Technical Appendix: Multi-Product Firms and Trade Liberalization (Not For Publication)

Transcription

1 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition (Not For Publiction) Anrew B. Bernr Tuck School of Business t Drtmouth, CEPR & NBER Stephen J. Reing Princeton University & CEPR Peter K. Schott Yle School of Mngement & NBER November 200

2 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 2 Contents. Introuction 3 2. Relte Literture on Multi-Prouct Firms 3 3. The Moel Weighte Averge Prouctivity Common n Country-Speci c Components of Prouct Attributes Stey-stte Prouct Aing n Dropping Multi-Prouct Firms n Comprtive Avntge Symmetric Countries Existence n Uniqueness of Equilibrium Proof of Proposition Reuctions in Vrible Tre Costs in the Open Economy Equilibrium Proof of Proposition Proof of Proposition Symmetric Country Close Form Solutions for Preto Distributions Generl Equilibrium with Symmetric Countries Mrgins of Tre with Symmetric Countries Preto Distributions n Heterogeneous Fixe Costs with Symmetric Countries Asymmetric Countries Asymmetric Country Equilibrium Prouct Attributes n Firm Ability Cuto s Free Entry Mss of rms Tre Shres Wges Price Inices Welfre Generl Equilibrium Properties of the Asymmetric Country Equilibrium Mrgins of Tre Across Countries Extensive Mrgin

3 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition Intensive Mrgin Mrgins of Tre Across Firms Extensive Mrgin Intensive Mrgin Observble Implictions Empiricl Evience Tre Liberlition n Prouct Rnge Mrgins of Tre Across Countries Mrgins of Tre Across Firms Mrgins of Tre Within Firms Hierrchies of Mrkets Dt Appenix Census of Mnufctures (CMF) Dt Linke/Longituinl Firm Tre Trnsction Dtbse (LFTTD) Other Dt Sources

4 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 4. Introuction This ppenix contins the technicl erivtions of expressions for ech section of the pper, the proofs of propositions n itionl supplementry mteril. Sections correspon to the sections with the sme title in the min pper. 2. Relte Literture on Multi-Prouct Firms No erivtions require. 3. The Moel In Section 3.., we give the expression for weighte verge prouctivity. In Section 3.2., we consier hybri speci ction of the moel where prouct ttributes hve both common n country-speci c components, s iscusse in footnote 7 in the pper. In Section 3.3., we exten the moel to incorporte stey-stte ing n ropping of proucts, s iscusse in footnote 8 in the pper. In Section 3.4., we exten the moel to incorporte multiple fctors of prouction n multiple inustries, s referre to in footnote 0 in the pper. 3.. Weighte Averge Prouctivity Weighte verge prouctivity epens on the rm bility cuto ( ) n weighte verge of prouct ttributes for ech rm bility ( ~ (')): 2 ~' 4 G i Z 3 ' ~ (') gi (') ' 5 ; () " ~ (') = Z (') Z (') () # ; (2) where ~ (') epens on the prouct ttributes cuto for ech rm bility ( (')) Common n Country-Speci c Components of Prouct Attributes Consier hybri speci ction of the moel, where prouct ttributes hve both common n country-speci c components: & = & k jk ; 0 & where k is common cross countries j for prouct k; jk vries cross both countries j n proucts k; n & prmeteries the reltive importnce of the common n country-speci c components.

5 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 5 Once the sunk entry cost is pi, rm observes the common component of prouct ttributes for ech prouct, k, n the country-speci c component of prouct ttributes, jk, for ech prouct n country. The common component is rwn seprtely for ech prouct from continuous istribution k ( k ) with cumultive istribution function Z k ( k ). The country-speci c component is rwn seprtely for ech prouct n country from continuous istribution jk ( jk ) with cumultive istribution function Z jk ( jk ). To mke use of lw of lrge numbers results, we ssume tht the rm bility n prouct ttributes istributions re inepenent of one nother n inepenently istribute cross rms. Similrly, we ssume tht the common prouct ttributes istribution, k ( k ), is inepenently istribute cross proucts, while the country-speci c prouct ttributes istribution, jk ( jk ), is inepenently istribute cross proucts n countries. With these ssumptions, rm s profitbility is now correlte cross proucts n countries for two resons. First, higher bility (') rises rm s pro tbility cross ll proucts n countries. Secon, higher common component of prouct ttributes ( k ) rises rm s pro tbility cross ll countries for given prouct. These correltions re however imperfect becuse of stochstic vrition in the country-speci c component of prouct ttributes. As &!, the hybri cse reuces to the common-prouct-ttributes speci- ction iscusse in the pper. As &! 0, the hybri cse reuces to the country-speci c-prouct ttributes speci ction iscusse in the pper Stey-stte Prouct Aing n Dropping As the focus of the pper is the cross-section istribution of exports cross rms, countries n proucts, we follow much of the literture on rm heterogeneity in interntionl tre in bstrcting from ynmics. In this section, we show tht the moel cn be extene to introuce stochstic vrition in rm bility n prouct ttributes over time. In this extension, we embe simpli e version of the ynmics from the close economy moel of Bernr, Reing n Schott (200) in n open economy setting. While we consier symmetric countries n the country-speci cprouct-ttributes speci ction, it is strightforwr to inste consier symmetric countries n the common-prouct-ttributes speci ction. The speci ction of entry, prouction n emn is similr to tht consiere in the pper, except tht we consier continuous time version of the moel with stey-stte entry n exit of rms n stey-stte ing n ropping of proucts within rms. Once rm incurs the sunk entry cost, f e, rm bility n prouct ttributes re rwn from the continuous istributions g (') n () respectively, with cumultive istributions G (') n Z (). After rm observes its initil vlues of bility n prouct ttributes, it ecies whether to prouce or exit. If the rm

