The Extensive Margin of Exporting Products: A Firm-level Analysis Online Supplement

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1 The Extensive Margin of Exporting Proucts: A Firm-level Analysis Online Supplement Costas Arkolakis Yale University, CESifo an NBER Marc-Anreas Muenler UC San Diego, CESifo an NBER September 2014 Sharat Ganapati Yale University Abstract This Online Supplement contains four parts. First, we generalize the Arkolakis, Ganapati an Muenler 2014) moel to neste consumer preferences. Each inner nest hols the proucts in a firm s prouct line with an elasticity of substitution that iffers from that of the outer nest over prouct lines of ifferent firms. Secon, we present tabulations of the unerlying Brazilian export ata for Thir, we conuct Monte Carlo simulations to assess ientification uner our estimation routine. Finally, we offer a variety of robustness checks for our main estimates, using alternative assumptions an restricte samples. Keywors: International trae; heterogeneous firms; multi-prouct firms; firm an prouct panel ata; Brazil JEL Classification: F12, L11, F14 We thank Elhanan Helpman an Sam Kortum for valuable comments on this note. Any remaining errors are ours. We acknowlege NSF support SES an SES ) with gratitue. costas.arkolakis@yale.eu ka265) sharat.ganapati@yale.eu sg596/) muenler@uc.eu econ.uc.eu/muenler). Ph: )

2 1 Neste Preferences with Different Elasticities We analyze a generalize version of the Arkolakis et al. 2014, henceforth AGM) moel of multi-prouct firms that allows for within-firm cannibalization effects. The main result is that the qualitative properties of the AGM moel are retaine: the size istribution of firm sales an the istribution of the firms numbers of proucts is consistent with regularities in Brazilian exporter ata as well as other ata sets. More importantly, the general equilibrium properties of the moel o not epen on the inner nests elasticity the elasticity across the proucts of a given firm) so that the general equilibrium of the moel can be easily characterize using the tools of Dekle, Eaton an Kortum 2007). While the moel is highly tractable, the introuction of one more eman elasticity as a further egree of freeom. This egree of freeom can be iscipline using inepenent estimates for the outer an inner nests elasticities, such as those of Broa an Weinstein 2006). Uner an accoring parametrization, the moel can be use for counterfactual exercises that simulate the impact of changes in trae costs on the firm size istribution an the istribution of the firms numbers of proucts. In the following subsection we present an solve the generalize moel. We erive its aggregate properties in subsection 1.2. Subsection 1.3 conclues the presentation of the moel with neste preferences. 1.1 Moel There is a countable number of countries. We label the source country of an export shipment with s an the export estination with. We aopt a two-tier neste CES utility function for consumer preferences. 1 Each inner nest of consumer preferences aggregates a firm s proucts with a CES utility function an an elasticity of substitution ε. Using marketing terminology, the prouct bunle of the inner nest can be calle a firm s prouct line or prouct mix). The prouct lines of ifferent firms are then aggregate using an outer CES utility nest with an elasticity σ. Each firm offers a countable number of proucts but there is a continuum of firms in the worl. We assume that every prouct line is uniquely offere by a single firm, but a firm may ship ifferent prouct lines to ifferent estinations. Formally, the representative consumer s utility function at estination is given by U = s Ω G ω) g=1 q g ω) ε 1 ε σ 1 σ ε ε 1 ω σ σ 1 1 Atkeson an Burstein 2008) use a similar neste CES form in a heterogeneous-firms moel of trae but their outer nest refers to ifferent inustries an the inner nests to ifferent firms within the inustry. Eaton an Kortum 2010) present a stochastic moel with neste CES preferences to characterize the firm size istribution an their proucts uner Cournot competition. In our moel, firms o not strategically interact with other firms. This property of the moel allows us to characterize general equilibrium beyon the behavior of iniviual firms. 2

3 where q g ω) is the quantity consume of the g-th prouct of firm ω, proucing in country s. Ω is the set of firms from source country s selling to country. The representative consumer s first orer conitions imply that eman for the g-th prouct of firm ω in market is where p g ω) is the price of that prouct, q g ω) = p g ω) ε P ω;g ) ε σ P σ 1 T, P ω;g ) G ω) g=1 p g ω) ε 1) 1/ε 1) is the ieal price inex for the prouct line of firm ω selling G ω) proucts in market, an [ ] 1/σ 1) P P ω;g ) σ 1) ω Ω s is the ieal consumer price inex in market. T is total consumption expeniture Firm optimization We assume that the firm has a linear prouction function for each prouct. A firm with overall prouctivity φ faces an efficiency φ/hg) in proucing its g th prouct, where hg) is an increasing function with h1) = 1. We call the firm s total number of proucts G at estination its exporter scope at. Prouctivity is the only source of firm heterogeneity so that, uner the moel assumptions below, firms of the same type φ from country s face an ientical optimization problem in every estination. Since all firms with prouctivity φ will make ientical ecisions in equilibrium, it is convenient to name them by their common characteristic φ from now on. 2 The firm also incurs local entry costs to sell its g-th prouct in market : f g) > 0 for g > 1, with f 0) = 0. These incremental prouct-specific fixe costs may increase or ecrease with exporter scope. The overall entry cost for market is enote by F G) G g=1 f g) an strictly increases in exporter scope by efinition. Profits of a firm with prouctivity φ from country s that sells proucts g = 1,...,G in at prices p g are G π φ) = p g w ) s φ/hg) τ p ε g P φ;g ) ε σ P σ 1 T F G ). 1) g=1 We consier the first-orer conitions with respect to the prices p g of each prouct g, consistentwithanoptimalprouct-linepricep φ;g ), analsowithrespecttoexporter 2 To simplify the exposition, we assume here that firms face no other iiosyncratic cost components, whereas the Arkolakis et al. 2014) moel also allows for a estination specific market access cost shock c so that a firm in that moel is characterize by a pair of shocks φ,c ). 3

