product dfferentaton n, respectvely, the traded and the non-traded sectors. These measures naturally change wth entry n the dfferent ndustres. In partcular, f they are ncreasng functons, an ncrease n the quantty ndces, u and v, or, conversely, lower quantty ndces, make varetes closer substtutes. Combnng (11) and (13), we readly obtan θ (z d ) [ 1 r θ (z d ) ] = µc, θ (z m ) [1 r θ (z m )] = µcτ, θ (z n ) [1 r ψ (z n )] = λc. (14) Condtons (14), whch equate margnal revenue and margnal cost, allow us to pn down the quanttes z d, z m, and z n as functons of the market aggregates µ and λ only. 9 The analogy wth the frst-order condtons n the addtve case s agan very clear from (13). See, e.g., Zhelodobko et al. (2012). 10 In addton, ths assumpton guarantees that the second-order condtons for proft maxmzaton hold. See Appendx B for detals. 10

13 3 Equlbrum n the non-traded sector We now study how the equlbrum n the non-traded sector changes wth trade lberalzaton, as measured by a decrease n trade costs τ. Usng (8), (13), and the sectoral budget constrant M y = 1 α, for a gven mass M of non-traded varetes the symmetrc equlbrum condtons can be expressed as follows: cy Pv = M 1 r ψ (z n ) = 1 α ( ) P, (15) P and Solvng (16) for z n yelds Mψ(z n ) = 1. (16) z n = ψ 1 (1/M). (17) Snce ψ s an ncreasng functon, the relatonshp (17) mples that the relatve consumpton of each varety of the non-traded good decreases wth an expandng range of varetes n the non-traded sector. Usng (13) and (17), we fnd that the equlbrum markup for a varety of the non-traded good s gven by r ψ [ψ 1 (1/M)]. Hence, the markups for non-traded varetes decrease n response to entry f and only f r ψ ( ) s ncreasng or, equvalently, f the elastcty of substtuton σ( ) s decreasng. We next analyze how the prce ndex P vares wth entry. Dvdng (15) by v, usng z n y/v and (17), we can express the prce ndex as a functon of the mass of frms: P = cmψ 1 (1/M) ( ). (18) 1 r ψ ψ 1 (1/M) As can be seen from (18), f r ψ ( ) s ncreasng, addtonal entry leads to a fall n prces. In other words, n that case more frms means tougher competton. 11 the varetes of the non-traded good s gven by The total expendture for E(M, P ) = L [1 α(p/p)], (19) where we use the result that the prce ndex n the non-traded sector depends on M only. By (18) and (19), E ncreases n response to an expanson of product dversty because consumers value varety. Furthermore, E naturally decreases when traded goods become cheaper due to the substtuton effect between traded and non-traded goods. 12 Ths effect wll be key n what 11 The numerator n (18) s always decreasng n M, for the elastcty of ψ 1 ( ) exceeds 1 by concavty of ψ( ). Actually, r ψ ( ) could also be moderately decreasng wthout changng that result. We henceforth consder mostly the case where t s ncreasng, as ths seems to be emprcally more relevant. 12 Manufactured goods and servces are arguably substtutes rather than complements, at least for fnal goods. 11

14 follows to understand how trade lberalzaton affects the economy as a whole and the ntersectoral allocaton n partcular. Fnally, (19) and the expressons for markups mply that the equlbrum frm sze, q n = E/(M ), and operatng proft, π n, n the non-traded sector are gven by q n (M, P/P) = L cm [1 α(p/p)] [ 1 r ψ ( ψ 1 (1/M) )], (20) and π n (M, P/P) = L M [1 α(p/p)] r ψ( ψ 1 (1/M) ), (21) respectvely. These expressons wll be helpful to analyze how frms szes change n response to trade lberalzaton. As mpled by (20), an ncreasng r ψ ( ) s suffcent (but not necessary) for entry to reduce frms szes. We now determne the equlbrum mass, M, of frms. To ths end, we assume that there s free-entry so that profts are zero n (21). The zero-proft condton s gven by L [ ] ( 1 α(p/p) rψ ψ 1 (1/M) ) = F. (22) M Condton (22) allows us to pn down the mass of frms n the non-traded sector as a functon of the prce ndex, P, of traded varetes. We can show the followng result: Proposton 2. (Traded prces and the mass of non-traded frms) Assume that r ψ ( ) s ncreasng and that the budget share of non-traded goods, α( ), decreases not too fast. Then there exsts a unque symmetrc free-entry equlbrum n the non-traded sector. Furthermore, a reducton n the prce ndex, P, for the traded goods leads to