NBER WORKING PAPER SERIES FIRM HETEROGENEITY AND AGGREGATE WELFARE. Marc J. Melitz Stephen J. Redding

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1 NBER WORKING PPER SERIES FIRM HEEROGENEIY ND GGREGE WELFRE Marc J. Melitz Stephen J. Reing Working Paper NIONL BUREU OF ECONOMIC RESERCH 050 Massachusetts venue Cambrige, M 0238 March 203 We are grateful to Harvar an Princeton Universities for research support. We woul like to thank Pol ntras, Costas rkolakis, riel Burstein, rnau Costinot, Gene Grossman, Elhanan Helpman, nres Roriguez-Clare, Esteban Rossi-Hansberg an Jon Vogel for helpful comments. We are also grateful to Davi Krisztian Nagy for research assistance. Responsibility for any results, opinions an errors is the authors' alone. he views epresse herein are those of the authors an o not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulate for iscussion an comment purposes. hey have not been peerreviewe or been subject to the review by the NBER Boar of Directors that accompanies official NBER publications. 203 by Marc J. Melitz an Stephen J. Reing. ll rights reserve. Short sections of tet, not to ecee two paragraphs, may be quote without eplicit permission provie that full creit, incluing notice, is given to the source.

2 Firm Heterogeneity an ggregate Welfare Marc J. Melitz an Stephen J. Reing NBER Working Paper No. 899 March 203 JEL No. F2,F5 BSRC We eamine how firm heterogeneity influences aggregate welfare through enogenous firm selection. We consier a homogeneous firm moel that is a special case of a heterogeneous firm moel with a egenerate prouctivity istribution. Keeping all structural parameters besies the prouctivity istribution the same, we show that the two moels have ifferent aggregate welfare implications, with larger welfare gains from reuctions in trae costs in the heterogenous firm moel. Calibrating parameters to key U.S. aggregate an firm statistics, we fin these ifferences in aggregate welfare to be quantitatively important up to a few percentage points of GDP). Uner the assumption of a Pareto prouctivity istribution, the two moels can be calibrate to the same observe trae share, trae elasticity with respect to variable trae costs, an hence welfare gains from trae as shown by rkolakis, Costinot an Roriguez-Clare, 202); but this requires assuming ifferent elasticities of substitution between varieties an ifferent fie an variable trae costs across the two moels. Marc J. Melitz Department of Economics Harvar University 25 Littauer Center Cambrige, M 0238 an NBER mmelitz@harvar.eu Stephen J. Reing Department of Economics an Woorow Wilson School Princeton University Fisher Hall Princeton, NJ an NBER reings@princeton.eu n online appeni is available at:

3 Introuction Over the last ecae, new theories of heterogeneous firms in ifferentiate prouct markets have been evelope to account for features of isaggregate trae ata. aking stock, rkolakis, Costinot an Roriguez-Clare 202) ask whether these new insights for micro ata have altere our unerstaning of the aggregate welfare gains from trae. hey show that there eists a class of heterogeneous an homogeneous firm moels in which a country s omestic trae share is a sufficient statistic for the aggregate welfare gains from trae. hus, if the ifferent moels generate the same omestic trae share, then they also eliver the same welfare gains from that trae. Base on this result, they summarize the contribution of new theories of heterogeneous firms to our unerstaning of the aggregate welfare implications of trae as So far, not much. In this paper, we compare a heterogeneous firm moel to a homogeneous firm moel that is a special case with a egenerate prouctivity istribution. ll other structural parameters are assume to be the same in the two moels. he heterogeneous firm moel features an aitional ajustment margin that is absent from the homogeneous firm moel, namely the enogenous changes in aggregate prouctivity that result from the entry an eit ecisions of heterogeneous firms. s a result, the two moels have ifferent aggregate welfare implications. Calibrating to an initial autarky equilibrium or to an initial open economy equilibrium, we show that welfare is the same in the two moels for the calibrate value of trae costs but is strictly higher in the heterogeneous firm moel than in the homogeneous firm moel for all other values of trae costs. he intuition for our results involves reveale preference arguments of the kin commonly use in international trae. We start from initial equilibria in the heterogeneous an homogeneous firm moels that feature ientical aggregate statistics incluing, crucially, welfare). In the homogeneous firm moel, aggregate prouctivity is eogenous, an hence remains unchange following changes in trae costs. In contrast, in the heterogeneous firm moel, aggregate prouctivity respons to changes in trae costs, associate with the enogenous ajustments in the prouctivity cutoffs for the omestic an eport market. However, the open economy equilibrium in the heterogeneous firm moel is efficient: a welfare-maimizing social planner face with the same prouction an entry technologies woul choose the same allocation as the market equilibrium. Since the social planner chooses a new allocation following the change in trae costs that has ifferent aggregate prouctivity from the initial allocation, this new allocation must yiel at least as high an in general higher) welfare than another feasible allocation in which aggregate prouctivity is hel constant. But this constant prouctivity allocation is ientical to the new equilibrium in the homogeneous firm moel following the change in trae costs. herefore the new equilibrium in the heterogeneous firm moel must yiel at least as high an in general higher) welfare than the new equilibrium in the homogeneous firm moel. he aitional ajustment margin of heterogeneous firms entry an eit ecisions epans the prouction set an enables higher welfare to be achieve in the heterogeneous firm moel than in the homogeneous firm moel following the change in trae costs. In the homogeneous firm moel, 2