6 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 6 exits, its prouction knowlege is lost, n the sunk cost must be incurre gin in orer for the rm to re-enter. If the rm enters, it fces Poisson probbility > 0 of shock to bility ', in which cse new vlue for bility ' 0 is re-rwn from the sme istribution s upon entry g (' 0 ). The rm lso fces Poisson probbility " > 0 of n iiosyncrtic shock to prouct ttributes for its vriety of given prouct in given country, in which cse new vlue for prouct ttributes 0 is re-rwn from the sme istribution s upon entry 0. As in Melit (2003), the rm lso fces constnt exogenous probbility of eth of > 0, s result of force mjeure events beyon its control. As prouct ttributes for rm s vriety of ech prouct chnge over time, previously profitble proucts n mrkets become unpro tble n re roppe when flls below the prouct ttributes cuto for the mrket { ('), x (')}. Similrly, previously unpro tble proucts become vible n re e when rises bove the prouct ttributes cuto for the mrket { ('), x (')}. Since the istribution from which prouct ttributes re rwn following stochstic shock is the sme s the istribution from which they re rwn upon entry, the sttionry istribution for prouct ttributes, (), tkes simple form: () = () ; (3) where the sttionry istribution for prouct ttributes conitionl on prouct being supplie to mrket is trunction of () t the prouct ttributes cuto for the mrket { ('), x (')}. As the istribution from which rm bility is rwn following stochstic shock is the sme s the istribution from which it is rwn upon entry, the sttionry istribution for rm bility, g ('), lso tkes simple form. The sttionry istribution for rm bility conitionl on supplying mrket is trunction of the istribution g (') t the rm bility cuto for serving the mrket: 8 g(') >< for ' ' G(' ) in the omestic mrket g (') = g(') G(' >: x ) for ' ' x in ech export mrket : (4) 0 otherwise The etermintion of generl equilibrium remins lrgely the sme s in the pper with few pproprite moi ctions. In the ynmic extension consiere here, the combintion of sunk entry cost n stochstic shocks to rm bility genertes n option vlue to rm entry. If rm chooses to exit, it forgoes both the net present vlue of its instntneous ow of pro ts n lso Assuming tht rm bility n prouct ttributes re re-rwn following stochstic shock from the sme istributions s upon entry is simplifying evice, which enbles us to introuce ynmics in s trctble wy s possible. The close economy moel of Bernr, Reing n Schott (200) consiers richer forms of ynmics tht llow for seril correltion in rm bility n prouct ttributes. While introucing seril correltion complictes the moel s ynmics, it oes not chnge its cross-sectionl preictions tht re our focus.

7 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 7 the option of experiencing stochstic bility shocks. Therefore, with this option vlue to rm entry, the threshol bility for rm entry n exit will in generl lie below the bility t which the instntneous ow of totl rm pro ts is equl to ero. 2 The vlue of rm with bility ' is etermine ccoring to the following Bellmn eqution: v (') = (') Z + + v ' 0 g ' 0 ' 0 : (5) + ' As the istribution from which new vlue for bility is rwn following stochstic shock is the sme s upon entry n is inepenent of rm s existing vlue for bility, the solution to this Bellmn eqution tkes simple form. Substituting for v (' 0 ) on the right-hn sie of (5) using the tril solution v (') = (') + R ' (' 0 ) g (' 0 ) ' 0, n solving for n, yiels the equilibrium vlue of rm: v (') = (')! " Z # G ' ' 0 g ' 0 ' 0 : (6) ' Therefore the equilibrium vlue of rm with bility ' is weighte verge of the current ow of rm pro ts n the expecte ow of rm pro ts following stochstic bility shock, where the weights epen on the probbility of rm eth, the probbility of n bility shock n the probbility tht rm remins ctive following n bility shock. Substituting the expression for the equilibrium vlue of rm into the free entry conition, n rerrnging, the free entry conition cn be re-written s follows: 2 Z V = 4 (') + R 3 + ' [ (' 0 ) (')] g (' 0 ) ' G ' g (') ' = f e ; (7) ' which cn be simpli e to yiel the following expression: Z! (') V = + G ' g (') ' = f e ; (8) ' which cn be in turn re-written s: Z! Z (') V = + G ' g (') ' + n ' x '! x (') + G ' g (') ' = f e : As in the pper, the moel hs recursive structure, n the etermintion of generl equilibrium is strightforwr. We begin by etermining {', ' x, (' ), x (' x)} using four equilibrium reltionships. To erive the rst of these reltionships, we use the expression for totl rm pro ts 2 In contrst, xe n mrginl prouction costs for iniviul proucts hve no sunk component, n prouct ttributes for rm s vriety of ech prouct evolve inepenently of whether or not the vriety is supplie. Therefore, once rm hs ecie to enter, the rm s ecision whether or not to supply ech prouct reuces to perio-byperio comprison of contemporneous revenue n prouction costs.

8 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 8 in the pper s well s (') = (' =') (' ) n x (') = (' x=') x (' x), which together imply tht the free entry conition (8) cn be written s: V = + Z ' x Z ' 2 4 " Z Z (' =') (' ) (' x =') x(' x ) ' ' ' x x (' x) 3 '! A f () F 5 g (') ' + G c # ng (')! f x () F x ' = f e : + G c ' ' ' (9) Since the only sunk cost in the moel is the entry cost, f e, n we consier n equilibrium with selection into export mrkets, the lowest bility rm tht enters serves only the omestic mrket: ' < ' x. Therefore only rm s ecision of whether or not to serve the omestic mrket is ecte by the option vlue of entry. In contrst, the rm s ecision whether or not to serve the export mrket is etermine by comprison of the instntneous ow of revenue n the xe exporting costs. The prouct n rm exporting cuto s re therefore etermine s in the pper: Z " # x (' f x () = F x (0) x) x(' x ) ' x = fx f (' ) x (' x) ' : () In eciing whether or not to enter n supply the omestic mrket, rm tkes into ccount the option vlue of entry. At the omestic cuto bility, ', the vlue of the rm is equl to ero, v (' ) = 0, which from (6) implies tht the ow of totl rm losses exctly equls the probbility of stochstic shock to rm bility times expecte pro ts conitionl on stochstic shock occurring: Z (' ) A f p () F 5 (' =') (' ) = + Z ' Z ' x ' 2 Z (' ) 4 (' =') (' ) " Z x (' x ) (' x =') x(' x ) 0 '! ' A f () F 5 g (') ' ' + G ' # ' ng (')! ' x x (' f x () F x ': x) + G c To chrcterie {', ' x, (' ), x (' x)}, we rst use (0) to etermine x (' x) inepenent of the other equtions of the moel. Secon, we use () to etermine ' x s function of ', (' ) n x (' x). Thir, substituting for ' x n using the vlue of x (' x) etermine bove, (9) n (2) provie two equtions tht together etermine {', (' )}. Hving chrcterie { x (' x), ', (' )}, the equilibrium vlue of ' x follows immeitely from (). Finlly, hving etermine ' (2)

