Web-Base Technical Appeni: Multi-Prouct Firms an Trae Liberalization Anrew B. Bernar Tuck School of Business at Dartmouth & NBER Stephen J. Reing LSE, Yale School of Management & CEPR Peter K. Schott Yale School of Management & NBER December 2008 This appeni contains aitional technical erivations an supplementary material for the main paper. Section presents moel solutions for the special case where rm prouctivity an consumer tastes are Pareto istribute. Section 2 contains the complete proof of Proposition 3 in the main paper. Section 3 shows that it is straightforwar to eten the moel to introuce epenence in consumer tastes. Section 4 augments the moel to inclue stochastic variation in rm prouctivity an consumer tastes over time, which inuces steay-state aing an ropping of proucts, but oes not change the moel s preictions for the cross-section istribution of eports across rms, proucts an countries that are our focus. Section 5 shows that the moel can be etene to incorporate comparative avantage base on cross-country i erences in factor enowments an cross-inustry i erences in factor intensity.. Solutions for the Special Case of Pareto Distributions In this section of the appeni, we solve the moel for the special case when rm prouctivity is rawn from the Pareto istribution g (') = a' a min' (a+) an consumer tastes are rawn from the Pareto istribution z () = z z min (z+). We assume ' min > 0, min > 0 an a > z > ( ) > 2, which ensures that the variance of total rm revenue is nite... General Equilibrium with Pareto Distributions We begin by solving for the equilibrium setuple {', ', (' ), (' ), P, R}. Using the Pareto istribution of consumer tastes in the zero-pro t prouctivity cuto conition (equation (4) in the main tet) an the eporting prouctivity cuto coni-
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 2 tion (equation (6) in the main tet), we have: (' ) = z ( ) (' ) = z ( ) f F f F z min : () z min : (2) where we focus on parameter values for which we have an interior equilibrium an selection into eport markets: (' ) > (' ) > min. Combining () an (2) with the relationship between the eporting an zero-pro t cuto abilities (equation (7) in the main tet), we obtain: ' = f f z ( ) z( ) F F z ' : (3) where we again focus on parameter values for which we have an interior equilibrium an selection into eport markets: ' > '. Using the Pareto istributions of prouctivity an consumer tastes in the free entry conition (equation (9) in the main tet) yiels: v e = a a z 'min 'min F + nf a z ' = f ' e ; (4) which together with (3) uniquely etermines ' as a function of parameters alone. With a Pareto istribution of prouctivity an consumer tastes, average rm revenue (equation (36) in the Appeni of the main paper) is: za r = a z F + nf ' ' a ; where from (3) (' =' ) is a function of parameters alone, which implies that we have etermine r as a function of parameters alone. From the steay-state stability conition (equation (20) in the main tet), free entry conition (equation (9) in the main tet) an labor market clearing conition (equation (2) in the main tet), we have R = L. The mass of rms follows immeiately from aggregate revenue an average revenue: M = R=r. Using the Pareto istribution of prouctivity an consumer tastes in equations (23) an (24) in the main tet, the masses of rms supplying each prouct to the omestic an eport markets are: M = a a z f z ( ) F M; M = a a z f M z ( ) F
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 3 where again we focus on an interior equilibrium for which < f z ( ) F. a a z < f z ( ) F < Finally, weighte-average prouctivity in the omestic an eport markets are: ~' = z z ( ) ' (' ) ; ~' = z z ( ) ' (' ) : With the mass of rms an weighte average prouctivity in both the omestic an eport markets etermine, the price ine for each prouct, P, follows immeiately from equation (25) in the main tet. Revenue for each prouct follows immeiately from the CES revenue function, the prouct price ine an aggregate revenue. This completes the characterization of the equilibrium setuple {', ', (' ), (' ), P, R}..2. Pareto Distributions an the Margins of Trae Consier rst the etensive margins of proucts an countries (the within- rm etensive margins). With a Pareto istribution of consumer tastes, the share of proucts eporte to a given country by a rm with prouctivity ' an the share of countries to which a rm with prouctivity ' eports a given prouct are both given by: z z [ Z ( min min ' ('))] = = (') ; (5) (' ) where we have use equation () in the main tet; (' ) is given by (2); an ' is etermine by (3) an (4). From the above epression, it follows immeiately that the etensive margins of proucts an estinations are both increasing in rm prouctivity. To etermine their relationship with variable trae costs, note that from (2) (' ) is inepenent of variable trae costs, while from (3) an (4) ' is increasing in variable trae costs. Therefore reuctions in variable trae costs increase the etensive margins of both proucts an estinations. Consier net average eports per rm, prouct an country (the intensive margin). With a Pareto istribution of consumer tastes, we have: Z r (') = Z ( (')) (') f z () ; (6) (') z = z ( ) f ; which is inepenent of both variable trae costs an rm prouctivity. '