4 either all firms eport or no firm eports. In contrast, in the heterogeneous firm moel, there is the possibility of reallocating resources from low prouctivity firms that only serve the omestic market to higher prouctivity firms that eport. herefore the level of welfare that the social planner can achieve in a moel with this etra ajustment margin must be at least as high an in general higher) than the level of welfare that can be achieve in a moel without it. fter eveloping these results, we iscuss their relationship with rkolakis, Costinot an Roriguez- Clare 202) s result that a country s omestic trae share is a sufficient statistic for the welfare gains from trae in the special case of a Pareto prouctivity istribution. he two sets of results are consistent with one another because they reflect funamentally ifferent approaches to comparing moels. Our approach, which we refer to as the micro approach, is to compare moels that iffer in their prouctivity istribution but retain the same values for all other structural parameters incluing trae costs. In contrast, the approach of rkolakis, Costinot an Roriguez-Clare 202), which we refer to as the macro approach, is to compare moels that are calibrate to have the same reuce-form elasticity of trae with respect to trae costs an the same enogenous omestic trae share. he macro approach has some avantages: it ensures that both moels match key features of international trae ata an compares the two moels at empirically-observe moments. However, calibrating the two moels to the same reuce-form trae elasticity involves assuming ifferent structural eman parameters ifferent elasticities of substitution between varieties). Furthermore, calibrating the two moels to the same enogenous omestic trae share involves assuming ifferent trae costs ifferent values of both fie an variable trae costs). herefore, the macro approach changes the egree of firm heterogeneity, the ability of consumers to substitute between varieties an the value of trae costs between the two moels. In contrast, our micro approach only changes the egree of firm heterogeneity between moels, which enables us to isolate the effect of the egree of firm heterogeneity on aggregate welfare. he macro approach s assumption that there is a single empirical moment summarizing the elasticity of trae with respect to trae costs is also quite restrictive. Even uner the assumption of a Pareto prouctivity istribution, the heterogeneous firm moel has ifferent elasticities of trae with respect to trae costs epening on whether these trae costs are variable or fie. More broaly for general continuous prouctivity istributions incluing Pareto istributions that are truncate from above), the elasticity of trae with respect to either variable or fie trae costs is an enogenous variable. In this general case, we show how this elasticity varies with the magnitue of trae costs an the relative importance of fie versus variable trae costs. Our micro approach can be applie to this more general case where the trae elasticity respons to the change in trae costs. Finally, the fact that calibrating to the same omestic trae share involves assuming ifferent trae costs in the two moels has important implications for the interpretation of the macro approach. In some cases, it may be precisely the consequences of a given level of trae costs for trae an welfare that is of ultimate interest. he micro approach highlights that the answer to this question epens on the egree of firm heterogeneity. 3