9 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 9 {', ' x, (' ), x (' x)}, the remining elements of the equilibrium vector cn be etermine s in the pper. In this extension of the moel in the pper, the generl equilibrium fetures stey-stte ing n ropping of proucts n estintions s well s stey-stte entry n exit of rms. Ech perio mesure of new rms incur the sunk entry cost. Of these new rms, those with n bility rw bove the omestic cuto (' ) enter, while those with n bility rw below ' exit. Among incumbent rms, rm with unchnge bility supplies constnt mesures of proucts to the omestic n export mrkets, but the ientity of these proucts chnges s stochstic shocks to prouct ttributes occur. As result of these stochstic shocks rm with unchnge bility rops mesure of the proucts previously supplie to ech mrket n s n equl mesure of the proucts not previously supplie to ech mrket. Finlly, s stochstic shocks to n incumbent rm s bility occur, the mesure of proucts supplie to ech mrket expns with increses in bility n contrcts with ecreses in bility. An incumbent rm enters export mrkets when its bility rises bove the exporting cuto (' x). An incumbent rm exits enogenously when its bility flls below ' or exogenously when eth occurs s result of force mjeure consiertions beyon the control of the rm. While the extene moel incorportes stey-stte ing n ropping of proucts n estintions, the moel s preictions for the cross-section istribution of exports cross rms, proucts n countries remin unchnge. As these cross-sectionl preictions re the focus of our nlysis, we o not pursue this ynmic extension further in the pper Multi-Prouct Firms n Comprtive Avntge In this section of the ppenix, we show tht the moel cn be extene to incorporte comprtive vntge bse on cross-country i erences in fctor bunnce n cross-inustry ifferences in fctor intensity. In this extension, we embe our moel of multi-prouct rms within the two-fctor, two-country n two-inustry heterogeneous rm frmework of Bernr, Reing n Schott (2007). The moel in the pper cn be viewe s cpturing single inustry contining mny proucts, with rms supplying i erentite vrieties of these proucts. We now generlie this frmework by nlying two inustries, i = ; 2, ech of which hs this structure. The two inustries enter n upper tier of the representtive consumer s utility tht tkes the Cobb-Dougls form: U = U U 2 2 ; + 2 = ; = ; (3) where U i is n inex for inustry i tht is e ne over the consumption C k of continuum of

10 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 0 proucts k within the inustry (s in eqution () in the pper), n C k is itself n inex tht is e ne over the consumption c k (!) of continuum of vrieties! within ech prouct (s in eqution (2) in the pper). Inustries now constitute n upper tier of utility, proucts form n intermeite tier n vrieties occupy lower tier. We ssume for simplicity tht the two inustries hve the sme elsticity of substitution cross proucts within inustries () n cross vrieties within proucts (). We lso ssume for simplicity tht there re only two countries: home n foreign. The home country is ssume to be skill-bunnt reltive to the foreign country n inustry is ssume to be skill-intensive reltive to inustry 2. The combintion of inustry-level vrition in fctor intensity n country-level vrition in fctor bunnce gives rise to enowment-riven comprtive vntge. The skille wge in the home country is chosen s the numerire. To enter n inustry i, rm must incur the sunk entry cost for tht inustry, which equls f ei (w S ) i (w L ) i, where ws enotes the skille wge, w L correspons to the unskille wge n i prmeteries inustry fctor intensity. After the sunk entry cost for n inustry is pi, the rm rws its bility n vlues of prouct ttributes for tht inustry. The istributions of rm bility n prouct ttributes re inepenently n ienticlly istribute cross inustries, so tht informtion bout bility within n inustry cn only be obtine by incurring the sunk entry cost for tht inustry. The istributions of rm bility n prouct ttributes re lso inepenently n ienticlly istribute cross countries, which ensures consistency with the Heckscher-Ohlin moel s ssumption of common technologies cross countries. The technology for prouction hs the sme fctor intensity s for entry. 3 While the fctor intensity of prouction vries cross inustries, ll proucts within n inustry re moelle symmetriclly n therefore hve the sme fctor intensity. 4 To supply vriety of prouct to the omestic mrket, rm must incur xe n vrible costs. The vrible cost epens upon the rm s bility s in the pper. The xe n vrible costs use the two fctors of prouction with the sme proportions. Hence the totl cost of serving the omestic mrket for rm in inustry i is: Z T C i = F i + f i + q! i (' i ; ik ) k (w S ) i (w L ) i; > k2 i(' i ) ' > 2 > 0; (4) i where i (' i ) enotes the (enogenous) rnge of proucts supplie to the omestic mrket by rm with bility ' i in inustry i. The xe costs of serving n export mrket n supplying 3 Allowing fctor intensity i erences between entry n prouction introuces itionl interctions with comprtive vntge s iscusse in Bernr, Reing n Schott (2007). 4 The symmetry of proucts within inustries is clerly simpli ction, but is useful for the lw of lrge numbers results tht etermine the frction of proucts supplie by rm with given bility.

11 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition prouct to n export mrket re moelle nlogously. We etermine generl equilibrium using n pproch very similr to tht use in the pper n therefore omit the reporting of similr equtions here to conserve spce. The mesure of proucts supplie to the omestic n export mrkets by rm with given bility cn be etermine using expressions nlogous to those in the pper. Similrly, the ero-pro t n exporting cuto s for bility cn be etermine using the sme line of resoning s in the pper. Entry, prouction n exporting costs ll hve the sme fctor intensity, n therefore terms in fctor prices cncel from the relevnt expressions. There is, however, n importnt generl equilibrium interction between comprtive vntge n rms prouct supply ecisions. Comprtive vntge n tre costs together generte cross-country i erences in inustry price inices. As in Bernr, Reing n Schott (2007), these i erences, in turn, generte greter export opportunities in comprtive-vntge inustries thn comprtive-isvntge inustries. The reltive price inices for the two inustries vry cross countries becuse of the combintion of comprtive-vntge-bse specilition n tre costs. Specilition les to lrger mss of omestic rms reltive to foreign rms in country s comprtive-vntge inustry thn in its comprtive-isvntge inustry. Vrible tre costs introuce wege between omestic n export prices for vriety. Aitionlly, the xe costs of becoming n exporter imply tht not ll rms export, n the xe costs for exporting iniviul proucts imply tht not ll the proucts supplie omesticlly re exporte. Combining specilition n tre costs, the comprtive-vntge inustry hs greter mss of lower-price omestic vrieties reltive to the mss of higher-price foreign vrieties thn the comprtive-isvntge inustry. As result, the price inex in the omestic mrket is lower reltive to the price inex in the export mrket in the comprtive-vntge inustry. Hence, the egree of competition in the omestic mrket is higher reltive to the egree of competition in the export mrket in the comprtive-vntge inustry. These i erences in the egree of competition in turn imply tht vrible pro ts in the export mrket re greter reltive to vrible pro ts in the omestic mrket in the comprtive-vntge inustry thn in the comprtive-isvntge inustry. Proposition A Other things equl, the opening of the close economy to tre les to: () greter reuction in the rnge of proucts supplie to the omestic mrket in the comprtivevntge inustry thn in the comprtive-isvntge inustry ( H F 2 (') > F (')), (') > H 2 (') n (b) lrger increse in the ero-pro t cuto for bility below which rms exit in the comprtivevntge inustry thn the comprtive-isvntge inustry (' H > ' H 2 n ' F 2 >