4 scope G. As shown in AppenixA, thefirst-orerconitionswith respect to pricesimply a constant markup over marginal cost for all proucts equal to σ σ/σ 1). Using the constant markup rule in eman for the g-th prouct of a firm with exporter scope G yiels optimal sales of the prouct p g φ)q g φ) = σ w ) ε 1) sτ P φ;g ) ε σ P σ 1 T. 2) φ/hg) Using this result an the efinition of P φ;g ), we can rewrite profits that a firm generates at estination selling G proucts as where π φ;g ) = P φ;g ) σ 1)Pσ 1 T F G ) σ = HG ) σ 1) σw s τ ) σ 1) φ σ 1 P σ 1 T F G ), 3) σ HG ) [ G ] 1/ε 1) hg) ε 1) g=1 is the firm s prouct efficiency inex. We impose the following assumption, which is necessary for optimal exporter scope to be well efine. Assumption 1 Parameters are such that Z G) = f G)/[HG) σ 1) HG 1) σ 1) ] strictly increases in G. The expression for Z G) reuces to Z G) = f G)hG) σ 1 when ε = σ. So, in that case, Assumption 1 is ientical to the one consiere in AGM. For a firm to enter a estination market, its prouctivity has to excee a threshol φ, where φ is implicitly efine by zero profits for the first prouct: P σ 1 T [P φ ;1)] σ 1) = σf 1). Using the convention h1) = 1 for G = 1 in 3) yiels φ ) σ 1 = σf 1) σw sτ ) σ 1 P σ 1 T. 4) Similarly, we can efine the threshol prouctivity of selling G proucts in market. The firm is inifferent between introucing a G-th prouct or stopping with an exporter scope of G 1 at the prouct-entry threshol φ,g if ) ) π φ,g ;G π φ,g ;G 1 = 0. 5) 4

5 Using equations 3) an 4) in this profit equivalence conition, we can solve out for the implicitly efine prouct-entry threshol φ,g, at which the firm sells G or more proucts, φ,g ) σ 1 φ = )σ 1 f G ) HG ) σ 1) HG 1) σ 1) f 1) = φ )σ 1 f 1) Z G ), 6) where we efine φ,1 φ. So, uner Assumption 1, the profit equivalence conition 5) impliesthattheprouct-entrythresholsφ,g strictlyincreasewithganmoreprouctive firms will weakly raise exporter scope compare to less prouctive firms. Base on these relationships, we can make the following statement Export sales can be written succinctly as t φ) = σ w ) 1 ε sτ G P σ 1 T hg) 1 ε [P G φ))] ε σ φ g=1 ) σ 1 φ = σf 1) HG φ)) σ 1) 7) φ using equation 4). This sales relationship is similar in both moels with ε σ an moels with ε = σ. The only ifference between the two types of moels is that HG ) epens on ε by 3). If the term HG ) converges to a constant for G, then export sales are Pareto istribute in the upper tail if φ is Pareto istribute. Proposition 1 Suppose Assumption 1 hols. Then for all s, : exporter scope G φ) is positive an weakly increases in φ for φ φ ; total firm exports t φ) are positive an strictly increase in φ for φ φ. Proof. The first statement follows irectly from the iscussion above. The secon statement follows because HG φ)) σ 1) strictly increases in G φ) an G φ) weakly increases in φ so that t φ) strictly increases in φ by 7). Similar to AGM, we efine exporter scale an exporter s mean sales) in market as ) σ 1 φ HG φ)) 1 σ a φ) = σf 1) φ G φ) Uner a mil conition, exporter scale a φ) increases with φ an thus with a firm s total sales t φ). The following sufficient conition ensures that exporter scale increases with total sales. Case 1 The function Z g) strictly increases in g with an elasticity lnz g) lng > 1. 5

6 Case 1 is more restrictive than Assumption 1 in that the conition not only requires Z to increase with g but that the increase be more than proportional. We can formally state the following result. Proposition 2 If Z g) satisfies Case 1, then sales per export prouct a φ) strictly increase at the iscrete points φ = φ, φ,2, φ,3,... Proof. Compare to AGM, Z g) is efine in more general terms, but it enters the relevant relationships in the same way as in AGM before. Case 1 therefore also suffices in the neste-utility moel, an the proposition hols see the Appenix in AGM for etails of the proof for non-neste utility) Within-firm sales istribution We revisit optimal sales per prouct an their relationship to exporter scope an the prouct s rank in a firm s sales istribution. The relationship lens itself to estimation in micro ata. Using the prouctivity threshols for firm entry 4) an prouct entry 6) in optimal sales 2) an simplifying yiels ) σ 1 p g φ)x g φ) = σz G )HG ) ε σ φ hg) ε 1) = σ φ,g f G )HG ) ε 1 1 [1 hg ) ε 1) /HG ) ε 1) ] σ 1 ε 1 φ φ,g ) σ 1 hg) ε 1). 8) Note that HG) ε σ strictly falls in G if ε > σ. Uner Case 1, the term Z G )HG ) ε σ must strictly increase in G, however, because iniviual prouct sales strictly rop as the prouct inex g increases an hg) ε 1) falls. So, if Z G )HG ) ε σ i not strictly increase in G, average sales per prouct woul not strictly increase, contrary to Proposition 2. Compare to AGM, the relationship 8) is not log-linear if ε σ an requires a nonlinear estimator, similar to the general case in continuous prouct space Arkolakis an Muenler 2010). 1.2 Aggregation To erive clear preictions for the moel equilibrium we specify a Pareto istribution of firm prouctivity following Helpman, Melitz an Yeaple 2004) an Chaney 2008). A firm s prouctivity φ is rawn from a Pareto istribution with a source-country epenent location parameter b s an a shape parameter θ over the support [b s,+ ) for all estinations s. The cumulative istribution function of φ is Pr = 1 b s ) θ /φ θ an the probability ensity function is θb s ) θ /φ θ+1, where more avance countries are thought to have a higher location parameter b s. Therefore the measure of firms selling to country, that is the measure of firms with prouctivity above the threshol φ, is b θ s M = J s. 9) φ )θ 6