less frms and hgher markups n the non-traded sector. Proof. When r ψ ( ) > 0, and α(p/p) decreases not too fast, then the operatng proft (21) s a decreasng functon of M. In ths case, (22) has a unque soluton M (P ). Pluggng M nto equatons (19) and (20) and usng the defnton of the markups then determnes a unque symmetrc free-entry equlbrum. Furthermore, because α( ) s a decreasng functon, a reducton n P mples an upward shft of the locus descrbed by the left-hand sde of (22). Hence, cheaper traded goods lead to fewer frms n the non-traded sector. What s the economc meanng of the assumptons underlyng Proposton 2? As dscussed above, the frst assumpton, r ψ > 0, s a necessary and suffcent condton for entry to generate pro-compettve effects. We beleve that ths s the emprcally plausble case. Moreover, ths condton has a purely demand sde-related nterpretaton: a hgher consumpton ndex s equvalent to more product dfferentaton. The second assumpton, namely that α( ) s a slowly For example, publc transportaton and cars are clearly substtutes. Whle we acknowledge that complementartes do exst, we beleve that substtutablty s overall the more plausble assumpton. 12

15 decreasng functon, means that traded and non-traded goods are relatvely poor substtutes for consumers. If we thnk about the former as ncludng mostly manufactured goods, whereas the latter consst mostly of servces, ths seems to be a farly natural assumpton to make. In what follows, we take the assumptons that r ψ ( ) > 0 and that α( ) decreases not too fast as our benchmark. 13 To llustrate the latter by means of a smple example, consder the ces upper-ter utlty: U = [ βu (σ 1)/σ + (1 β)v (σ 1)/σ] σ/(σ 1), 0 < β < 1 < σ. In that case, we have α (P/P) = (P/P) 1 σ. (23) [(1 β)/β] σ + (P/P) 1 σ From equaton (23), we see that when σ 1 (.e., goods become ndependent), the slope of α( ) s less and less steep. Hence, the operatng proft (21) s more lkely to decrease n M snce traded and non-traded goods are poor substtutes. Gven the prce ndex P for traded goods, equatons (18) and (22) unquely pn down the equlbrum prce ndex, P (P ), and the equlbrum mass of frms, M (P ). More precsely, we can solve (22) for M and plug the soluton nto (18). We then obtan a decreasng functon P = P (P ). Ths allows us to express the equlbrum expendture share for non-traded goods as a functon of P only: a(p ) α(p/p (P )). Snce the expendture share s the only channel through whch the traded sector affects the non-traded sector, the mpact of a fall n trade costs on the non-traded sector s fully captured by changes n the prce ndex n the traded sector. How does a change n P affect the consumers expendture shares? Two effects are at work. Frst, there s the standard substtuton effect between traded and non-traded goods, whch ncreases α( ) when traded goods become cheaper. Second, there s the ncome effect that arses because an ncrease n α( ) leads to more demand for traded goods, thus entcng more frms to enter the traded sector. Ths makes competton tougher n that sector and leads to an addtonal fall n P. Snce both effects work n the same drecton, a(p ) unambguously decreases wth P. In other words, f trade lberalzaton reduces the prce ndex P for traded goods, expendtures on non-traded goods always decrease. The key fndng n Proposton 2 s that cheaper traded goods lead to fewer frms n the non-traded sector. Ths, n turn, makes competton n that sector less tough (because r ψ ( ) s ncreasng functon), whch allows non-traded frms to charge hgher markups. In Secton 5.1, we show that trade lberalzaton reduces the prce ndex for traded goods under farly plausble assumptons, thereby reducng competton n the non-traded sector. 13 If α decreases "too steeply", we may loose unqueness. However, for the exstence of a stable free-entry equlbrum, t suffces (though t s not necessary) that both α and r ψ are bounded away from 1. All the comparatve statcs results reman vald for stable equlbra. 13