5 he remainer of the paper is organize as follows. In Sections 2 an 3, we begin by contrasting the close an open economy equilibria of moels with an without firm heterogeneity. Given the same structural parameters an aggregate prouctivity, Section 2 shows that the heterogeneous an homogeneous firm moels have the same close economy welfare. Section 3 shows that the welfare gains from trae are higher in the heterogeneous firm moel than in the homogeneous firm moel. Section 4 provies further economic intuition for our results by showing that the market equilibrium in the heterogeneous firm moel is efficient an eveloping our reveale preference argument. Section 5 eamines the relationship between welfare an a country s trae share with itself. Section 6 shows that our results apply for a comparison of two open economy equilibria with ifferent values of trae costs. Section 7 consiers the special case of a Pareto prouctivity istribution. Section 8 shows that the ifferences in the aggregate implications of heterogeneous an homogeneous firm moels are quantitatively relevant. Section 9 conclues. 2 Close Economy We compare the canonical heterogeneous an homogeneous firm moels of Melitz 2003) an Krugman 980). he homogeneous firm moel is a special case of the heterogeneous firm moel with a egenerate prouctivity istribution. 2. Heterogeneous Firm Moel he specification of preferences, prouction an entry is the same as Melitz 2003). 2 here is a continuum of firms that are heterogeneous in terms of their prouctivity ϕ 0, ), which is rawn from a common istribution g ϕ) after incurring a sunk entry cost of units of labor. Labor is the sole factor of prouction. Prouction involves a fie prouction cost an a constant marginal cost that epens on firm prouctivity, so that l ϕ) = f + q ϕ) /ϕ units of labor are require to supply q ϕ) units of output. Consumers have constant elasticity of substitution CES) preferences efine over the ifferentiate varieties supplie by firms. he equilibrium revenue for a firm with prouctivity ϕ is then: rϕ) = RP σ pϕ) σ, where R is aggregate revenue; P is the aggregate CES price ine; an pϕ) is the price chosen by a firm with prouctivity ϕ. Profit maimization implies that equilibrium prices are a constant mark-up over marginal cost: pϕ) = σ w σ ϕ. Hence profits are a constant fraction of revenue minus the fie prouction cost: πϕ) = rϕ) σ wf. web-base technical appeni contains the erivations of all epressions in the paper. 2 Following most of the subsequent international trae literature, incluing rkolakis, Costinot an Roriguez-Clare 202), we consier a static version of Melitz 2003) in which there is zero probability of firm eath. 4

6 Fie prouction costs imply a prouctivity cutoff below which firms eit ϕ ) efine by the following zero-profit conition: where the superscript enotes autarky. ) σ σ r ϕ ) = R σ P ϕ w σ = σwf, ) he equilibrium value of this zero-profit prouctivity is uniquely etermine by the free entry conition that requires that the probability of successful entry times average profits conitional on successful entry is equal the sunk entry cost: [ G ϕ )] π = wfe. Using the epression for profits above, this free entry conition can be epresse as: ϕ f J ϕ ) = fe, 2) J [ ϕ ) ) ϕ σ = ϕ ] Gϕ) = [ [ Gϕ )] ϕ ], 3) where J ϕ ) is a monotonically ecreasing function an is a weighte average of firm prouctivities: = [ ϕ ϕ σ ] σ G ϕ) G ) ϕ. 4) he mass ontrants M e ) equals the mass of proucing firms M) ivie by the probability of successful entry Gϕ )): M e = M Gϕ ) = R σw [ + [ G )] ], ϕ 5) f where the secon equation uses the relationship between the mass of firms, aggregate revenue an average revenue M = R/ r ), the relationship between average revenue an average profits r = σ π + wf )) an free entry [ G ϕ )] π = wfe ). Combining the mass ontrants 5) an the free entry conition 2), aggregate revenue equals total labor payments R = wl), where we choose labor as the numeraire an hence w =. Using this equality of aggregate revenue an total labor payments in the zero-profit conition ), welfare can be written solely in terms of the zero-profit prouctivity ϕ ) an parameters: W Het = w ) L P = σ σ σf σ ϕ. 6) herefore the zero-profit prouctivity ϕ ) is a sufficient statistic for welfare. 2.2 Homogeneous Firm Moel We construct a homogeneous firm moel that replicates the same aggregate equilibrium as the heterogeneous firm equilibrium that we just escribe. Firms pay a sunk entry cost of units of labor an raw a prouctivity oither zero or ϕ with eogenous probabilities Ḡ an Ḡ) respectively. 5