12 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 2 ' F ). Proof. The reltionship between the exporting cuto bility, ' xi n the ero-pro t-cuto bility, ' i, in the home country is: ' H xi = H i ' H i ; H fxi R H i i f i R F P H i P F i i (' i ) xi (' xi ) ; (5) where n nlogous expression hols for the foreign country. Compring the free entry conitions in the open n close economies, the expecte vlue of entry in the open economy is equl to the vlue for the close economy plus n itionl positive term which cptures the expecte pro ts from the export mrket. Since xi (' i ) = (' xi =' i) xi (' xi ) n ' xi = positive term is lrger, the smller the vlue of ' H x ='H ' H x2 ='H 2 = H H 2 = 2 fx =f f x2 =f 2 i' i, this itionl i. Diviing eqution (5) for the two inustries:! P (' ) = x (' x ) H =P2 H 2 = x2 (' x2 ) P F =P 2 F The reminer of the proof follows the sme structure s the proof of Proposition 4 in Bernr, Reing n Schott (2007) n we present n bbrevite version here. The price inex for the skill-intensive inustry reltive to the lbor-intensive inustry is lower in the skill-bunnt country thn the lbor-bunnt country: P H =P H 2 ' 2 : < P F =P 2 F. Therefore, in the bsence of other i erences in prmeters cross inustries except fctor intensity (common vlues of i, F, f, F x, f x, f e, g (') n () cross inustries): H < H 2 n similrly F 2 < F. Hence, the itionl positive term in the free entry conition cpturing expecte pro ts in the export mrket is lrger in the comprtive-vntge inustry thn the comprtive-isvntge inustry. Noting tht i (') = (' i =' i) i (' i ), xi (') = (' xi =' i) xi (' xi ), n 'H xi = H i 'H i, the expecte vlue of entry in the free entry conition (9) in the pper is monotoniclly ecresing in ' i. Therefore, the comprtive-vntge inustry s lrger increse in the expecte vlue of entry following the opening of tre requires lrger rise in the ero-pro t cuto for rm bility ' i in orer to restore equlity between the expecte vlue of entry n the unchnge sunk entry cost: ' H > 'H 2 n 'F 2 > 'F. Since i (') = (' i =' i) i (' i ) n i (' i ) is unchnge by the opening of tre, the comprtive-vntge inustry s lrger rise in the ero-pro t cuto for rm bility ' i implies greter reuction in the rnge of proucts supplie to the omestic mrket in the comprtivevntge inustry thn in the comprtive-isvntge inustry: H F 2 (') > F ('). (') > H 2 (') n

13 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 3 4. Symmetric Countries In Section 4.., we estblish the existence n uniqueness of equilibrium with symmetric countries. In Section 4.2., we prove Proposition in the pper, which exmines the opening of the close economy to tre. In Section 4.3., we emonstrte tht similr, though more nunce, preictions hol for reuctions in vrible tre costs in the open economy equilibrium. In Section 4.4., we prove Proposition 2 in the pper, which exmines the reltionship between the mrgins of tre n vrible tre costs. In Section 4.5., we prove Proposition 3 in the pper, which is concerne with the reltionship between the mrgins of tre n rm bility. In Section 4.6., we report close form solutions for symmetric countries with Preto istributions of rm bility n prouct ttributes. 4.. Existence n Uniqueness of Equilibrium The symmetric country generl equilibrium is reference by the sextuple: {', ' x, (' ), x (' x), R, P }. As the moel hs recursive structure, the etermintion of generl equilibrium is strightforwr. In rst bloc of two equtions, (' ) n x (' x) cn be etermine inepenently of the other enogenous vribles of the moel. Using the ero-pro t n exporting cuto s for rm bility n prouct ttributes, the unique equilibrium vlues of (' ) n x (' x) re implicitly e ne by: 2 Z 4 (' ) Z x(' x ) ' 3! 5 f () = F (6) " # x (' f x () = F x : (7) x) Hving etermine (' ) n x (' x) s function of prmeters, secon bloc of two equtions etermines ' n ' x. The rst of these equtions is the reltionship between the exporting n ero-pro t cuto s for rm bility, which tkes the following form for symmetric countries: where ' x = ' ; fx f of equtions. (' ) x (' x) ; (8) hs been etermine s function of prmeters from the solutions to the rst bloc The secon eqution is the free entry conition, which with symmetric countries

14 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 4 becomes: 2 2 Z Z! 3 V = ' ' (' )=' ' 5 ' ' f () F 5 g (') ' { } Term A Z " Z " # ' +n # ' x ' x x (' x )=' x (' x) ' f x () F x g (') ' = f e ; x { } Term B where Term A cptures expecte pro ts in the omestic mrket; Term B cptures expecte pro ts in the export mrket; n we hve use: (9) (') = (' =') (' ) (20) x (') = (' x=') x (' x) : (2) Using the reltionship between the cuto s for bility (8) to substitute for ' x, the free entry conition (9) cn be expresse in terms of single unknown, ' : 2 2 Z Z! 3 V = ' ' (' )=' ' 5 ' ' f () F 5 g (') ' (22) Z " Z " # ' +n # ' ' x (' x )=' x (' x) ' f x () F x g (') ' = f e ; where, (' ) n x (' x) hve been etermine s function of prmeters from the solutions to the rst bloc of equtions n re invrint with respect to '. By inspection of (22), noting tht, (' ) n x (' x) hve been etermine s function of prmeters lone, it is evient tht the free entry conition hs the following properties: () The right-hn sie is the constnt sunk entry cost: f e > 0; (b) As '! 0, the left-hn sie converges towrs in nity: V! ; (c) As '!, the left-hn sie converges towrs ero: V! 0; () The left-hn sie is monotoniclly ecresing in ', since higher vlue of ' rises the lower limits of both of the ouble integrls n lso reuces the vlue of the terms insie the ouble integrls. It follows tht there exists unique xe point, where the expecte vlue of entry (V ) is equl to the constnt sunk entry cost (f e ), s shown in Figure. Hving etermine the unique equilibrium vlue of ', ' x follows immeitely from the reltionship between the cuto s (8) n the solutions to the rst bloc of equtions. Thus, the rst two blocs of equtions etermine { (' ), x (' x), ', ' x}, from which we cn solve for the omestic n exporting cuto s for prouct ttributes for ll rm bilities, (') n x (') respectively, using (20) n (2).