16 4 Equlbrum n the traded sector We now analyze the equlbrum n the traded sector. We proceed n two steps. Frst, we look at the mpacts of freer trade on the traded sector for the general case of Kmall preferences. We derve suffcent condtons for the prce ndex n the traded sector to fall wth trade lberalzaton. Ths comparatve statc result s crucal to analyze the effects of freer trade n Secton 5.1. Second, we ntroduce ε-ces preferences, whch are a specal but relevant case of Kmball preferences, to derve sharper results. We show that these preferences can get arbtrarly close to ces preference so that the equlbrum dsplays comparatve statc propertes smlar to those prevalng under the ces (see Proposton 5 below). Ths result s mportant for the subsequent analyss snce t allows us to derve many propertes of equlbrum by contnuty n the vcnty of ces preferences for whch results are relatvely easy to compute. 4.1 General case A symmetrc equlbrum n the traded sector satsfes the followng four equlbrum condtons: () zero proft condton: x d r θ ( z d ) 1 r θ (z d ) + τxm r θ (z m ) 1 r θ (z m ) = F cl ; (24) () Kmball preference representaton: N [ θ ( z d) + θ (z m ) ] = 1; (25) () proft maxmzaton: (v) sectoral budget constrant: [ P u = a(p ) = Nu θ ( z d) (1 r θ ( z d ) ) θ (z m ) (1 r θ (z m )) = 1 τ ; (26) cz d 1 r θ (z d ) + cτz m 1 r θ (z m ) ]. (27) We frst solve (25) (26) for the relatve consumpton levels, z d and z m. Gven the mass of frms, N, n each country, the locus n (25) s downward-slopng n (z d, z m )-space, because θ( ) s an ncreasng functon. The slope of the locus n (26) s postve, whch comes from the secondorder condton for proft maxmzaton (that condton s gven by r θ < 2; see Appendx B). Consequently, the two curves have a unque ntersecton denoted by ( z d (N, τ), z m (N, τ) ). How does that ntersecton vary wth the mass of frms, N, and trade costs, τ? When N ncreases, the locus n (25) shfts downwards, whle the locus n (26) remans unchanged. Thus, both z d 14

17 and z m decrease n the mass of frms. An ntutve explanaton for ths result s love for varety: addtonal entry leads consumers to spread ther budget over a larger range of varetes. Hgher trade costs, τ, lead to a downward shft of the locus n (26), whle the locus n (25) remans unchanged. Thus, z d ncreases wth trade costs, whle z m decreases wth trade cost. Intutvely, freer trade shfts relatve demands from domestcally produced varetes towards mported ones. Observe that domestc markups r d θ (zd ) and foregn markups r m θ (zm ) depend on the mass of frms, N, and trade costs, τ, va the relatve consumpton levels z d and z m. Entry drves both markups downward f and only f r θ ( ) s an ncreasng functon. In other words, whether the effect of entry s pro- or ant-compettve s fully determned by the nature of consumers varetylovng behavor. Note that for a gven mass of frms, trade lberalzaton always shfts domestc and foregn markups n opposte drectons. More precsely, a decrease n τ drves domestc markups downwards and foregn markups upwards f and only f r θ ( ) s an ncreasng functon. Otherwse, the result s reversed. In the emprcally plausble case wth pro-compettve effects, domestc markups and foregn markups converge as trade becomes freer: n the lmt, when trade s costless, domestc and export markups are equalzed. We can summarze the foregong results n the followng proposton: Proposton 3. (Equlbrum n the traded sector) Assume that r θ (z) > 0. Then: () there exsts a ( unque symmetrc equlbrum for any gven N; () the equlbrum domestc and foregn markups r ) θ d z d ( and rθ m (zm ) both decrease wth N; and () the domestc markups r ) θ d z d decrease wth τ, whereas the foregn markups rθ m (zm ) ncrease wth τ. Proof. In the text. How does the prce ndex P vary wth the mass of frms N and trade cost τ? To answer that queston, we frst rewrte (27) as follows: [ P = P (N, τ) cn z d (N, τ) 1 r θ (z d (N, τ)) + τz m (N, τ) 1 r θ (z m (N, τ)) and use the defnton of homothetc preferences to get u = u(n, τ) a ( P (N, τ) ) /P (N, τ). We stll need to pn down N, whch s endogenously determned by free entry. To ths end, we use the zero proft condton (24), whch now takes the followng form: [ ] u(n, τ) z d r θ (z d (N, τ)) (N, τ) 1 r θ (z d (N, τ)) + r θ (z m (N, τ)) τzm (N, τ) 1 r θ (z m = F (N, τ)) cl. (28) How the left-hand sde of (28), the traded good producers operatng profts π(n, τ), vares wth N s generally ambguous, so that multple equlbra may a pror exst. Let N denote a soluton to (28). In what follows, we focus on stable equlbra only,.e., those where π/ N < 0 ], 15