7 Fie prouction costs imply that only firms rawing a prouctivity of ϕ fin it profitable to prouce. herefore proucing firms are homogeneous an there is a egenerate prouctivity istribution conitional on prouction at ϕ. Note that the only ifference between the homogeneous an heterogeneous firm moels is the prouctivity istribution ontrants: the istribution G.) is replace by the egenerate istribution with parameters Ḡ an ϕ. ll other parameters are the same. his homogeneous firm moel is isomorphic to Krugman 980), in which the representative firm s prouctivity is set equal to ϕ an the fie prouction cost is scale to incorporate the epecte value ontry costs F = f + / [ Ḡ] ). hese values for the representative firm s prouctivity an the fie prouction cost are eogenous an hel constant in Krugman 980). o simplify the eposition, we aopt this Krugman 980) interpretation. he representative firm s prouction technology is: l = q ϕ + F. 7) Consumers again have constant elasticity of substitution CES) preferences efine over the ifferentiate varieties supplie by firms. Profit maimization implies that equilibrium prices are a constant markup over marginal cost: p = σ w. σ ϕ Profit maimization an free entry imply that equilibrium output an employment for the representative variety are proportional to the fie prouction cost: q = ϕ F σ ), l = σf. Using the common employment for each variety, the mass of firms can be etermine from the labor market clearing conition: M = L σf. 8) Using the equilibrium pricing rule an the mass of firms, the CES price ine is: ) σ P σ w σ = M, 9) σ ϕ where we again choose labor as the numeraire an hence w =. Using the price ine 9), the mass of firms 8), an our choice of numeraire, welfare can be written in terms of prouctivity an other parameters: 2.3 ggregate Equilibrium Equivalence W Hom = w ) L P = σ σ ϕ. 0) σf σ We now pick the parameters Ḡ an ϕ of the egenerate prouctivity istribution with homogeneous firms such that the autarky equilibrium is isomorphic to the heterogeneous firm equilibrium, in the following sense: 6

8 Proposition Consier a homogeneous firm moel that is a special case of the heterogeneous firm moel with an eogenous probability of successful entry [ Ḡ] = [ Gϕ ) ] an an eogenous egenerate istribution of prouctivity conitional on successful entry ϕ =. Given the same value for all remaining parameters {f,, L, σ}, all aggregate variables welfare, wage, price ine, mass of firms, an aggregate revenue) are the same in the close economy equilibria of the two moels. Proof. Combining F = f + / [ Gϕ )] with the free entry conition 2) in the heterogeneous firm moel, we obtain: F = f ϕ. ) Substituting this result into close economy welfare in the homogeneous firm moel 0), we obtain the same welfare in the two moels: W Hom = w ) L P = σ σ σf σ ϕ = W Het. Equal wages follow from our choice of numeraire w = ). Equal welfare an equal wages in turn imply equal price inices. Equal masses of firms follow from 8), 5) an ). Equal aggregate revenue follows from R = wl = L in both moels. 3 Open Economy We consier the canonical case of trae between two symmetric countries, as eamine in Krugman 980) an Melitz 2003). We assume that the heterogeneous an homogeneous firm moels feature the same trae costs, so that there is a fie eporting cost of f units of labor an an iceberg variable trae cost, where τ > units of a variety must be shippe from one country in orer for one unit to arrive in the other country. We compare the effect of moving from the close economy to the open economy on welfare in the two moels, keeping all structural parameters other than the prouctivity istribution the same in the two moels. 3. Heterogeneous Firm Moel Equilibrium firm revenues in the omestic an eport markets are: r ϕ) = RP σ p ϕ) σ, r ϕ) = τ σ r ϕ), where the subscript inicates the omestic market an the subscript inicates the eport market. Profit maimization implies that equilibrium prices are again a constant mark-up over marginal costs, with eport prices a constant multiple of omestic prices ue to the variable costs of trae: p ϕ) = σ w σ ϕ, p ϕ) = τp ϕ), 2) 7