15 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 5 Figure : Existence of Unique Equilibrium ' A thir bloc of equtions etermines the remining components of the equilibrium sextuple: {R, P }. From the free entry conition, V = [ G (' )] = f e, n the reltionship between the mss of rms proucing n the mss of entrnts, M = [ G (' )] M e, totl pyments to lbor use in entry equl totl pro ts: M = M e f e = L e ; where we hve use the choice of numerire n country symmetry: w i = w =. Totl pyments to lbor use in prouction equl totl revenue minus totl pro ts: R M = L p ; from which it follows tht totl revenue equls the economy s lbor enowment n the lbor mrket clers: R = L. To etermine the price inex for ech prouct, we use the following reltionship: P = " m ~' + nm x ~' x # ; (23) where we now show tht ech of the terms on the right-hn sie cn be written in terms of {', ' x, (' ), x (' x), R}, which hve been lrey etermine bove. Weighte verge bility in the omestic mrket is: " ~' G ' Z ' # ' ~ (') g (') ' ;

16 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 6 " ~ (') = Z ( (')) Z (') () which cn be uniquely etermine from {', (' )} using (20). Weighte verge bility in the export mrket is: " ~' x G (' x) Z ' x " ~ x (') = Z ( x (')) # # ' ~ x (') g (') ' ; Z x(') () # which cn be uniquely etermine from {' x, x (' x)} using (2). The mesures of rms supplying ech prouct to the omestic n export mrkets re: " Z! # m = [ Z ( ('))] g (') G ' ' M; (24) m x = ' " Z ' x [ Z ( x ('))] where the mss of rms proucing is: g (') G (' x) # ' M; (25) ; ; M = R r ; (26) n verge rm revenue is: Z " Z r = ' (') (') f () #! Z " + G Z (' x) G ' ' x x(') x (')! g (') G ' ' (27) # g (') f x () G (' ': x) From (20) n (2), r is uniquely etermine by {', ' x, (' ), x (' x)}. Given r n R = L, we immeitely obtin M using (26). Given M n {', ' x, (' ), x (' x)}, the nl two components of the price inex, m n m x in (24) n (25), cn be etermine using (20) n (2). We hve therefore etermine the price inex, which completes the chrcterition of the equilibrium sextuple {', ' x, (' ), x (' x), R, P } Proof of Proposition Proof. We rst exmine the impct of the opening of the close economy to tre on '. With symmetric countries, the open economy free entry conition cn be written s in (9), which is

17 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 7 reprouce below for clrity: 2 2 Z Z! 3 V = ' ' (' )=' ' 5 ' ' f () F 5 g (') ' { } Term A Z " Z " # ' +n # ' x ' x x (' x )=' x (' x) ' f x () F x g (') ' = f e ; x { } Term B where (' ) n x (' x) re etermine by prmeters lone in (6) n (7) n hence re invrint to (', ' x). Therefore the free entry conition e nes ownwr-sloping reltionship in (', ' x) spce, s shown in Figure 2. (28) Figure 2: Free Entry Conition The close economy free entry conition correspons to the limiting cse of in nitely lrge tre costs, where ' x! n Term B! 0. As the close economy is opene to tre n the exporting cuto for rm bility (' x) flls to nite vlue, Term B becomes strictly positive. It follows tht the omestic cuto for rm bility (' ) must necessrily rise, in orer to reuce Term A n leve the expecte vlue of entry (V ) equl to the unchnge sunk entry cost (f e ). Hving estblishe the impct of the opening of tre on ', we now turn to the impct on the rnges of proucts supplie to the omestic n export mrkets. The omestic prouct ttributes cuto for ech rm bility, (') = (' =') (' ), is monotoniclly incresing in ', while (' )

18 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 8 epens solely on prmeters. Therefore, s the opening of the close economy to tre rises ', it increses the omestic prouct ttributes cuto for ech rm bility, which implies tht ll surviving rms rop proucts with lower vlues of prouct ttributes from the omestic mrket. As the close economy is opene to tre n ' x flls to nite vlue, some surviving rms begin to export. The rnge of proucts exporte is etermine by the export prouct ttributes cuto for ech rm bility, which is given by x (') = (' x=') x (' x), where x (' x) epens solely on prmeters. Given selection into export mrkets, the new entrnts into exporting re high-bility rms, ' x > ', n the proucts e in export mrkets hve high ttributes, x (') > ('). We now exmine the implictions of the chnge in the rnge of proucts supplie to ech mrket for rm prouctivity. As iscusse in the pper, rm prouctivity epens on rm bility (') n revenue-shre (~r ('; )) weighte verge of prouct ttributes () cross proucts n mrkets: " JX Z # i (') ('; ) ~r ('; ) ; (29) (') j= where ('; ) = '; ~r ('; ) I (') r ('; ) () ; r i (') n I (') = if ' n mrket is serve or ero otherwise, n we re concerne here with symmetric countries. Uner utrky, the revenue shre of proucts with ttributes 2 [ (') ; ) is: ~r r ('; ) () (') () ('; ) = = R r (') (') (') () ; 2 [ (') ; ); (30) where the superscript enotes utrky; n the superscript t will be use to enote the open economy below. To chrcterie the revenue shre of proucts with i erent ttributes in the open economy, consier non-exporters n exporters in turn. () For non-exporters, proucts with ttributes 2 [ (') ; t (')) re roppe from the omestic mrket n hence experience ecline in their shre of rm revenue. In contrst, proucts with ttributes 2 [ t (') ; ) experience rise in their shre of rm revenue. Therefore the istribution ~r ('; ) in the open economy rst-orer stochsticlly omintes tht in the close economy, n prouctivity (29) rises for non-exporters. (b) For exporters, proucts with ttributes 2 [ (') ; t (')) re roppe from the omestic mrket n hence experience ecline in their shre of rm revenue. Proucts with ttributes 2 [ t (') ; t x (')) experience n mbiguous chnge in their shre of rm revenue: ~r t ('; ) = (') () (') (') ; 2 [ t () R Z t (') (') (') () n Z t x (') (') (') ; t x (')); () :