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 4 Consier nally the share of rms that eport (the across- rm etensive margin). With a Pareto istribution of consumer tastes, we have: [ G (' )] ' a [ G (' )] = = ' f which is ecreasing in variable trae costs. increase the share of rms that eport. f z ( ) z( ) F F z ; (7) Hence reuctions in variable trae costs Note that the within- rm an across- rm etensive margins are also ecreasing in prouct e eporting costs, f (from (5) an (7) using (2), (3) an (4)). In contrast, the intensive margin is increasing in f (from (6))..3. Pareto Distributions an Heterogeneous Fie Costs The result that the intensive margin is inepenent of rm prouctivity requires both a Pareto istribution of consumer tastes an a prouct e eporting cost that is inepenent of consumer tastes. Suppose instea that prouct e eporting costs vary with consumer tastes: f = 0. In this case, we obtain: Z (' ) r (') = (') = " (' ) z z ( ) 0 ( (')) ; ' (' ) ; ' ( (' )) # 0 z () = F : Therefore if higher consumer taste proucts have higher e eporting costs, > 0, the intensive margin is ecreasing in rm prouctivity, '. In contrast, if higher consumer taste proucts have lower e eporting costs, < 0, the intensive margin is increasing in rm prouctivity, '. If is interprete as a component of rm prouctivity that is speci c to iniviual proucts (see Footnote 6 in the main paper), < 0 is perhaps more natural, since it implies that lower values of are associate with higher variable an e costs. 2. Complete Proof of Proposition 3 Proof. We rst characterize ' =. From the free entry conition (equation (9) in the main tet), e ne = V f e. By the implicit function theorem, ' = =
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 5 (=) = (=' ). Substituting for ', (') an (') in equation (9) in the main tet using equations (9), (4), (), (6), an (7) in the main tet, we obtain V= < 0 an V=' < 0. Therefore, we have establishe that ' = < 0. We net characterize ' =. Di erentiating with respect to in equation (7) in the main tet, we obtain: ' = + ' ' ' ; (8) It follows that to establish ' = > 0, it su ces to show that (' =) = (=' ) >. To o so, we again use the implicit function theorem to evaluate ' = = (=) = (=' ). Aitionally, equations (9), (4), (), (6), an (7) in the main tet imply the following: (') = = (') =, (') =' = (') =', ' = = ' = an ' =' = ' ='. Combining these results with ' = = (=) = (=' ), we obtain (' =) = (=' ) >. Therefore we have establishe that ' = > 0. Since (') = (' =') (' ), where (' ) is invariant to, an since ' = < 0, we have establishe that (') = < 0. Aitionally, since (') = (' =') (' ), where (' ) is invariant to, an since ' = > 0, we have establishe that (') = > 0. To evaluate the impact of the reuction in variable trae costs on measure rm prouctivity, enote the values of variables before the reuction by the superscript T an the values of variables after the reuction by the superscript T T. T T (') > T (') an T T (') < T ('). (a) For omestic rms, proucts with consumer tastes 2 [ T From the above: (')) are roppe from the omestic market an therefore eperience a ecline in their share of rm revenue. In contrast, proucts with consumer tastes 2 [ T T (') ; ) eperience a rise in their share of rm revenue. Therefore the istribution ~r T T ('; ) rst-orer stochastically ominates the istribution ~r T ('; ), where ~r ('; ) = r ('; ) z () =r ('). Hence the reuction in variable trae costs raises measure rm prouctivity (equation (33) in the Appeni of the main paper) for omestic rms. To characterize the magnitue of the rise, note that before the change in variable trae costs, the share of proucts in rm revenue for omestic rms is: ~r T D ('; ) = (') z () R T (') (') z () (') z () ; 2 [ T (') ; ): (9) AA
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 6 After the change in variable trae costs, the share of proucts in rm revenue for omestic rms can be written as: ~r T T D ('; ) = (') z () BB BB AA 3 ; 3 Z T T (') ; 2 [ T T (') ; ); (0) T (') (') z () > 0: Therefore, from (9) an (0), the ratio of rm revenue shares after an before the change in variable trae costs for omestic rms is: ~r D T T ~r D T ('; ) AA = ('; ) BB > ; 2 [T T (') ; ): () (b) For new eporters, the share of proucts in rm revenue before the reuction in variable trae costs, ~r NE T ('; ), is the same as for omestic rms in (9) for all 2 [ T 2 [ T (') ; ). After the reuction in variable trae costs, proucts with consumer tastes (')) are roppe from the omestic market an therefore eperience a ecline in their share of rm revenue. On the other han, proucts with consumer tastes 2 [ T T revenue: ~r T T NE ('; ) = (') z () CC CC AA 4 > BB; 4 Z T T (') (')) eperience an ambiguous change in their share of rm ; 2 [ T T T (') (') z () Z T T (') (')); n T T (') z () ; Since 4 is ambiguous in sign, we have ~r NE T T ('; )? ~rt T NE ('; ) for 2 [T (')). Finally, proucts with consumer tastes 2 [ T T of rm revenue: ~r T T NE ('; ) = (') ; ) eperience a rise in their share h + n T T i (') z () + n ( T T ) AA 5 ; 2 [ T T (') ; ); 5 h + n i Z T T T T (') T (') (') z () +n T T Z T T (') T T (') (') z () > 0; where we have re-written the enominator of ~r T T NE ('; ) in a i erent form. As 5 > 0, we have ~r NE T T ('; ) > ~rt T NE ('; ) for 2 [T (') ; ). Therefore, irrespective of whether the revenue share of proucts with consumer tastes