18 at N = N. 14 The mpact of a change n τ on frms profts s twofold: () trade lberalzaton reduces costs, whch leads to rsng profts; and () trade lberalzaton shfts the prce ndex P (N, τ), whch may result n tougher competton and hence n lower profts. In any case, (28) mples that at any stable equlbrum trade lberalzaton leads to an ncrease n the equlbrum mass of frms N f and only f the former effect domnates the latter,.e., when π/ τ < 0. If, n addton, P (N, τ) decreases n N and ncreases n τ, then trade lberalzaton also drves down the equlbrum value of the prce ndex P P (N, τ). Thus, we have the followng result. Proposton 4. (Decreasng prce ndex) A suffcent condton for dp /dτ > 0 s gven by: () P / N < 0; and () dn /dτ < 0. Proof. In the text. Under what condtons on the model s prmtves are we sure that P / N < 0 and dn /dτ < 0? As we show n the next secton, these two propertes always hold n the ces model. Hence, they should ntutvely also hold for preferences that are suffcently close to the ces n a sense that we need to make precse. We hence now ntroduce the concept of ε-ces preferences and show that an ε-ces equlbrum exsts. When ε s small enough, that equlbrum has the same qualtatve propertes than the ces equlbrum, safe that markups are varable. The latter property s very useful to nvestgate the behavor of models wth Kmball preferences that are, however, close to ces preferences and nhert many of ther propertes by contnuty. 4.2 ε-ces preferences As dscussed n Secton 2.1, Kmball preferences nclude the ces as a specal case. Gven that there s a contnuum of Kmball-type preferences, we can choose preferences that are arbtrarly close to the ces wthn that class. Let θ(z) = z ρ exp ( ϕ(z) ), ρ σ σ 1 (0, 1), (29) where ϕ( ) s suffcently small n a sense that ϕ( ) < ε. Intutvely, f ϕ( ) s close to zero then (29) wll be close to ces preferences. 15 We hence call these preferences ε-ces preferences. We relegate the techncal detals to the appendx and descrbe n Appendx A.2 condtons whch ensure these preferences are close to standard ces preferences. We also derve the equlbrum condtons of the model for that case. We can prove the followng result. Proposton 5. (Exstence and contnuty of ε-ces equlbrum) For ε-ces preferences wth ε suffcently small, an equlbrum: () always exsts; and () s a small perturbaton of the ces equlbrum. 14 By dong so, we do not rule out the possblty of multple stable equlbra. We wll do so n the next subsecton, where we focus on a specfc nstance of preferences that lead to a unque stable equlbrum. 15 Consder the case where ϕ( ) 0, then (29) bols down exactly to ces preferences. 16

19 Proof. See Appendx A.3. The key result for the subsequent analyss s the contnuty property of ε-ces equlbra wth respect to the ces, as stated by clam () of Proposton 5. Ths property explans why our model largely retans the tractablty of the ces case whle allowng for departures from t along several emprcally relevant dmensons varable markups and frm szes. As shown n the next secton, t provdes a powerful tool for comparatve statcs analyses to reveal, e.g., the mpacts of trade lberalzaton under non-ces Kmball preferences. 5 The mpacts of trade lberalzaton We are now equpped to nvestgate how trade lberalzaton,.e., a decrease n τ, nfluences the equlbrum n the traded and n the non-traded sectors. Frst, we look at how trade lberalzaton drectly affects the traded sector. We then take nto account the ndrect effects of trade lberalzaton on the non-traded sector. Ths orderng s natural snce, as shown before, the effects n the non-traded sector stem entrely from changes n the prce ndex of the traded sector. 5.1 Traded sector We show n Appendx A.4 that when preferences are ces, the followng comparatve statcs hold: d(z d ) dτ > 0 > d(zm ), dτ dp dτ > 0, and dn dτ < 0. (30) Moreover, ntuton suggests that (30) must stll hold by contnuty for the ε-ces case when ε s small enough. Appendx A.4 provdes a formal proof relyng on Proposton 5. These comparatve statc results are mportant for assessng how markups and frm szes n the traded sector react to trade lberalzaton. Markups. We frst look at how domestc and export markups n the traded sector change n response to trade lberalzaton. Recall that n the ces case, markups are nvarant to changes n the mass of frms and to changes n trade costs. Ths s, however, no longer the case under ε-ces preferences, even when ε s small. Put dfferently, even small departures from the ces lead to varable markups, and we can nvestgate ther behavor n response to freer trade. 16 In what follows, we focus on small perturbatons, ϕ, n (29) that generate r θ (z) > 0. In that 16 It s worth notng that the method of establshng comparatve statcs by contnuty lke n Appendx A.3 does not work wth markups, for under the ces we have rθ d / τ = rm θ / τ = 0. Hence, we can say nothng usng contnuty arguments. 17