7 As a resilt, the probability ensity function of the conitional prouctivity istribution for entrants is given by µ φ) = { θφ ) θ /φ θ+1 if φ φ 0 otherwise. 10) We efine the resulting Pareto shape parameter of the total sales istribution as θ θ/σ 1). With these istributional assumptions we can compute a number of aggregate statistics from the moel. We enote aggregate bilateral sales of firms from s to country as T. The corresponing average sales are efine as T, so that T = M T an T = t φ)µ φ)φ. 11) Similarly, we efine average local entry costs as F = F G φ))µ φ)φ. To compute T, we impose two aitional assumptions. Assumption 2 Parameters are such that θ > σ 1. φ φ Assumption 3 Parametersare such that F G=1 f G) 1 θ [ HG) 1 σ HG 1) 1 σ] θ is strictly positive an finite. Then we can make the following statement. Proposition 3 Suppose Assumptions 1, 2 an 3 hol. Then average sales T per firm are a constant multiple of average local entry costs F T = θσ F = f 1) θ F. θ 1 Proof. See Appenix C. As a result, bilateral expeniture trae shares can be expresse as λ = M T k M = J sb s )θ w s τ ) θ f 1) θ F k T k k J, 12) kb k ) θ w k τ k ) θ f k 1) θ Fk an expression that epens on the values of ε an σ only insofar as these parameters affect F through HG). 7

8 We can also compute mean exporter scope at a estination: Ḡ = G φ)µ φ)φ φ = φ ) θ θ = φ,2 [ φ,2 φ φ θ 1 φ+ ) θ φ ) θ + φ ) θ φ,3 φ,3 φ,2 2φ θ 1 φ+... ) θ ) φ,2 θ φ ) θ Completing the integration, rearranging terms an using equation 6), we obtain Ḡ = f 1) θ Z g) θ. 13) g=1 For the average number of proucts to be a well efine an finite number we require Assumption 4 Parameters are such that g=1 Z g) θ is strictly positive an finite. In Appenix C we show that fixe costs expeniture is a constant share of firm sales where we enote means using a bar), as summarize in Proposition 3: F T = θ 1 θσ. We erive aggregate welfare in Appenix D an emonstrate in Appenix E that wage income an profit income can be expresse as a constant share of total output y s per capita: π s = ηy s, w s = 1 η)y s, where η 1/ θσ). Since aggregates of the moel o not epen on ε, the equilibrium efinition is the same as in AGM. 1.3 Conclusion We have characterize an extension of the AGM moel when the elasticity of substitution between a firm s iniviual proucts oes not equal the elasticity of substitution across proucts lines of ifferent firms. The extene moel retains the main qualitative implications of the baseline AGM moel, in which the two elasticities are the same. Future work using the structure of the generalize moel to obtain estimates of the two elasticities may lea to a better unerstaning of the substitution effects within an across firms. ] 8

9 2 Data: Exporte Proucts an Export Destinations Tables 1 an 2 show Brazil s top ten export estinations by number of exporters an the top ten exporte HS 6-igit prouct coes by total value in In Table 1, Argentina is the most common export estination an the Unite States receives most Brazilian exports in value. In Table 2, meium-size aircraft is the leaing export prouct in value, followe by woo pulp an biofuel proucts. Table 1: Top Brazilian Export Destinations Destination # Exporters Export Value USD) Argentina 4,590 5,472,333,618 Uruguay 3, ,642,201 USA 3,083 9,772,577,557 Chile 2,342 1,145,161,210 Paraguay 2, ,065,104 Bolivia 1, ,543,791 Mexico 1,336 1,554,452,204 Venezuela 1, ,281,591 Germany 1,217 1,364,610,059 Peru 1, ,896,577 Source: SECEX 2000, manufacturing firms an their manufacture proucts. Table 2: Top Brazilian Exporte Items HS Coe Description Value USD) Airplanes between 2 an 15 tons 2,618,856, Bleache non-coniferous chemical woo pulp 1,523,403, Soybean oil-cake an other soli resiues 1,245,752, Passenger vehicles between 1,500 an 3,000 cc 1,197,222, Transmission apparatus incorporating reception apparatus 926,618, Footwear, with outer soles 854,950, Semifinishe proucts of iron or nonalloy steel 802,801, Unwrought aluminum, not alloye 765,195, Orange juice, frozen 561,103, Raw soli cane sugar 520,544,094 Source: SECEX 2000, manufacturing firms an their manufacture proucts. 9

10 3 Sensitivity of Monte Carlo Estimates To ocument ientification uner our simulate metho of moments estimator, we run Monte Carlo tests with generate ata. We generate 333,000 Brazilian firms uner the initial parameters Θ, where } Θ = {δ 1,δ 2,, α, θ,σ ξ,σ c = { 1.17, 0.90, 1.73, 1.84, 1.89, 0.53} reflects the baseline estimates from Table 2 in the main text. The generate ata have approximately 10,000 exporters. We then apply our simulate metho of moments routine to the generate ata an fin the optimum, recovering an estimate of the parameter vector ˆΘ. We repeat the ata generation an estimation proceure 30 times an report in Table 3 the mean an stanar eviation of the elements of ˆΘ. The Monte Carlo results in Table 3 ocument that our proceure accurately pinpoints all parameters of interest. In particular ˆ α,ˆσ ξ,ˆσ c ) are precisely estimate, with point estimates close to the initial parameters behin the generate ata an with stanar errors less than 2 percent of the true value. Similarly, ˆδ an ˆ θ are estimate close to their true values, their stanar errors are uner 4 percent of their true values. The proximity of our parameter estimates to the initial parameters unerlying the ata generation, an their precision, substantiate that our simulate metho of moments estimator ientifies the AGM moel s parameters of interest. Table 3: Monte Carlo Results Θ δ 1 δ 2 α θ σξ σ c δ 1 δ 2 Parameter of generate ata Estimate mean) s.e.) 0.03) 0.03) 0.02) 0.05) 0.03) 0.01) 0.01) 10