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 7 2 [ T T (')) rises or falls, the i erence between ~r T T ('; ) an ~r T ('; ) goes from being negative at low values of to being positive at high values of. This is a su cient conition for the istribution ~r T T ('; ) to rst-orer stochastically ominate the istribution ~r T ('; ). Hence the reuction in variable trae costs raises measure rm prouctivity (equation (33) in the Appeni of the main paper) for new eporters. Aitionally, the magnitue of the change in the shares of proucts in rm revenue for new eporters can be characterize as: ~r NE T T ('; ) ~r NE T ('; ) = AA CC < AA ~r NE T T ('; ) ~r NE T ('; ) = BB ; h + n i T T AA BB + DD DD n T T Z DD < n T T BB: T T (') > AA BB > ; (') z () ; 2 [T T 2 [T T (') ; ); (')); Therefore, from the above epressions an (), the change in revenue shares of proucts with low consumer tastes 2 [ T (')) is the same for new eporters as for omestic rms, the change in revenue shares of proucts with intermeiate consumer tastes 2 [ T T (')) is smaller for new eporters than for omestic rms, an the change in revenue shares for proucts with high consumer tastes 2 [ T T (') ; ) is larger for new eporters than for omestic rms. These are su cient conitions for the istribution ~r NE T T T ('; ) to rst-orer stochastically ominate the istribution ~rt D ('; ). Hence new eporters eperience greater measure prouctivity growth than omestic rms following the reuction in variable trae costs. (c) For continuing eporters, proucts with consumer tastes 2 [ T (')) are roppe from the omestic market an therefore eperience a ecline in their share of rm revenue. On the other han, proucts with consumer tastes 2 [ T T an 2 [ T T (') ; T (')) (')) eperience an ambiguous change in their share of rm rev-
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 8 enue: F F ~r E T ('; ) = (') z () AA + EE ; 2 [T Z EE n T (') z () ; T (') E ('; ) = (') z () AA F F ; 2 [T T h ~r T T ~r T T E ('; ) = Z T T (') T (') (') z () + n T T i (') z () AA F F T T T ; 2 [ T T (') ; T (') ; T EE + n Z T T T (') AA F F? AA + EE ; an + n T T AA F F? T T (') (')); (') AA + EE : (')); (')); z () Therefore we have ~r NE T T ('; )? ~rt T E ('; ) for 2 [T (')) an 2 [ T T (') ; T (')). Finally, proucts with consumer tastes 2 [ T change in their share of rm revenue: h + n i T (') z () ~r E T ('; ) = AA + EE h + n i T T (') z () ~r E T T ('; ) = As = (AA AA F F (') ; ) also eperience an ambiguous ; 2 [ T (') ; ); ; 2 [ T (') ; ): F F )? = (AA + EE), we have ~r NE T T ('; )? ~rt E ('; ) for 2 [T (') ; ). To characterize the magnitue of the change in revenue shares, consier the ratio of prouct rm revenue shares after an before the reuction in variable trae costs:! : ~r T T E ('; ) ~r E T ('; ) = ~r T T E ('; ) ~r E T ('; ) = ~r T T E ('; ) ~r E T ('; ) = AA + EE AA F F ; 2 [T T h + n i T T (AA + EE) ; 2 [ T T (') ; T AA F F h + n i T T (AA + EE) + n ( T ) (AA F F ) ; 2 [T (') ; ): (')); (')); As +n T T > +n T >, the change in prouct rm revenue shares for continuing eporters is smallest for 2 [ T T an of intermeiate size for 2 [ T (')), largest for 2 [ T T (') ; T (')), (') ; ). With this orering of changes in the shares
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 9 of proucts in rm revenue, the change in measure prouctivity (equation (33) in the Appeni of the main paper) for continuing eporters is in general ambiguous. Aitionally, from the above epressions an (), the change in measure prouctivity (equation (33) in the Appeni of the main paper) for continuing eporters can be higher or lower than for omestic rms. 3. Depenence in Consumer Tastes While the moel s simplifying assumption that the consumer tastes istributions are inepenently istribute provies a tractable way to introuce heterogeneity across proucts an countries within rms, it is straightforwar to eten the analysis to introuce epenence in consumer tastes. For eample, suppose that consumer tastes for a rm s variety of a prouct inclue a common component, p, which is the same across countries for a given prouct, an an iiosyncratic component, c, which varies across both countries an proucts: = & p & pc ; 0 & : The common component of consumer tastes plays a similar role to rm prouctivity which, as in the paper, is common across proucts an countries within rms. Once the sunk entry cost is pai, a rm observes the common component of consumer tastes for each prouct, p, an the iiosyncratic component of consumer tastes, pc, for each prouct an country. The common component of consumer tastes is rawn separately for each prouct from a continuous istribution z p ( p ) with cumulative istribution function Z p ( p ). The iiosyncratic component of consumer tastes is rawn separately for each prouct an country from a continuous istribution z pc ( pc ) with cumulative istribution function Z pc ( pc ). To make use of law of large numbers results, we assume that the prouctivity an consumer tastes istributions are inepenent of one another an inepenently istribute across rms. Similarly, we assume that the common consumer tastes istribution, z p ( p ), is inepenently istribute across proucts, while the iiosyncratic consumer tastes istribution, z pc ( pc ), is inepenently istribute across proucts an countries. Even with these assumptions, a rm s pro tability is now correlate across proucts an countries for two reasons. First, higher prouctivity, ', raises a rm s pro tability across