20 case, d(z d ) /dτ > 0 > d(z m ) /dτ drectly mples d(r d θ ) dτ > 0 > d(rm θ ). (31) dτ In words, trade lberalzaton leads to a reducton n markups for locally produced varetes, and to an ncrease n the markups of mported varetes. Snce domestc markups exceed export markups, ths means that freer trade leads to the convergence of markups for traded goods across countres. Yet, as we show n the next subsecton, t also leads to a dvergence of markups across ndustres wthn countres. Frm sze. We now study how trade lberalzaton affects frm sze, q L(x d + τx m ). In the ces case, t s well known that trade lberalzaton does not affect frm sze but leads to a decrease n domestc sales, x d, and an ncrease n exports, τx m. Slghtly perturbng the ces preferences reveals addtonal effects of trade lberalzaton on frm sze. To see ths, we combne (A-6) and (A-8) n Appendx A.2 to obtan: r θ (z d ) q 1 r θ (z d ) Lτxm [ rθ (z d ) 1 r θ (z d ) r ] θ(z m ) = F 1 r θ (z m ) c. (32) As we have shown before, a reducton n trade costs τ leads to: () an ncrease n output for the foregn market, τx m, by contnuty wth the ces case; and () an ncrease n the markups of mported varetes, r θ (z m ), and a decrease n the markup of domestc varetes, r θ (z d ). Note also that, because we are close to the ces, the latter effect s of second order compared to the former effect. More generally, all new effects that arse under ε-ces preferences are of second order and thus are neglgble compared to the frst-order effects for suffcently small values of ε. Furthermore, as shown n Appendx A.2., we have z m < z d, so that the expresson n square brackets n (32) s postve. To sum up, the second term on the left-hand sde of (32) ncreases wth trade lberalzaton, because the postve effect of trade lberalzaton on output (τx m ) domnates the negatve effect on markups. Ths mples that the frst term must also ncrease for the zero-proft condton to hold. Snce domestc markups, r θ (z d ), decrease wth a fall n trade costs, we can conclude that frm output q must hence ncrease wth trade lberalzaton. 17 Sze of the manufacturng sector. We now nvestgate how trade lberalzaton affects the relatve sze of the traded sector, gven by N (cq + F )/L. Note frstly that the labor force of ndvdual frms, cq + F, ncreases, yet that ths s a second-order effect under ε-ces preferences (because frm sze s constant under the ces). Note secondly that the mass of frms n the traded sector N ncreases because of changes n a(p ). Ths consttutes the frst-order effect for changes 17 Evdence of ncreasng frm szes n the export sector followng trade lberalzaton s provded by, e.g., Levnsohn (1999). 18

21 Table 1: Trade openness and the share of manufacturng employment. Dependent varable: log(manufacturng share) (1) (2) (3) (4) (5) (6) ln(openness) (0.034) (0.026) (0.024) ln(openness, mports) a a a (0.033) (0.033) (0.029) ln(openness, exports) a a a (0.027) (0.027) (0.027) ln(value-added per employee) a a (0.021) (0.019) ln(output per employee) a a (0.025) (0.023) p-value: ln(openness, mports) ln(openness, exports) = 0 Observatons 2,147 1,988 2,003 2,147 1,988 2,003 R-squared Notes: Unbalanced panel of 145 countres between 1980 and See Appendx C for addtonal nformaton on the data. All specfcatons nclude country and year fxed effects. Huber-Whte robust standard errors are reported n parentheses. Sgnfcance levels: a : p < 0.01, b : p < 0.05, c : p < 0.1. n the sze of the traded sector. If we assume that a(p ) s almost flat, then the amount of labor allocated to the traded sector ncreases only a lttle n the wake of trade lberalzaton (see the dscusson n Secton 3). In other words, the model