11 4 Sensitivity Analysis To assess the robustness of our baseline estimates in AGM, we perform a number of moifications to our main specification. Overall, we fin that our baseline results are remarkably robust to sample restrictions an alternative variable efinitions. 4.1 Ajuste sales Our baseline estimates imply a large an statistically significant ifference between δ LAC an δ ROW. In a first robustness check, we strive to rule out that this ifference coul be riven by ifferent typical sales across sets of proucts that Brazilian firms ship to LAC an non-lac countries. We therefore correct sales an control for the mean sales of prouct groups at the HS 2-igit level. Concretely, we take the upwar or ownwar eviation of a firm s HS 6-igit prouct sales to a estination logy p ω from the worlwie prouct-group sales mean of Brazilian exporters: ỹ ωg = exp { logy ωg 1 M ω 1 N g HS2 logy ωg } This ajustment oes not reuce the sample size. We report the results in the row labele 1. Ajuste sales in Table 4. The estimates are broaly consistent with the baseline, but the estimate scope elasticities of market access costs δ an of prouct efficiency α are lower in absolute magnitue, an so is the estimate Pareto shape parameter θ. These estimates imply that both the within-firm prouct istribution is more concentrate in the top prouct an the between-firm sales istribution has more firms in the upper tail with extremely high sales. An intuitive explanation is that emeaning sales by the average exporter s typical sale in a prouct group exacerbates sales eviations of specific proucts, thus making istributions appear more extreme. However, signs of our estimates stay the same an broa magnitues remain qualitatively similar to the baseline estimates. 4.2 Avance manufacturing A relate robustness concern is that estimation coul be riven by ifferent feasible exporter scopes across prouct groups that Brazilian firms ship to LAC an non-lac estination. For example, more ifferentiate inustries, or more technology riven inustries, might allow for the export of more potential varieties, or the HS classification system might simply provie more iniviual HS 6-igit proucts within more refine HS 2-igit prouct groups. In a secon robustness exercise we therefore split the sample into firms that are active in relatively avance manufacturing inustries. We present results from our efinition of avance manufacturing as three top-level NACE sectors Manufacture of machinery an equipment n.e.c, Manufacture of electrical an optical equipment an Manufacture of transport equipment coes DK, DL an DM). Uner this sectoral restriction, a markely reuce sample size of only 2,539 Brazilian manufacturing exporters remains. Despite the consierable rop in sample size, however,. 11

12 results in the row labele 2. Avance manufacturing in Table 4 are broaly consistent with the baseline. In avance manufacturing inustries, the ifference in scope elasticities of market access costs δ is larger between LAC an non-lac countries. While market access costs rop off even faster with scope in LAC countries in avance manufacturing inustries, in non-lac estination the converse is the case: market access costs remain more elevate at higher scopes than in the average manufacturing inustry. The gap between the LAC an non-lac scope elasticities of market access costs is almost ouble as wie in avance manufacturing inustries as it is in the average inustry. For our counterfactual exercise, this wier ifference of market access cost elasticities between LAC an non-lac countries implies even more pronounce benefits of harmonizing market access costs across the worl. Other parameter estimates are similar to the baseline, an specially the scope elasticity of prouct efficiency α is not statistically significantly ifferent in avance inustries compare to the average inustry. Overall, every sign remains the same an estimates that are statistically ifferent from the baseline remain comparable in their qualitative economic implications. 4.3 Eight-igit NCM prouct categories To make our results closely comparable to evience from other countries, in our main text we efine a prouct as a Harmonize System HS) 6-igit coe, which is internationally comparable by requirement of the Worl Customs Organization WCO) across its 200 member countries. To query the potential sensitivity of our results to a refine prouct classification, we use the Mercosur 8-igit level Nomenclatura Comum o Mercosul NCM8), which roughly correspons to the 8-igit HS level by the Worl Customs Organization. As the row 3. NCM 8-igit manufacturing in Table 4 shows, our results are harly sensitive at all to the change in level of isaggregation. No single estimate is statistically ifferent from our baseline estimates. 4.4 Sensitivity to estinations Argentina an Unite States Two estination markets ominate Brazilian manufacturing exports: Argentina the top estination in terms of exporter counts) an the Unite Statesthe top estination in terms of export value). To assure ourselves that the estimates are not riven by potential outlier behavior of export flows to those two estinations, we remove them from the sample. As Table 4 in the row labele 4. Dropping ARG, USA shows, only the estimate of the firm size istribution s shape parameter θ becomes statistically significantly ifferent from the baseline estimate. All other estimates are statistically inistinguishable from the baseline estimates. When we omit Argentina an the Unite States, the higher estimate for the Paretotailinex θ impliesalowerprobabilitymassintheuppertailoffirmswithextremely high sales. Even though Argentina an the Unite States attract a large number of export entrants from Brazil, these markets also exhibit a stronger concentration of exports among just a few top-selling firms than the average Brazilian export estination. 12