9 his equilibrium pricing rule implies that profits in each market are a constant proportion of revenues minus the fie costs: π ϕ) = r ϕ) σ wf, π ϕ) = r ϕ) σ wf. Here, we assume that fie eporting costs are incurre in the source country an we apportion the fie prouction cost to the omestic market. he prouctivity cutoffs for serving the omestic market ϕ ) an eport market ϕ ) are efine by the following zero-profit conitions: ) σ σ r ϕ ) = R σ P ϕ w σ = σwf. 3) ) σ σ r ϕ ) = R σ P ϕ τw) σ = σwf, 4) where the superscript inicates the open economy equilibrium. ogether these two zero-profit conitions imply that the eport cutoff is a constant multiple of the omestic cutoff that epens on the fie an variable costs of trae: ϕ = τ f f ) σ ϕ. 5) For sufficiently high fie an variable trae costs τ f /f ) σ > ), only the most prouctive firms eport, consistent with an etensive empirical literature see for eample the review in Bernar, Jensen, Reing an Schott 2007). he free entry conition again equates the epecte value ontry to the sunk entry cost, [ G ϕ )] π = wfe, an can be re-written as follows: f J ϕ ) + f J ϕ ) = fe, 6) where J ) is efine in 3) an we can efine weighte average prouctivity in the eport market ) in an analogous way to weighte average prouctivity in the omestic market ) in 4). Using the relationship between the prouctivity cutoffs 5), an noting that J ) is a ecreasing function, the free entry conition 6) etermines a unique equilibrium value of the omestic cutoff ϕ ), which in turn etermines the eport cutoff ϕ ). Furthermore, since J ) is a ecreasing function, the omestic cutoff in the open economy is strictly greater than the omestic cutoff in the close economy ϕ > ϕ ) for positive values of fie eporting costs. s in the close economy, the mass ontrants M e ) equals the mass of firms M) ivie by the probability of successful entry G ϕ ) ): M e = M Gϕ ) = R σw [ + [ G )] ], ϕ f + [ G ϕ 7) )] f an aggregate revenue equals total labor payments R = wl). 8

10 Using the equilibrium pricing rule an the mass of firms, the CES price ine in the open economy can be written as: P σ = M [ ) σ + χτ σ ] ) σ σ w σ, 8) σ where χ = [ G ϕ )] / [ G ϕ )] is the proportion oporting firms. We choose labor in one country as the numeraire. With symmetric countries, this implies a common unit wage in each country w = ). Rearranging the price ine 8), an using the mass of firms 7) an our choice of numeraire, welfare can be epresse in terms of prouctivity an parameters: W Het = w P = [ σ L ) σ σ σ ] Gϕ ) + f + χf σ [ + χτ σ ] σ. 9) In an open economy equilibrium with selection into eport markets ϕ > ϕ ), the zero-profit conition for the omestic market 3) implies that open economy welfare can be written equivalently in terms of the omestic prouctivity cutoff an parameters: W Het = w ) L P = σ σ σf σ ϕ. 20) Comparing 6) an 20), an noting that the omestic cutoff is higher in the open economy than in the close economy ϕ > ϕ ), there are necessarily welfare gains from trae. In contrast, in an open economy equilibrium in which all firms eport, the omestic an eport prouctivity cutoffs are equal to one another ϕ = ϕ ), an are etermine by the requirement that the sum of variable profits in the omestic an eport markets is equal to the sum of fie prouction an eporting costs. Using this zero-profit conition, open economy welfare also can be written in terms of the omestic prouctivity cutoff an parameters: W Het = w + τ σ ) ) P = L σ σ σ f + f ) σ ϕ. 2) Comparing 6) an 2), an noting that f /f τ σ in an open economy equilibrium in which all firms eport, there are again necessarily welfare gains from trae. 3.2 Homogeneous Firm Moel In the homogeneous firm moel, the probability of successful entry an prouctivity conitional on successful entry are eogenous an remain unchange an equal to [ Ḡ] an ϕ respectively. For sufficiently high fie an variable trae costs τ σ f /F > ), the representative firm oes not fin it profitable to eport. In this case, welfare in the open economy equilibrium is necessarily higher in the heterogeneous firm moel than in the homogeneous firm moel, because the two moels have the same 9

11 close economy welfare, there are welfare gains from trae, an trae only occurs in the heterogeneous firm moel. In contrast, for sufficiently low fie an variable trae costs τ σ f /F < ), the representative firm fins it profitable to eport. In this case, there is positive trae in both moels, an we now compare their relative welfare in such an open economy equilibrium. Profit maimization again implies that equilibrium prices are a constant mark-up over marginal costs, with eport prices a constant multiple of omestic prices ue to the variable costs of trae: p = σ w, p = τp. 22) σ ϕ ogether profit maimization an free entry imply that equilibrium output an employment for the representative variety are proportional to fie costs: q = ϕ F + f ) σ ), l = σ F + f ). herefore both output an employment rise for the representative firm following the opening of trae to cover the aitional fie costs oporting. From the labor market clearing conition, this rise in employment for the representative firm implies a fall in the mass of omestically-prouce varieties: M = L σ F + f ). 23) Using the equilibrium pricing rule an the mass of firms, the CES price ine in the open economy is: P σ = [ + τ σ] M where we again choose labor as the numeraire an hence w =. ) σ w σ, 24) σ ϕ Rearranging the price ine 24), an using the mass of firms 23) an our choice of numeraire, welfare can be again epresse in terms of prouctivity an parameters: [ W Hom = w + τ σ ] ) P = L σ σ ϕ. 25) σ F + f ) σ 3.3 Relative Welfare In the homogeneous firm moel, aggregate prouctivity is eogenous an hence constant by assumption. In contrast, in the heterogeneous firm moel, aggregate prouctivity is etermine by the enogenous entry an eit ecisions of heterogeneous firms. his provies a new ajustment margin through which the economy can respon to the opening of trae. he presence of this new ajustment margin implies that the relative change in welfare following the opening of trae is strictly larger in the heterogeneous firm moel than in the homogeneous firm moel. Since the homogeneous firm moel is a special case of the heterogeneous firm moel, this comparison of the welfare gains from trae across 0