19 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 9 As the sign of is in generl mbiguous, ~r t ('; )? ~r ('; ) for 2 [ t (') ; t x (')). Finlly, proucts with ttributes 2 [ t x (') ; ) experience rise in their shre of rm revenue: + n ~r t (') () ('; ) = [ + n ] R (') (') ; 2 [ t x (') ; ); () n R 3 t (') 2 4 (') (') () + R 5 t x (') t (') n (') > 0: () where we hve re-written the enomintor of ~r t ('; ) in i erent form. As 2 > 0, ~r t ('; ) > ~r ('; ) for 2 [ t x (') ; ). Irrespective of whether the revenue shre of proucts with intermeite ttributes 2 [ (') ; t (')) rises or flls, the i erence between the open economy vlue of ~r ('; ) n the close economy vlue goes from being negtive t low vlues of 2 [ (') ; t (')) to being positive t high vlues of 2 [ t x (') ; ). This is su cient conition for the istribution ~r ('; ) in the open economy to rst-orer stochsticlly ominte tht in the close economy. Therefore rm prouctivity (29) rises for exporters Reuctions in Vrible Tre Costs in the Open Economy Equilibrium While in the min pper we focus on the impct of opening the close economy to tre, similr, though more nunce, preictions hol for reuctions in vrible tre costs in the open-economy equilibrium. 5 Proposition B Reuctions in vrible tre costs result in within- rm relloction tht les surviving multiple-prouct rms to focus on their most successful proucts: () ll surviving rms rop proucts with lower ttributes from the omestic mrket, which rises their prouctivity, (b) surviving rms tht enter the export mrket rop proucts with low ttributes from the omestic mrket n proucts with higher ttributes in the export mrket, which rises their prouctivity, (c) surviving exporters rop proucts with low ttributes from the omestic mrket n proucts in the export mrket. As these proucts e in the export mrket hve higher ttributes thn those roppe from the omestic mrket, but hve lower ttributes thn those previously exporte, the e ect on their prouctivity is mbiguous. 5 While we concentrte on reuctions in vrible tre costs, reuctions in prouct xe exporting costs hve similr e ects, except where otherwise inicte, s long s there remins selection into export mrkets: x (') > (') n ' x > '.

20 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 20 Proof. To estblish the proposition, we rst chrcterie ' =. Using the free entry conition for symmetric countries (9), we e ne = V f e. Applying the implicit function theorem, ' = = (=) = (=' ). Substituting for ' x, (') n x (') in (9) using (20), (6), (2), (7), n (8), we obtin V= < 0 n V=' < 0. Therefore, we hve estblishe tht ' = < 0. We next chrcterie ' x=. Di erentiting with respect to in eqution (8), we obtin: ' x ' = + : (3) ' x ' It follows tht to estblish ' x= > 0, it su ces to show tht (' =) = (=' ) >. To o so, we gin use the implicit function theorem to evlute ' = = (=) = (=' ). From (20), (6), (2), (7), n (8), we hve: x (') = = x (') =, x (') =' = x (') =', ' x= = ' x= n ' x=' = ' x='. Combining these results with ' = = obtin (' =) = (=' ) >. Therefore we hve estblishe tht ' x= > 0. (=) = (=' ), we Since (') = (' =') (' ), where (' ) is invrint to, n since ' = < 0, we hve estblishe tht (') = < 0. Since x (') = (' x=') x (' x), where x (' x) is invrint to, n since ' x= > 0, we hve lso estblishe tht x (') = > 0. To etermine the impct of the reuction in vrible tre costs on rm prouctivity, enote the vlues of vribles before the reuction by the superscript t n the vlues of vribles fter the reuction by the superscript tt. From the bove: tt () For omestic rms, proucts with ttributes 2 [ t (') > t (') n tt x (') < t x ('). (') ; tt (')) re roppe from the omestic mrket n hence experience ecline in their shre of rm revenue. In contrst, proucts with ttributes 2 [ tt (') ; ) experience rise in their shre of rm revenue. Therefore the istribution of rm revenue shres ~r tt ('; ) rst-orer stochsticlly omintes the istribution ~r t ('; ). It follows tht the reuction in vrible tre costs rises rm prouctivity for omestic rms. To chrcterie the mgnitue of the rise, note tht the shre of proucts in rm revenue for omestic rms before the chnge in vrible tre costs is: ~r D t (') () ('; ) = R t (') (') () (') () ; 2 [ t (') ; ): (32) AA

21 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 2 After the chnge in vrible tre costs, the shre of proucts in rm revenue for omestic rms cn be written s: ~r tt D ('; ) = (') () BB BB AA 3 ; 3 Z tt (') ; 2 [ tt (') ; ); (33) t (') (') () > 0: Therefore, from (32) n (33), the rtio of rm revenue shres fter n before the chnge in vrible tre costs for omestic rms is: ~r D tt ~r D tt ('; ) AA = ('; ) BB > ; 2 [tt (') ; ): (34) (b) For new exporters, the shre of proucts in rm revenue before the reuction in vrible tre costs, ~r t NE ('; ), is the sme s for omestic rms in (32) for ll 2 [t (') ; ). reuction in vrible tre costs, proucts with ttributes 2 [ t After the (') ; tt (')) re roppe from the omestic mrket n therefore experience ecline in their shre of rm revenue. On the other hn, proucts with ttributes 2 [ tt shre of rm revenue: ~r tt NE ('; ) = (') () CC CC AA 4 > BB; 4 Z tt (') (') ; tt (')) experience n mbiguous chnge in their x ; 2 [ tt t (') (') () (') ; tt (')); Z Since 4 is mbiguous in sign, we hve ~r tt NE ('; )? ~rt NE proucts with ttributes 2 [ tt x ~r tt NE ('; ) = x tt x (') n tt (') () : ('; ) for 2 [tt (') ; tt (')). Finlly, (') ; ) experience rise in their shre of rm revenue: h + n tt i (') () h + n ( tt ) i ; 2 [ tt x (') ; ); AA 5 x 5 h + n tt i Z tt (') t (') (') () + n tt Z tt x (') tt (') (') () > 0; where we hve re-written the enomintor of ~r tt NE ('; ) in i erent form. As 5 > 0, we hve ~r NE tt ('; ) > ~rt NE ('; ) for 2 [tt x (') ; ). Therefore, irrespective of whether the revenue shre of proucts with ttributes 2 [ tt (') ; tt (')) rises or flls, the i erence between ~r tt ('; ) n ~r t ('; ) goes from being negtive t low vlues of to being positive t high vlues of. This is su cient conition for the istribution ~r tt ('; ) to rst-orer stochsticlly ominte the istribution ~r t ('; ). It follows tht the reuction in x