13 Table 4: Robustness for Select Subsamples estimate s.e.) δ LAC δ ROW α θ σξ σ c δ LAC δ ROW Baseline all manufacturing) 0.09) 0.11) 0.09) 0.09) 0.04) 0.03) 0.05) 1. Ajuste sales 0.06) 0.06) 0.06) 0.07) 0.03) 0.02) 0.03) 2. Avance manufacturing 0.14) 0.17) 0.12) 0.23) 0.08) 0.06) 0.13) 3. NCM 8-igit manufacturing 0.06) 0.08) 0.05) 0.10) 0.04) 0.03) 0.05) 4. Dropping ARG, USA all manufacturing) 0.08) 0.07) 0.07) 0.09) 0.04) 0.02) 0.07) 5. Exporter share percent 0.11) 0.14) 0.10) 0.11) 0.04) 0.03) 0.05) Source: SECEX 2000, manufacturing firms an their manufacture proucts. Note: Proucts at the HS 6-igit level. Estimates of δ LAC inicates the scope elasticity for incremental prouct access costs for Brazilian firms shipping to other LAC estinations. Similarly δ ROW perform the same role for exports to non-lac estinations. See text for full escription of various specifications. 4.5 Exporter share An arguably important moment for our simulate metho of moments is the share of formally establishe Brazilian manufacturing firms that export. Among the universe of Brazilian firms with at least one employee, only three percent of firms are exporters in This share is similar to that observe in other countries, for which ata on the universe of firms with at least one employee is available. However, censuses an surveys in most eveloping an some inustrialize countries truncate their target population of firms from below with threshols up to 20 employees. To query sensitivity of our estimates to the share of exporters, we hypothetically consier an alternative share of 10 percent of Brazilian firms exporting. This exercise serves two purposes. First, comparisons of our finings to future results in other countries may epen on using a hypothetically truncate target population of firms from below. Secon, we can check how our results might epen on a hypothetically more export oriente manufacturing sector such as, for instance, the U.S. manufacturing sector. Table 4 reports the results in the row labele 5. Exporter share 10 percent. Compare to the baseline, the scope elasticities of market access costs δ an of prouct efficiency α increase in absolute magnitue. Intuitively, the estimator tries to explain the hypothetically higher share of exporters with relatively faster eclines in market access costs as exporter scope increases but to offset those access cost reuctions with relatively steeper eclines in prouct efficiency away from core competency, so as to keep matching the overall pattern of exporter scopes across estinations. The other three parameter esti- 13

14 Table 5: Alternative Regional Aggregates δ 1 δ 2 δ 1 δ 2 Baseline Latin America an CaribbeanLAC) Baseline Rest of Worl 1. Mercosur Rest of LAC Non-Mercosur) 2. Mercosur Rest of Worl Non-Mercosur) ) ) ) Source: SECEX 2000, manufacturing firms an their manufacture proucts. Note: Proucts at the HS 6-igit level. mates remain similar to the baseline estimates. This final robustness exercise therefore clarifies how the firm entry margin influences ientification: if firm entry with the first prouct were hypothetically more prevalent, then for a given common market access cost component f 1) the access cost scheule woul nee to ecline faster with scope, leaing to wier exporter scopes everywhere, unless prouction efficiency also eclines faster with scope. 4.6 Sensitivity to Mercosur In a final set of robustness exercises, we alternate the pairings of regional aggregates. In the baseline, we split the worl into LAC Latin American an the Caribbean) an the Rest of the Worl non-lac). In a first alteration, we rop estination countries outsie of LAC from our sample an split LAC into Mercosur estinations in 2000 Argentina, Paraguay, Uruguay) an non-mercosur estinations. In the row labele 1. Mercosur Rest of LAC, Table 5 reports the results for the ifference in the scope elasticities of market access costs between the two sub-regions within LAC, an the ifference is negative as in the baseline but small an not statistically ifferent from zero). This fining justifies our treatment of LAC in the baseline as a relatively homogeneous region for Brazilian exporters. In a secon alteration of the regional split, we iscern between Mercosur estinations in 2000 Argentina, Paraguay, Uruguay) an the Rest of the Worl, where the Rest of the Worl inclues LAC countries outsie Mercosur as well as non-lac estinations. Expectely, given the earlier results in the baseline an in the first alteration, the ifference is negative but not quite as pronounce in magnitue as the ifference between LAC an the Rest of the Worl. We therefore conclue that LAC countries outsie Mercosur are more similar to Mercosur than to the Rest of the Worl an consier our baseline split of estinations into LAC an non-lac an aequate country grouping. 14

15 Appenix A Optimal prouct prices We characterize the first-orer conitions for the firm s optimal pricing rules at every estination. There are G φ) first-orer conitions with respect to p g. For any G φ), taking the first erivative of profits π φ) from 1) with respect to p g an iviing by p ε g P φ;g ) ε σ P σ 1 T yiels π φ) p g { = P σ 1 T P φ;g ) ε σ p ε g 1 ε 1 w ) s φ/hg) τ p 1 g G φ) +ε σ)p φ;g ) ε 1 k=1 A.1) p k w ) } s φ/hk) τ p ε k. The first-orer conitions require that A.1) equals zero for all proucts g = 1,...,G φ). Use the first-orer conitions for any two proucts g an g an reformulate to fin p g /p g = hg)/hg ). So the firm must optimally charge an ientical markup over the marginal costs for all proucts. Define this optimal markup as m. To solve out for m in terms of primitives, use p g = mw s τ /[φ/hg)] in the first-orer conition above an simplify: 1 ε 1 m +ε σ)p φ;g ) ε 1 m 1 m G φ) k=1 p ε 1) k = 0. Note that G φ) k=1 p ε 1) k = P φ;g ) ε 1). Solving the first-orer conition for m, we fin the optimal markup over each prouct g s marginal cost m = σ σ/σ 1). A firm with prouctivity φ optimally charges a price p g φ) = σw s τ /[φ/hg)] A.2) for its proucts g = 1,...,G φ). B Secon-orer conitions We now turn to the secon-orer conitions for price choice. To fin the entries along the iagonal of the Hessian matrix, take the first erivative of conition A.1) with respect to 15