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 0 all proucts an countries. Secon, a higher common component of consumer tastes for a prouct, p, raises a rm s pro tability across all countries for that prouct. These correlations are however imperfect because of stochastic variation in the iiosyncratic component of consumer tastes across both proucts an countries. As &! 0, the etene moel consiere here reuces to the moel in the paper, where consumer tastes are inepenently istribute across proucts an countries. As &!, the etene moel consiere here reuces to the special case iscusse in footnote in the paper, where there is perfect correlation of consumer tastes across countries. 4. Steay-state Prouct Aing an Dropping As the focus of our paper is the cross-section istribution of eports across rms, countries an proucts, we follow much of the literature on rm heterogeneity in international trae in abstracting from ynamics. In this section of the appeni, we show that the moel can be etene to incorporate stochastic variation in rm prouctivity an consumer tastes over time, which inuces steay-state aing an ropping of proucts within rms. In this etension, we embe a simpli e version of the ynamics from the close economy moel of Bernar, Reing an Schott (2008) in the open economy moel consiere in the main tet of our paper. The speci cation of entry, prouction an eman is similar to that consiere in the main tet. Once a rm incurs the sunk entry cost, f e, prouctivity an consumer tastes are rawn from the continuous istributions g (') an z () respectively, with cumulative istributions G (') an Z (). After a rm observes its initial values of prouctivity an consumer tastes, it ecies whether to prouce or eit. If the rm eits, its prouction knowlege is lost, an the sunk cost must be incurre again in orer for the rm to re-enter. If the rm enters, it faces a Poisson probability > 0 of a shock to prouctivity ', in which case a new value for prouctivity ' 0 is re-rawn from the same istribution as upon entry g (' 0 ). Similarly, the rm faces a Poisson probability " > 0 of an iiosyncratic shock to consumer tastes for its variety of a given prouct in a given country, in which case a new value for consumer tastes 0 is re-rawn from the same istribution as upon entry z ( 0 ). The assumption that following a stochastic shock prouctivity an consumer tastes are re-rawn from the same istributions as upon entry is a simplifying evice, which enables us to introuce ynamics in as tractable a way as possible. In the close economy moel of Bernar, Reing an Schott (2008),
Technical Appeni: Multi-Prouct Firms an Trae Liberalization As consumer tastes for a rm s variety of each prouct change over time, previously pro table proucts an markets become unpro table an are roppe when falls below the zero-pro t cuto for a market { ('), (')}. Similarly, previously unpro table proucts become viable an are ae when rises above the zero-pro t cuto for a market { ('), (')}. Since the istribution from which consumer tastes are rawn following a stochastic shock is the same as the istributions from which they are rawn upon entry, the stationary istribution for consumer tastes, z (), takes a particularly simple form: z () = z () ; (2) where the stationary istribution for consumer tastes conitional on a prouct being supplie to a market is a truncation of z () at the zero-pro t cuto for consumer tastes for a market { ('), (')}. As the istribution from which prouctivity is rawn following a stochastic shock is also the same as the istribution from which it is rawn upon entry, the stationary istribution for rm prouctivity, g ('), also takes a particularly simple form. stationary istribution for rm prouctivity conitional on supplying a market is a truncation of the istribution g (') at the cuto prouctivity above which rms serve the market: 8 >< g (') = >: g(') G(' ) for ' ' in the omestic market g(') G(' ) for ' ' in each eport market 0 otherwise The : (3) The etermination of general equilibrium remains largely the same as in the main tet of the paper. The only eception is the etermination of the value of the rm. In the ynamic etension consiere here, the combination of a sunk entry cost an stochastic shocks to rm prouctivity generates an option value to rm entry. If a rm chooses to eit, it forgoes both the net present value of its instantaneous ow of pro ts an also the option of eperiencing stochastic prouctivity shocks. Therefore, with this option value to rm entry, the threshol prouctivity for rm entry an eit will in general lie below the prouctivity at which the instantaneous ow of total rm pro ts is equal to we consier richer forms of ynamics that allow for serial correlation in prouctivity an consumer tastes. While the introuction of serial correlation complicates the moel s ynamics, it oes not change the key preictions of the moel for the cross-section istribution of eports.