22 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 22 vrible tre costs rises rm prouctivity for new exporters. The mgnitue of the chnge in the shres of proucts in rm revenue for new exporters cn be chrcterie s: ~r NE tt ('; ) ~r NE t ('; ) = AA CC < AA ~r NE tt ('; ) ~r NE t ('; ) = BB ; h + n tt i AA DD n tt BB + DD Z tt x (') DD < n tt BB: (') > AA BB > ; () ; 2 [tt (') ; tt (')); 2 [tt x (') ; ); Therefore, from the bove expressions n (34), the chnge in revenue shres of proucts with low ttributes 2 [ t (') ; tt (')) is the sme for new exporters s for omestic rms, the chnge in revenue shres of proucts with intermeite ttributes 2 [ tt x (') ; tt (')) is smller for new exporters thn for omestic rms, n the chnge in revenue shres for proucts with high ttributes 2 [ tt x (') ; ) is lrger for new exporters thn for omestic rms. These re su cient conitions for the istribution ~r NE tt ('; ) to rst-orer stochsticlly ominte the istribution ~rtt D ('; ). It follows tht new exporters experience greter prouctivity growth thn omestic rms following the reuction in vrible tre costs. (c) For continuing exporters, proucts with ttributes 2 [ t x (') ; tt (')) re roppe from the omestic mrket n therefore experience ecline in their shre of rm revenue. On the other hn, proucts with ttributes 2 [ tt (') ; tt mbiguous chnge in their shre of rm revenue: ~r E t ('; ) = (') () AA + EE ; 2 [t EE n t Z t x (') (') x () ; (')) n 2 [ tt x (') ; t x (')) experience n (') ; t x (')); ~r E tt ('; ) = (') () AA F F ; 2 [tt (') ; tt x (')); h + n tt i (') () ~r E tt ('; ) = ; 2 [ tt x (') ; t x (')); AA F F F F Z tt (') t (') (') () tt EE + n tt Z t AA F F? AA + EE ; n + n tt AA F F Therefore we hve ~r tt NE ('; )? ~rt E t ('; ) for 2 [tt (') ; tt x x (') tt x (')? (') AA + EE : ()! (')) n 2 [ tt x (') ; t x (')). Finlly, proucts with ttributes 2 [ t x (') ; ) lso experience n mbiguous chnge in their

23 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 23 shre of rm revenue: h + n t i (') () ~r E t ('; ) = AA + EE h + n tt i (') () ~r E tt ('; ) = As = (AA AA F F ; 2 [ t x (') ; ); ; 2 [ t x (') ; ): F F )? = (AA + EE), we hve ~r NE tt ('; )? ~rt E ('; ) for 2 [t x (') ; ). chrcterie the mgnitue of the chnge in rm revenue shres, consier the rtio of rm revenue shres fter n before the reuction in vrible tre costs: To ~r E tt ('; ) ~r E t ('; ) = ~r E tt ('; ) ~r E t ('; ) = ~r E tt ('; ) ~r E t ('; ) = AA + EE AA F F ; 2 [tt (') ; tt x (')); h + n tt i (AA + EE) ; 2 [ tt x (') ; t x (')); AA F F h + n tt i (AA + EE) h + n ( t ) i ; 2 [ t x (') ; ): (AA F F ) As + n tt > + n t >, the chnge in prouct rm revenue shres for continuing exporters is smllest for 2 [ tt sie for 2 [ t x (') ; ). (') ; tt x (')), lrgest for 2 [ tt x (') ; t x (')), n of intermeite With this orering of chnges in rm revenue shres, the chnge in rm prouctivity for continuing exporters is in generl mbiguous. Aitionlly, from the bove expressions n (34), the chnge in prouctivity for continuing exporters cn be higher or lower thn for omestic rms Proof of Proposition 2 Proof. () With continuum of symmetric proucts, the shre of proucts exporte to given mrket by existing exporters with bility ' ' x is [ Z ( x ('))]. From (2) n (7), x (') = (' x=') x (' x), where x (' x) is invrint to n Section 4.3. of this ppenix estblishe tht ' x= > 0. It follows tht reuctions in vrible tre costs reuce ' x n x ('), n hence rise [ Z ( x ('))], which estblishes prt () of the proposition. (b) With symmetric countries n common prouct ttributes, the probbility tht prouct is exporte to ll countries worlwie is [ Z ( x ('))]. With symmetric countries n countryspeci c prouct ttributes, the probbility tht prouct is exporte to ny given country is [ Z ( x ('))]. In both cses, the expecte number of countries to which prouct is exporte is [ Z ( x ('))] n. From (2) n (7), x (') = (' x=') x (' x), where x (' x) is invrint to n ' x= > 0. It follows tht reuctions in vrible tre costs reuce ' x n x ('), n hence

24 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 24 rise [ Z ( x ('))], which estblishes prt (b) of the proposition. (c) With symmetric countries, the shre of rms tht export equls [ G (' x)] = [ G (' )]. From Section 4.3. of this ppenix, ' = < 0 n ' x= > 0, which implies tht = < 0. It follows tht reuctions in vrible tre costs increse, which estblishes prt (c) of the proposition. () With symmetric countries, exports of rm with given bility n given prouct ttribute to given mrket cn be written s r x ('; ) = (= (' )) ('=' ) f, which is monotoniclly ecresing in, since (' ) is invrint to n Section 4.3. estblishe tht (' =) = (=' ) >. In contrst, verge exports per rm-prouct-country cn be written s: Z r x (') = Z ( x (')) x (') f x () : (35) x (') Since x (') = (' x=') x (' x), where x (' x) is invrint to n ' x= > 0, we hve x (') = > 0. Now note tht from (35): 2 r x (') = 6 4 Z ( x (')) = x (') Z ( x (')) { } >0 r x (') ( ) x (') r x (') 3 f x ( x (')) Z ( 7 x (')) 5 x (') { } >0 which is in generl mbiguous in sign epening on the vlue of x (') n the functionl form of the cumultive istribution function for prouct ttributes Proof of Proposition 3 Proof. () With continuum of symmetric proucts, the shre of proucts exporte to given country by existing exporters with bility ' ' x is [ Z ( x ('))]. In this expression, x (') = (' x=') x (' x) is monotoniclly ecresing in ' n Z () is continuous cumultive istribution function tht is incresing in. It follows tht [ estblishes prt () of the proposition. Z ( x ('))] is incresing in ', which (b) With symmetric countries n common prouct ttributes, the probbility tht prouct is exporte to ll countries worlwie is [ Z ( x ('))]. With symmetric countries n countryspeci c prouct ttributes, the probbility tht prouct is exporte to ny given country is [ Z ( x ('))]. In both cses, the expecte number of countries to which prouct is exporte is [ Z ( x ('))] n, where we showe in prt () tht [ Z ( x ('))] is incresing in '. We hve therefore estblishe prt (b) of the proposition. (c) With symmetric countries, exports of rm with given bility n given prouct ttribute to given mrket cn be written s r x ('; ) = (= (' )) ('=' ) f, which ;