16 the own price p g an then replace w s τ /[φ/hg)] = p g φ)/ σ by the first-orer conition to fin { 2 π φ) = P σ 1 p g ) 2 T P φ;g ) ε σ p ε g p 1 g ε σ +ε σ)p φ;g ) ε 1 [ ε 1)+ε/ σ]p ε g G +ε σ)ε 1)P φ;g ) 2ε 1) p ε g k=1 { = P φ;g ) ε σ P σ 1 T εp ε 1 g This term is strictly negative if an only if ε σ)p φ;g ) ε 1 p ε 1) g < ε. 1 1/ σ)p ε 1) k } +ε σ)p φ;g ) ε 1 p 2ε g } / σ. B.3) If ε σ, this last conition is satisfie because the left-han sie is weakly negative an ε > 0. If ε > σ, then we can rewrite the conition as p ε 1) g /[ G k=1 p ε 1) g ] < 1 < ε/ε σ) so that the conition is satisfie. So, the iagonal entries of the Hessian matrix are strictly negative. To erive the entries off the iagonal of the Hessian matrix, we take the erivative of conition A.1) for prouct g with respect to any other price p g an then replace w s τ /[φ/hg )] = p g φ)/ σ by the first-orer conition to fin { 2 π φ) = P σ 1 T P φ;g ) ε σ p ε g ε σ)p φ;g ) ε 1 [ ε 1)+ε/ σ]p ε g p g p g G +ε σ)ε 1)P φ;g ) 2ε 1) p ε g = P σ 1 T ε σ)p φ;g ) ε σ+ε 1 p ε g p ε g / σ. k=1 1 1/ σ)p ε 1) k } B.4) This term is strictly positive if an only if ε > σ. Having erive the entries of the Hessian matrix, it remains to establish the conitions uner which the Hessian is negative efinite. We iscern two cases. First the case of ε σ an then the case ε > σ. B.1 Negative efiniteness of Hessian if ε σ By B.3) an B.4), the Hessian matrix can be written as where H A εp ε 1 1 H = P φ;g ) ε σ P σ 1 T [ HA +ε σ)p φ;g ) ε 1 H B ], 0 εp ε εp ε an H B p ε 1 p ε 1 p ε 2 p ε 1 p ε 2 p ε 2 p ε 3 p ε 1 p ε 3 p ε 2 p ε 3 p ε

17 The Hessian matrix H is negative efinite if an only if the negative Hessian H = P φ;g ) ε σ [ P σ 1 T HA +σ ε)p φ;g ) ε 1 ] H B is positive efinite. Note that the sum of one positive efinite matrix an any number of positive semiefinite matrices is positive efinite. So, if H A an H B are positive semiefinite an at least one of the two matrices is positive efinite given ε σ), then the Hessian is negative efinite. A necessary an sufficient conition for a matrix to be positive efinite is that the leaing principal minors of the matrix are positive. The leaing principal minors of H A are positive, so H A is positive efinite. For H B, the first leaing principal minor is positive, an all remaining principal minors are equal to zero. So H B is positive semiefinite. Therefore the Hessian matrix H is negative efinite. B.2 Negative efiniteness of Hessian if ε > σ Another necessary an sufficient conition for the Hessian matrix H to be negative efinite is that the leaing principal minors alternate sign, with the first principal minor being negative. The first principal minor the first iagonal entry is strictly negative as is any iagonal entry by B.3). An application of the leaing principal minor test in our case requires a recursive computation of the eterminants of G φ) submatrices a solution of polynomials with orer up to G φ)). We choose to check for negative efiniteness of the Hessian in two separate ways when ε > σ. First, we erive a sufficient but not necessary) conition for negative efiniteness of the Hessian an query its empirical valiity. Secon, we present a necessarybut not sufficient) conition for negative efiniteness of the Hessian for any pair of two proucts. Sufficiency. A sufficient conition for the Hessian to be negative efinite is ue to McKenzie 1960): a symmetric iagonally ominant matrix with strictly negative iagonal entries is negative efinite. A matrix is iagonally ominant if, in every row, the absolute value of the iagonal entry strictly excees the sum of the absolute values of all off-iagonal entries. By our erivations above, all iagonal entries of the Hessian are strictly negative. For ε > σ, the conition for the Hessian to be iagonally ominant is G k g ε σ)p φ;g ) ε 1 p ε k p ε g < εp ε 1 g ε σ)p φ;g ) ε 1 p 2ε g for all of a firm φ s proucts rows of its Hessian), where we cancelle the strictly positive terms P σ 1 T P φ;g ) ε σ / σ from the inequality. So, for ε = σ the Hessian is iagonally ominant. Using the optimal price A.2) of prouct g from the first-orer conition an rearranging terms yiels the following conition G k=1 hk) ε G k=1 hk) ε 1) < ε ε σ hg) 1 B.5) 17