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 2 zero. 2 The value of a rm with prouctivity ' is etermine accoring to the following Bellman equation: v (') = (') + + Z + ' v (' 0 ) g (' 0 ) ' 0 ; (4) As the istribution from which a new value of prouctivity is rawn following a stochastic shock is the same as upon entry an is inepenent of a rm s eisting value of prouctivity, the solution to the Bellman equation takes a particularly simple form. Substituting for v (' 0 ) on the right-han sie of (4) using the trial solution v (') = (') + R (' 0 ) g (' 0 ) ' 0, an solving for an, yiels the equilibrium value of ' a rm: v (') = (') " Z # + + + + G (' ) (' 0 ) g (' 0 ) ' 0 ; (5) Therefore the equilibrium value of a rm with prouctivity ' is a weighte average of the current ow of rm pro ts an the epecte ow of rm pro ts following a stochastic prouctivity shock, where the weights epen on the probability of rm eath, the probability of a prouctivity shock an the probability that a rm remains active following a prouctivity shock. Substituting the epression for the equilibrium value of a rm into the free entry conition, an rearranging, the free entry conition can be re-written as follows: Z " R (') + # + [ (' 0 ) (')] g (' 0 ) ' 0 ' V = + G (' ) g (') ' = f e ; (6) ' which can be simpli e to yiel the following epressions: Z (') V = + G (' ) V = ' Z ' ' g (') ' = f e ; (7) (') + G (' ) g (') ' + n Z ' (') + G (' ) g (') ' = f e As in the main tet, the moel has a recursive structure, an the etermination of general equilibrium is straightforwar. We begin by etermining {', ', (' ), (' )} using 2 In contrast, e an marginal prouction costs for iniviual proucts have no sunk component, an consumer tastes for a rm s variety of each prouct evolve inepenently of whether or not the variety is supplie. Therefore, once a rm has ecie to enter, the rm s ecision whether or not to supply each prouct reuces to a perio-by-perio comparison of contemporaneous revenue an prouction costs.
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 3 the four equations below. To erive the rst of these equations, we use the epression for total rm pro ts in the main tet as well as (') = (' =') (' ) an (') = (' =') (' ), which together imply that the free entry conition (7) can be written as: V = + Z ' Z " Z " Z (' =') (' ) ' (' =') (' ) # ' g (')! ' (' ) f z () F (8) + G c (' )' # ' ng (') '! ' f (' z () F ) + G c (' ) = f e: Since the only sunk cost in the moel is the entry cost, f e, an we consier an equilibrium with selection into eport markets, the lowest prouctivity rm that enters serves only the omestic market: ' < '. Therefore only a rm s ecision of whether or not to serve the omestic market is a ecte by the option value of entry. In contrast, the rm s ecision whether or not to serve the eport market is etermine by a comparison of the instantaneous ow of revenue an the e eporting costs. The prouct an rm eporting cuto s are therefore etermine as in the main tet: Z " # f (' z () = F : (9) ) (' ) ' = f f (' ) (' ) ' ; (20) In eciing whether or not to enter an supply the omestic market, a rm takes into account the option value of entry. Therefore at the zero-pro t cuto prouctivity, ', the ow of total rm losses eactly equals the probability of a stochastic shock to rm prouctivity times epecte pro ts conitional on a stochastic shock occurring: " Z # (' )! (' ) f p z () F (' =') (' ) = + Z ' Z ' " Z (' ) (' =') (' ) " Z (' ) (' =') (' ) # ' g (')! ' (' ) f z () F + G (' )' # ' ng (') '! ' f (' z () F ) + G c (' ): (2)
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 4 To characterize {', ', (' ), (' )}, we rst use (9) to etermine (' ) inepenent of the other equations of the moel. Secon, we use (20) to etermine ' as a function of '. Thir, substituting for ' as a function of ' an using the equilibrium value of (' ), (8) an (2) provie two equations that together etermine {', (' )}. Having characterize { (' ), ', (' )}, the equilibrium value of ' follows immeiately from (20). Finally, having etermine {', ', (' ), (' )}, the remaining elements of the equilibrium vector can be etermine eactly as in the main tet. In this etension of the moel in the main tet, the general equilibrium features steay-state aing an ropping of proucts an estinations as well as steay-state entry an eit of rms. Each perio a measure of new rms incur the sunk entry cost. Of these new rms, those with a prouctivity raw above the zero-value cuto enter, while those with a prouctivity raw below the zero-value cuto eit. Among incumbent rms, a rm with unchange prouctivity supplies constant measures of proucts to the omestic an eport markets, but the ientity of these proucts changes as stochastic shocks to consumer tastes occur. These stochastic shocks inuce a rm with unchange prouctivity to rop a measure of the proucts previously supplie to each market an a an equal measure of the proucts not previously supplie to each market. Finally, as stochastic shocks to an incumbent rm s prouctivity occur, the measure of proucts supplie to each market epans with increases in prouctivity an contracts with ecreases in prouctivity. An incumbent rm eits enogenously when prouctivity falls below the zero-value cuto or eogenously when eath occurs as a result of force majeure consierations beyon the control of the rm. While the etene moel incorporates steay-state aing an ropping of proucts an estinations, the moel s preictions for the cross-section istribution of eports across rms, proucts an countries are unchange, an are therefore robust to this etension. Although we have mae the simplifying assumption that following a stochastic shock prouctivity an consumer tastes are re-rawn from the same istributions as upon entry, richer forms of ynamics can also be introuce, as consiere in the close economy moel of Bernar, Reing an Schott (2008). Again the moel s cross-sectional preictions are robust to the introuction of these richer forms of ynamics.