predcts that the traded sector expands slghtly as trade gets freer. Ths predcton may seem counterfactual snce the traded sector whch s mostly manufacturng has decreased as a share of employment n all developed countres over the last 40 years, despte ncreasng trade lberalzaton. One reason for that change s the ncrease of manufacturng productvty va the substtuton of captal for labor. Ths effect s absent from our model where frm-level productvty s nvarant. Columns (1) (3) n Table 1 report emprcal evdence consstent wth those observatons. As can be seen from the table, the mpact of trade openness measured by the rato of mports plus exports over gdp on the share of manufacturng employment n the economy s very small and nsgnfcant, whle productvty gans are assocated wth fallng shares of manufacturng employment. In our model, trade costs are symmetrc so that trade lberalzaton ncreases both mport competton and export opportuntes. When breakng down trade openness n terms of mports and exports n the data, we see that a 1% ncrease n both mports and exports has a negatve and statstcally sgnfcant effect on the share of manufacturng employment n specfcaton (4). Ths result seems to run aganst the expanson of manufacturng wth ncreasng trade lberalzaton as predcted by the model. However, the negatve effect dsappears once we control for productvty n regressons (5) and (6),.e., a symmetrc trade expanson has an nsgnfcant mpact on manufacturng 19

22 employment. 18 The followng proposton summarzes our key results concernng the mpacts of trade lberalzaton on frms n the traded sector. Proposton. (Effects of trade lberalzaton on the traded sector) Consder the case of ε-ces preference wth r θ > 0. Then there exsts ε such that for every 0 < ε < ε, trade lberalzaton leads to: () lower markups for domestcally produced traded varetes; () hgher markups for mported traded varetes; () more frms n the traded sector; (v) larger frms, as measured by ther total output, n the traded sector; and (v) more labor allocated to the traded sector. Proof. In the text. Observe that the same qualtatve propertes also hold n other models of monopolstc competton, such as those by Krugman (1979), Behrens and Murata (2007, 2012), Zhelobodko et al. (2012), and Kchko et al. (2014). The novelty of our results s to brng together multple sectors and costly trade n general equlbrum wth varable elastcty of substtuton for a rch class of homothetc preferences. We further show n Secton 6 that the results of Proposton 5.1 may stll hold outsde of a small neghborhood,.e., when we move farther away from the ces case whle keepng preferences as gven by (4). 5.2 Non-traded sector We now turn to the non-traded sector. As shown n Secton 3, the mpacts of trade costs on the non-traded sector are fully captured by the changes n the prce ndex assocated wth the traded sector. Snce we have establshed the mpact of trade costs on the prce ndex n the foregong subsecton, the analyss s now straghtforward. Markups. Because trade lberalzaton makes competton n the traded sector tougher, as shown n Secton 5.1, the prce ndex for traded goods falls as trade gets freer. Furthermore, when traded goods get relatvely cheaper, the expendture share on non-traded goods falls. As a consequence, f r ψ > 0 and α( ) s a slowly decreasng functon, a decrease n P reduces the equlbrum mass of frms, M, n the non-traded sector. Ths, n turn, leads to an ncrease n both the prce ndex and the markups n that sector, as shown n Secton 3. It s worth pontng out here that ths effect s entrely drven by the general equlbrum nature of the model, whch leads to 18 These results are llustratve only and we do not clam that they have any causal nterpretaton. For example, the productvty changes may pck up trade lberalzaton effects as emphaszed n the heterogeneous frms lterature followng Meltz (2003). When frms are heterogeneous, the ntersectoral allocaton effects may well be dfferent. For example, Trefler (2004) documents that the Canada-US free trade agreement led to a decrease of Canadan manufacturng employment by 5%, despte substantal productvty gans and plant expansons for the more productve frms. Levnsohn (1999) fnds only lttle between-sector reallocatons usng epsodes of trade lberalzaton n Chle. 20