25 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 25 is monotoniclly incresing in '. In contrst, verge exports per rm-prouct-country re (35). Therefore: 2 3 r x (') ' = 6 4 Z ( x (')) = x (') Z ( x (')) { } >0 r x (') ( ) x (') r x (') f x ( x (')) Z ( 7 x (')) 5 x (') ' { } <0 which is in generl mbiguous in sign epening on the vlue of x (') n the functionl form of the cumultive istribution function for prouct ttributes Symmetric Country Close Form Solutions for Preto Distributions In this section, we chrcterie the symmetric country equilibrium for the specil cse where rm bility is rwn from the Preto istribution g (') = ' min ' (+) n prouct ttributes re rwn from the Preto istribution () = min (+). We ssume ' min > 0, min > 0 n > > >, which ensures tht rm revenue hs nite men. Agin we choose the wge in one country s the numerire, which, together with country symmetry, implies tht wge in ll countries is equl to one Generl Equilibrium with Symmetric Countries We begin by solving for the equilibrium sextuple {', ' x, (' ), x (' x), P, R}. Using the Preto istribution of prouct ttributes in the ero-pro t n exporting cuto conitions for rm bility (6) n (7), we hve: (' ) = ( ) x (' x) = ( ) f F f x F x min (36) min (37) where we focus on prmeter vlues for which we hve n interior equilibrium n selection into export mrkets: x (' x) > (' ) > min. Combining (36) n (37) with the reltionship between the exporting n ero-pro t cuto rm bilities (8), we obtin: ' x = fx f ( ) ( ) F F x ' (38) where we gin focus on prmeter vlues for which we hve n interior equilibrium n selection into export mrkets: ' x > '. Using the Preto istributions of rm bility n prouct ttributes in the free entry conition (9) yiels: v e = 'min 'min F ' + nf x ' = f e ; (39) x

26 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 26 which together with (38) uniquely etermines ' s function of prmeters lone. Uner the ssumption of Preto istributions of rm bility n prouct ttributes, verge rm revenue (27) becomes: r = ' F + nf x ' ; x where, from (38), (' =' x) is function of prmeters lone, which implies tht we hve etermine r s function of prmeters lone. Using the choice of numerire, the reltionship between the mss of rms proucing n the mss of entrnts (M = [ G (' )] M e) n the free entry conition (V = [ G (' )] = f e), it follows tht L e = M, s in Section 4.. bove. Since L p = R M, we hve R = L. The mss of rms proucing follows immeitely from ggregte revenue n verge revenue: M = R=r = L=r, where r ws etermine s function of prmeters bove. Using the Preto istributions of rm bility n prouct ttributes, the mesures of rms supplying ech prouct to the omestic n export mrkets, (24) n (25), re: m = f ( ) F M; M x = f x M ( ) F x where gin we focus on n interior equilibrium for which ws etermine s function of prmeters bove. < ( ) f F < f x ( ) F x Finlly, weighte-verge prouctivities in the omestic n export mrkets re: ~' = ( ) ' (' ) ; ~' x = ( ) ' x x (' x) ; n M where ', (' ), ' x n x (' x) were chrcterie s function of prmeters bove. Hving etermine the mss of rms proucing (M), the mesures of rms supplying ech prouct to the omestic n export mrkets, n weighte verge prouctivity in the omestic n export mrkets (~' n ~' x ), the price inex for ech prouct, P, follows immeitely from (23). This completes the chrcterition of the equilibrium sextuple {', ' x, (' ), x (' x), P, R} Mrgins of Tre with Symmetric Countries As iscusse in the pper, the totl exports of rm with given bility to given mrket cn be ecompose into the shre of proucts exporte (extensive mrgin) n verge exports per rm-prouct-country (intensive mrgin).

27 Technicl Appenix: Multi-Prouct Firms n Tre Liberlition 27 With Preto istribution of prouct ttributes, the shre of proucts exporte to given country by rm with bility ' is given by: [ Z ( min min ' x ('))] = = x (') x (' ; (40) x) ' x where x (' x) is given by (37); n ' x is etermine by (38) n (39). From the bove expression, it follows immeitely tht the extensive mrgin is monotoniclly incresing in rm bility. To etermine its reltionship with vrible tre costs, note tht from (37) x (' x) is inepenent of vrible tre costs, while from (38) n (39) ' x is incresing in vrible tre costs. Therefore reuctions in vrible tre costs increse the extensive mrgin. With Preto istribution of prouct ttributes, verge exports per rm-prouct-country re: Z r x ('; ) = Z ( x (')) x (') f x () ; (4) x (') = ( ) f x; which implies tht the intensive mrgin is inepenent of both vrible tre costs n rm bility. Finlly, uner the ssumption of Preto istribution of rm bility, the extensive mrgin of the shre of rms tht export is given by: [ G (' x)] ' = = G ' ' x f f x ( ) ( ) Fx F ; (42) which is ecresing in vrible tre costs. Hence reuctions in vrible tre costs increse the shre of rms tht export. Note tht the extensive mrgins of the shre of proucts exporte n the shre of rms tht export re lso ecresing in prouct xe exporting costs, f x (from (40) n (42) using (37), (38) n (39)). In contrst, the intensive mrgin of verge exports per rm-prouct-country is incresing in f x (from (4)) Preto Distributions n Heterogeneous Fixe Costs with Symmetric Countries The result tht the intensive mrgin of verge exports per rm-prouct-country (4) is inepenent of rm bility requires both Preto istribution of prouct ttributes n prouct xe exporting cost tht is inepenent of prouct ttributes. Suppose inste tht prouct xe exporting costs vry with prouct ttributes: f x = 0 x. In this cse: Z x(' x ) r x ('; ) = ( ) 0 ( x (')) ; ' x (') = x ' x (' ) ; " # x (' ( x (' x)) 0 () = F x : x)