18 for the Hessian to be a iagonally ominant matrix at the optimum. By convention an without loss of generality h1) = 1 for a firm with prouctivity φ. So the prouct efficiency scheule hg) strictly excees unity for the secon prouct an subsequent proucts. As a result, the left-han sie of the inequality is boune above for an exporter with a scope of at least two proucts at a estination: G k=1 hk) ε < G k=1 hk) ε 1) G k=1 hk) ε 1) G k=1 hk) ε 1) = 1. A sufficientbut not necessary) conition for the Hessian to be negative efinite is therefore 1 hg) < ε ε σ for all of the firm s proucts. However, the Hessian can still be negative efinite even if this conition fails. Clearly, the Hessian becomes negative efinite the closer is ε to σ because then the off-iagonal entries approach zero an the Hessian is trivially iagonally ominant. Moreover, the Hessian can be negative efinite even if it is not a iagonally ominant matrix. To query the empirical valiity of the sufficient conition hg) < ε/ε σ), consier evience on proucts an brans in Broa an Weinstein2006). Their preferre estimates for ε an σ within an across omestic U.S. bran moules are 11.5 an 7.5. Estimates in Arkolakis et al. 2014) suggest that αε 1) is aroun 1.84 uner the specification that hg) = g α. These parameters imply that the conition hg) < ε/ε σ) is satisfie for Hessians with up to 414 proucts. In the Arkolakis et al. 2014) ata, no firm-country observations involve 415 or more proucts in a market with a meian of one prouct an a mean of 3.52). Even if aitional proucts iniviually violate the sufficient conition, Hessians with more proucts may still be negative efinite. Necessity. Consier any two proucts g an g. Negative efiniteness of the Hessian mustbeinepenentoftheoreringofproucts, sothesetwoprouctscanbeassignethe first an secon row in the Hessian without loss of generality. As state before, a necessary an sufficient conition for the Hessian to be negative efinite is that the leaing principal minors of the Hessian alternate sign, with the first principal minor being negative. So a necessary conition for the Hessian to be negative efinite is that the principal minors of any two proucts first an secon in the Hessian) alternate sign, with the first principal minor negative an the secon positive. The first principle minor is strictly negative because all iagonal entries are strictly negative by B.3). The secon principal minor is strictly positive if an only if the eterminant satisfies 2ε 2 P φ;g ) ε 1) εε σ) p ε 1) g +p ε 1) g ) ε σ) 2 p g p g ) ε 1) P φ;g ) ε 1 > 0, where we cancelle the strictly positive terms P 2σ 1) B.6) T P φ;g ) 2ε σ) / σ 2 from the in- g P φ;g ) ε 1). equality an multiplie both sies by p ε 1 g pε 1 18

19 To buil intuition, consier the ual-prouct case with G φ) = 2. Then conition B.6) simplifies to hg) ε 1) G k=1 hk) ε 1) hg ) ε 1) < ε G k=1 hk) ε 1) ε σ ε+σ ε σ. For ε > σ, both terms in the prouct on the right-han sie strictly excee unity while the terms in the prouct on the left-han sie are strictly less than one, an the conition is satisfie. Inthemulti-prouctcasewithG φ) > 2, replacep ε 1) k g,g p ε 1) k hg) ε 1) G in conition B.6) an simplify to fin k=1 hk) ε 1) hg ) ε 1) < ε G k=1 hk) ε 1) ε σ g +p ε 1) g = P φ;g ) ε 1) ε+σ ε σ + ε k g,g hk) ε 1) ε σ. G k=1 hk) ε 1) For ε > σ, the necessary conition on any two proucts of a multi-prouct firm is trivially satisfie by the above erivations because the aitional aitive term on the right-han sie is strictly positive. In summary, parameters of our moel are such that, for any two proucts of a multiprouct firm, the secon-orer conition is satisfie. C Proof of Proposition 3 Average sales from s to are T = y G ) θφ )θ φ = σf φ θ+1 1)θ φ The proof of the proposition follows from the following Lemma. Lemma 1 Suppose Assumptions 1, 2 an 3 hol. Then φ σ 2 θ /φ )σ 1 θ HG φ)) σ 1 φ = f θ 1 1) θ σ 1) F, where φ F υ=1 Proof. Note that φ σ 2 θ /φ )σ 1 θ HG φ)) σ 1 φ = H1) 1 σ φ φ [f υ)] 1 θ [ Hυ) 1 σ Hυ 1) 1 σ] θ. φ,2 φ [ φ,2 = H1) 1 σ φ σ 2 θ /φ )σ 1 θ HG φ)) σ 1 φ. φ,3 φ σ 2 θ φ+h2) 1 σ ) σ 1 θ ] φ ) σ 1 θ φ,2 [θ σ 1)]φ [ )σ 1 θ ) φ,3 σ 1 θ +H2) 1 σ φ,2 [θ σ 1)]φ )σ 1 θ ) σ 1 θ ] φ σ 2 θ φ

20 Alsonotethat,usingequations4)an6),theratio can be rewritten as ) σ 1 θ φ,g φ,g 1 ) σ 1 θ [ φ ),2 σ 1 θ φ ) σ 1 θ] /φ )σ 1 θ = φ )σ 1 θ [ φ ) σ 1 ]σ 1 θ [ σ 1 f g) φ Hg) 1 σ Hg 1) 1 σ f ) σ 1 ]σ 1 θ σ 1 f g 1) 1) Hg 1) 1 σ Hg 2) 1 σ f 1) = [ φ ) σ 1] σ 1 θ σ 1 f g) 1 θ = f 1) θ 1 [ Hg) 1 σ Hg 1) 1 σ] f g 1) 1 θ 1 θ [ Hg 1) 1 σ Hg 2) 1 σ] 1 θ. We efine 3 F υ=1hυ) 1 σ [f υ +1)] 1 θ [ Hυ +1) 1 σ Hυ) 1 σ] [f υ)] 1 θ 1 θ [ Hυ) 1 σ Hυ 1) 1 σ] 1 θ = [ Hυ) 1 σ Hυ 1) 1 σ] [f υ)] 1 θ [ υ=1 Hυ) 1 σ Hυ 1) 1 σ] 1 θ = υ=1 [f υ)] 1 θ [ Hυ) 1 σ Hυ 1) 1 σ] θ. With this efinition we obtain φ σ 2 θ /φ )σ 1 θ HG φ)) σ 1 φ = f θ 1 1) θ σ 1) F. φ 3 In the special case with ε = σ, we can rearrange the terms an fin F = υ=1 [f υ)] 1 θ [hυ) σ 1] θ = υ=1 [f υ)] 1 θ hυ) θ. 20