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 5 5. Multi-Prouct Firms an Comparative Avantage In this section of the appeni, we show that the moel can be etene to incorporate comparative avantage base on cross-country i erences in relative factor abunance an cross-inustry i erences in factor intensity. In this etension, we embe our moel of multi-prouct rms within the two-factor, two-country an two-inustry heterogeneous rm framework of Bernar, Reing an Schott (2007). The moel escribe thus far can be viewe as capturing a single inustry containing many proucts, with rms supplying i erentiate varieties of these proucts. We now generalize the framework by analyzing two inustries, j = ; 2, each of which has this structure. The two inustries enter an upper tier of the representative consumer s utility that takes the Cobb-Douglas form: = U U 2 2 ; + 2 = ; = ; (22) where U j is an ine for inustry j that is e ne over the consumption C k of a continuum of proucts k within the inustry (as in equation () in the main tet), an C k is itself an ine that is e ne over the consumption c k (!) of a continuum of varieties! within each prouct (as in equation (2) in the main tet). Inustries now constitute an upper tier of utility, proucts form an intermeiate tier an varieties occupy a lower tier. While not central to the analysis, we assume for simplicity that the two inustries have the same elasticity of substitution across proucts within inustries () an across varieties within proucts (). We also assume for simplicity that there are only two countries: home an foreign. The home country is assume to be skill-abunant relative to the foreign country an inustry is assume to be skill-intensive relative to inustry 2. The combination of inustry-level variation in factor intensity an country-level variation in factor abunance gives rise to enowment-riven comparative avantage. The skille wage in the home country is chosen as the numeraire. The inustries i er in the intensity with which they use two factors of prouction: skille an unskille labor. To enter an inustry j, a rm from the competitive fringe must incur the sunk entry cost for that inustry, which equals f ej (w S ) j (w L ) j, where w S enotes the skille wage, w L correspons to the unskille wage an j parameterizes inustry factor intensity. After the sunk entry cost is pai, the rm raws
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 6 its prouctivity an values of consumer tastes for the inustry that it enters as above. The istributions of rm prouctivity an consumer tastes are ientically an inepenently istribute across inustries, so that information about prouctivity within an inustry can only be obtaine by incurring the sunk entry cost for that inustry. The istributions of rm prouctivity an consumer tastes are also ientically an inepenently istribute across countries, which ensures consistency with the Heckscher-Ohlin moel s assumption of common technologies across countries. The technology for prouction has the same factor intensity as for entry. 3 While the factor intensity of prouction varies across inustries, all proucts within an inustry are moelle symmetrically an therefore have the same factor intensity. 4 To manufacture a variety of a prouct, a rm must incur a e an variable cost of prouction as above. The variable cost epens upon the rm s prouctivity in manufacturing the prouct, which is again etermine by both the rm s prouctivity an consumer tastes for that prouct. Fie an variable costs use the two factors of prouction with the same proportions, so that the total cost of prouction for a rm in inustry j that serves the omestic market alone is: Z T C j = F j + "f j + q #! j ' j ; jk k (w S ) j (w L ) j ; > > ' j 2 > 0;(23) jk k2 (') where (') enotes the (enogenous) range of proucts supplie to the omestic market by a rm with prouctivity '. The e costs of serving an eport market an supplying a prouct to an eport market are moelle analogously, an use skille an unskille labor with the same factor intensity as prouction an entry. We etermine general equilibrium using an approach very similar to that use in the main tet an therefore omit the reporting of similar equations here to conserve space. The measure of proucts supplie to the omestic an eport markets by a rm with a particular prouctivity can be etermine using epressions analogous to those above. Similarly, the zero-pro t cuto an eporting cuto for prouctivity can be etermine following the same line of reasoning as previously. Entry, prouction an eporting costs all have the same factor intensity, an therefore terms in factor prices cancel from 3 Allowing factor intensity i erences between entry an prouction introuces aitional interactions with comparative avantage as iscusse in Bernar, Reing an Schott (2007). 4 The symmetry of proucts within inustries is clearly a simpli cation, but is useful for the law of large numbers results that etermine the fraction of proucts manufacture by a rm with a particular ability.