21 D Welfare We have that P 1 σ = s = s = s = = s φ φ [P φ)] 1 σ µφ)φ M G φ) υ=1 σw s τ ) 1 σ b θ sθ σ w s φ/hg) τ [ φ,2 H1) 1 σ σw s τ ) θ b θ f 1) sθ 1 σ T 1 σ ) 1 ε 1 ε θφ ) θ φ φ θ+1 ) σ 1 θ φ,1 θ σ 1) ) 1 θ[h1) 1 σ φ,2 ) σ 1 θ ) +... ] ) σ 1 θ ) φ,1 σ 1 θ ] ) φ,1 σ 1 θ )+..., where we use the efinition of φ,1 for the last step. The final term in parentheses equals F so P θ θ σ) θ = 1 1 θ/σ 1)T b 1 θ σ) θ sw s τ ) θ F. s Using this relationship in equation 12), we obtain T P ) θ = T w ) θ θ σ) θ σ) θ 1 b θ λ θ F 1). T 1 θ If trae is balance then T = y, where y is output per capita. Since, by the efinition of F 1), this variable is homogeneous of egree 1 θ in wages an w /y is constant in all equilibria see proof below) we arrive at the same welfare expression as in Arkolakis, Costinot an Roriguez-Clare 2012): the share of omestic sales in expeniture λ an the coefficient of the Pareto istribution are sufficient statistics to characterize aggregate welfare in the case of balance trae. The final step is to verify that the wage w is a constant fraction of per-capita output y so that the first ratio on the right-han sie is constant. We emonstrate this next. E Constant wage share in output per capita We show that the ratio w /y is a constant number. We first look at the share of fixe costs in bilateral sales. Average fixe costs incurre by firms from s selling to are F = φ,2 φ φ,3 F 1)θ φ )θ φ+ F φθ+1 2)θ φ )θ φ,2 = F 1)φ ) θ[ φ,2 φ θ φ+ ) θ φ ) θ] F 2)φ ) θ[ φ,3 ) θ ) φ,2 θ ]... 21

22 Using the efinition F G ) G g=1 f g) an collecting terms with respect to φ,g we can write the above expression as F = f 1)+ φ,2 ) θφ ) θ f 2)+ ) φ,3 θφ ) θ f 3)+.... Using the efinition of φ,g from equation 6) to replace terms in the above equation, we obtain φ,g ) σ 1 = φ )σ 1 HG ) σ 1) HG 1) σ 1) f G ) f 1). Therefore [ ] 1/σ 1) f 2) 1/σ 1) H2) σ 1) θ H1) σ 1) F = f 1)+ f 1) 1/σ 1) [H1) σ 1) ] 1/σ 1) f 2)+... [ [ ] ) ] 1/σ 1) θ = f 1)+f 1) θ f 2) 1/σ 1) H2) σ 1) H1) σ 1) f 2)+... = [f 1)] θ f 1) 1 θ f 2) 1 θ f 3) 1 θ + ] + θ [H2) σ 1) H1) [H3) σ 1) σ 1) H2) σ 1)]... θ = [f 1)] θ F an hence F T = ] f 1) θ [f 1) 1 θ + f 2) 1 θ +... h2) θ f 1) θθσ = f g) 1 θ θ σ 1) g=1 hg) θ θ σ 1) g=1 θσ g=1 f g) 1 θ hg) θ f g) 1 θ hg) θ < 1. Finally, the share of profits generate by the corresponing bilateral sales is the share of variable profits in total sales 1/σ) minus the average fixe costs pai, as erive above. So π = 1 T σ θ σ 1) = σ 1 θσ θσ = 1 θσ η. This fining implies that the wage is a constant fraction of per capita income. The reason is that total profits for country s are π s L s = k λ skt k / θσ), where k λ skt k is the country s total income because total manufacturing sales of a country s equal its total sales across all estinations. So profit income an wage income can be expresse as constant shares of total income: π s L s = 1 θσ Y s an w s L s = θσ 1 θσ Y s. 22

23 References Arkolakis, Costas an Marc-Anreas Muenler, The Extensive Margin of Exporting Proucts: The Continuum Case, University of California, San Diego, unpublishe manuscript, 2010., Arnau Costinot, an Anres Roriguez-Clare, New Trae Moels, Same Ol Gains?, American Economic Review, February 2012, 102 1), , Sharat Ganapati, an Marc-Anreas Muenler, The Extensive Margin of Exporting Proucts: A Firm-level Analysis, Unpublishe manuscript, Atkeson, Anrew an Ariel Burstein, Pricing-to-Market, Trae Costs, an International Relative Prices, American Economic Review, December 2008, 98 5), Broa, Christian an Davi E. Weinstein, Globalization an the Gains From Variety, The Quarterly Journal of Economics, 2006, 121 2), Chaney, Thomas, Distorte Gravity: The Intensive an Extensive Margins of International Trae, American Economic Review, September 2008, 98 4), Dekle, Robert, Jonathan Eaton, an Samuel Kortum, Unbalance Trae, American Economic Review, May 2007, 97 2), Eaton, Jonathan an Samuel Kortum, Technology in the Global Economy: A Framework for Quantitative Analysis, University of Chicago, unpublishe manuscript, Helpman, Elhanan, Marc J. Melitz, an Stephen R. Yeaple, Export versus FDI with Heterogeneous Firms, American Economic Review, March 2004, 94 1), McKenzie, Lionel W., Matrices with Dominant Diagonal in Economic Theory, in K. J. Arrow, S. Karlin, an P. Suppes, es., Mathematical Methos in the Social Sciences, 1959: Proceeings of the first Stanfor symposium, Palo Alto: Stanfor University Press, 1960, chapter 2, p