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 7 the relevant epressions. There is however an important general equilibrium interaction between comparative avantage an rms prouct scope an entry ecisions. Because countries are no longer symmetric, comparative avantage an trae costs generate cross-country i erences in inustry price inices. As in Bernar, Reing an Schott (2007), these i erences, in turn, generate greater eport opportunities in comparativeavantage inustries than comparative-isavantage inustries. The relative price inices for the two inustries vary across countries because of the combination of comparative-avantage-base specialization an trae costs. Specialization leas to a larger mass of omestic rms relative to foreign rms in a country s comparative-avantage inustry than in its comparative-isavantage inustry. Variable trae costs introuce a wege between omestic an eport prices for a variety. Aitionally, the e costs of becoming an eporter imply that not all rms eport, an the e costs for eporting iniviual proucts imply that not all the proucts supplie omestically are eporte. Combining specialization an trae costs, the comparativeavantage inustry has a greater mass of lower-price omestic varieties relative to the mass of higher-price foreign varieties than the comparative-isavantage inustry. As a result, the price ine in the omestic market is lower relative to the price ine in the eport market in the comparative-avantage inustry. Hence, the egree of competition in the omestic market is higher relative to the egree of competition in the eport market in the comparative-avantage inustry. These i erences in the egree of competition in turn imply that variable pro ts in the eport market are greater relative to variable pro ts in the omestic market in the comparative-avantage inustry than in the comparative-isavantage inustry. Proposition CA: Other things equal, the opening of trae leas to: (a) greater focusing on core competencies in the comparative-avantage inustry than the comparative-isavantage inustry ( H (') > H 2 (') an F 2 (') > F (')), (b) a larger increase in the zero-pro t cuto for prouctivity below which rms eit in the comparative-avantage inustry than the comparative-isavantage inustry (' H > ' H 2 an 'F 2 > 'F ), (c) a larger increase in measure inustry prouctivity in the comparative-avantage inustry than in the comparative-isavantage inustry (~' H > ~' H 2 an ~' F 2 > ~' F ).
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 8 Proof. See Appeni at the en of this ocument. Following the opening of trae, countries specialize accoring to comparative avantage, which leas to a rise in the mass of rms in the comparative-avantage inustry relative to the comparative-isavantage inustry. While the comparative-avantage inustry epans an the comparative-isavantage inustry contracts, the sheing of low prouctivity proucts (i.e., the focusing on core competencies ) is greater in the comparative-avantage inustry. The opening of trae therefore leas to greater measure rm prouctivity growth in the comparative-avantage inustry. In aition to the greater focusing on core competencies in the comparative-avantage inustry, there is also a larger rise in the zero-pro t cuto for prouctivity below which rms eit. Hence, the opening of trae causes a larger increase in measure inustry prouctivity in the comparative-avantage inustry because of both stronger measure rm prouctivity growth an greater across- rm reallocations of resources. Consieration of the greater eport opportunities that eist in comparative-avantage inustries is the key to unerstaning these results. Following the opening of trae, there is a larger increase in labor eman at eporte proucts in the comparative-avantage inustry than in the comparative-isavantage inustry. This larger increase in labor eman leas to a rise in the relative price of the abunant factor, because this factor is use intensively in the comparative-avantage inustry. As the relative price of the abunant factor rises, the i erence in factor intensity between the two sectors implies that prouction costs in the comparative-avantage inustry increase relative to prouction costs in the comparative-isavantage inustry. This change in relative prouction costs reuces the pro ts erive from proucts that are supplie only to the omestic market in the comparative-avantage inustry relative to the comparative-isavantage inustry. The larger reuction in omestic market pro ts inuces greater sheing of low-prouctivity proucts an hence greater measure rm prouctivity growth in the comparative-avantage inustry. Similarly, the larger reuction in omestic market pro ts leas to a greater rise in the zero-pro t cuto for prouctivity an hence greater between- rm reallocations of resources in the comparative-avantage inustry than in the comparative-isavantage inustry. As a result, the interaction of comparative avantage an enogenous prouct selection causes a number of istinctive rm-level responses to trae liberalization that vary
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 9 with comparative avantage. The measure prouctivity of a rm epens on comparative avantage because it shapes the enogenous range of proucts that the rm chooses to supply to each market. Signi cantly, multi-prouct rms enogenous etensivemargin ajustments have general equilibrium consequences. The greater measure rm prouctivity growth an between- rm reallocations of resources in the comparativeavantage inustry inuce Ricarian i erences in measure prouctivity which are non-neutral across rms an inustries an provie an aitional source of welfare gains from trae. A Appeni A. Proof of Proposition CA Proof. The introuction of comparative avantage implies that countries are no longer symmetric. Therefore, the relative revenue from a prouct k in the omestic an eport markets in inustry j epens on price inices an aggregate revenue in the two countries: r jk ' j ; jk = j P F jk =P H jk R F =R H r jk ' j ; jk. In equilibrium, the price inices are the same for all proucts within an inustry an country ue to symmetry: P F jk = P F j an P H jk = P H j for all k. The equations etermining the equilibrium range of proucts supplie to the omestic market as a function of the zero-pro t-cuto rm prouctivity ' (equations (9) an (4) in the main tet) an the equations etermining the equilibrium range of proucts eporte as a function of the eporting cuto rm prouctivity ' (equations () an (6) in the main tet) remain eactly the same as in the moel with a single inustry. These relationships only compare the relative revenue of proucts within a particular market within an iniviual inustry, an are therefore unchange by the introuction of country asymmetries. The aitional terms in factor prices ue to the introuction of skille an unskille labor cancel from the left an right-han sies of equations (4) an (6) in the main tet. The free entry conition also takes the same form as in the moel with a single inustry (equation (9) in the main tet), since average pro ts in the omestic an eport market are evaluate separately relative to the lowest consumer tastes prouct supplie by a rm in each market, an terms in factor prices again cancel from the left an right-han sies of the equation.
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 20 However, the introuction of country asymmetries changes the relationship between the eporting cuto prouctivity ' j an the zero-pro t-cuto prouctivity ' j, which instea of equation (7) in the main tet is given by the following epression for the home country: ' H j = H j ' H j ; H j fj R H Pj H j f pj R F Pj F where an analogous epression hols for the foreign country. Since j the change in the relationship between ' j an ' j a ects the eporting cuto s for consumer tastes j! j j! ' j ; (24) ' j ' j = ' j =' j j ' j, ' j an hence average pro ts from the eport market in the free entry conition (equation (9) in the main tet). Comparing the free entry conitions in the open an close economies, the epecte value of entry in the open economy is equal to the value for the close economy plus an aitional positive term which captures the epecte pro ts from the eport market. Since j ' j = ' j =' j j ' j an ' j = j ' j, this aitional positive term is larger, the smaller the value of ' H =' H ' H 2 =' H 2 = H H 2 = 2 j. Diviing equation (24) for the two inustries: f =f p f 2 =f p2 (' ) = (' ) P H =P2 H 2 (' 2 ) = 2 (' 2) P F =P2 F The remainer of the proof follows the same structure as the proof of Proposition 4 in Bernar, Reing an Schott (2007) an we present an abbreviate version here. The price ine for the skill-intensive inustry relative to the labor-intensive inustry is lower in the skill-abunant country than the labor-abunant country: P H =P H 2 < P F =P F 2. Therefore, in the absence of other i erences in parameters across inustries ecept factor intensity (common values of j, F, f, F, f an f e across inustries): H < H 2 an similarly F 2 < F. Hence, the aitional positive term in the free entry conition capturing epecte pro ts in the eport market is larger in the comparative-avantage inustry than the comparative-isavantage inustry. Noting that j (') = ' j =' j j, j (') = H j ' H j ' j ' j=' j j ' j, an ' H j =, the epecte value of entry in the free entry conition (9) in the main tet is monotonically ecreasing in ' j. Therefore, the comparative-avantage inustry s larger increase in the epecte value of entry following the opening of trae requires a larger rise in the zero-pro t cuto for rm prouctivity ' j in orer to restore equality between the epecte value of entry an the unchange sunk entry cost: ' H > 'H 2
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 2 an ' F 2 > 'F. Since j (') = ' j =' j j ' j an j ' j is unchange by the opening of trae, the comparative-avantage inustry s larger rise in the zero-pro t cuto for rm prouctivity ' j implies a greater focusing on core competencies in the comparative-avantage inustry than in the comparative-isavantage inustry: H (') > H 2 (') an F 2 (') > F ('). From the e nition of weighte average prouctivity in Section B of the Appeni in the paper, the comparative-avantage inustry s larger rise in the zero-pro t cuto for prouctivity an greater focusing on core competencies implies a larger increase in weighte average prouctivity in the comparative-avantage inustry than in the comparative-isavantage inustry: ~' H > ~' H 2 an ~' F 2 > ~' F.
Technical Appeni: Multi-Prouct Firms an Trae Liberalization 22 References Bernar, Anrew B., Stephen J. Reing an Peter K. Schott (2007) Comparative Avantage an Heterogeneous Firms, Review of Economic Stuies, 74, 3-66. Bernar, Anrew B., Stephen J. Reing an Peter K. Schott (2008) Multi-Prouct Firms an Prouct Switching, revise version of NBER Working Paper, No. 2293, 2006.