12 the two moels is equivalent to a comparative static within the heterogeneous firm moel on the prouctivity istribution from a non-egenerate to a egenerate istribution). his comparative static interpretation requires that we hol all other parameters equal when comparing the two moels same f,, f, τ, L, σ). We maintain this assumption throughout the paper whenever we compare the heterogeneous an homogeneous firm moels. We pick the parameters of the egenerate prouctivity istribution uner homogeneous firms Ḡ an ϕ ) such that the autarky equilibrium is isomorphic to the heterogeneous firm one as outline in Proposition ): Ḡ = Gϕ ) an ϕ = Proposition 2 he proportional welfare gains from trae are strictly larger in the heterogeneous firm moel than in the homogeneous firm moel W Het /W Het > W Hom /W Hom ), ecept in the special case with no fie eporting cost. In this special case, the proportional welfare gains from trae are the same in the two moels. Proof. See the ppeni. In the special case with no fie eporting cost, the omestic prouctivity cutoff oes not respon to the opening of trae in the heterogeneous firm moel. s a result, the aitional ajustment margin provie by heterogeneous firms entry an eit ecisions is inoperable, an the welfare gains from trae are the same in the two moels. But this special case is uninteresting, because firm prouctivity ispersion plays no role in the heterogeneous firm moel the eit threshol an average prouctivity are the same in the close an open economies). Furthermore, this special case stans at os with an etensive boy ompirical evience that only some firms eport, eporters are larger an more prouctive than non-eporters, an there are substantial fie eporting costs. 3 Since the proportional welfare gains from trae are strictly lower in the homogeneous firm moel than in the heterogeneous firm moel for positive fie eporting costs, an since open economy welfare in the homogeneous firm moel is monotonically ecreasing in trae costs, we also obtain the following result. Proposition 3 o achieve the same proportional welfare gains from trae requires strictly lower trae costs either lower f an/or lower τ) in the homogeneous firm moel than in the heterogeneous firm moel, ecept in the special case with no fie eporting cost. Proof. he proposition follows immeiately from W Het /W Het > W Hom /W Hom from W Hom f < 0 an W Hom τ < 0 in 25). in Proposition 2 an lthough we chose the prouctivity of the representative firm ) to ensure the same close economy welfare in both moels, the ratio of open to close economy welfare in the homogeneous firm moel W Hom /W Hom is inepenent of the representative firm s prouctivity from 0) an 25)). It 3 For reviews of the etensive empirical literatures on firm eport market participation, see Bernar, Jensen, Reing an Schott 2007) an Melitz an Reing 202). For evience of substantial fie eporting costs, see Roberts an ybout 997) an Das, Roberts an ybout 2007).

13 follows that both of the above propositions hol for any value of the representative firm s prouctivity. Both propositions also hol for general continuous prouctivity istributions. 4 Reveale Preference he theoretical results throughout the paper are prove using the free entry conition in the market equilibrium of the heterogeneous firm moel. But to provie further economic intuition for these results, we consier the problem of a social planner choosing the prouctivity cutoffs an the mass of entrants to maimize welfare in the heterogeneous firm moel. We begin with the planner s problem in the close economy, in which welfare in the homogeneous an heterogeneous firm moels is the same. We net consier the planner s problem in the open economy. he planner is assume to maimize worl welfare. of the representative consumer in each country. 4 With symmetric countries, this is equivalent to maimizing the welfare We show that the planner s choices in the close an open economies coincie with the market allocations, an hence the market allocations in the heterogeneous firm moel are efficient. 5 We also show that the social planner in general chooses to ajust the prouctivity cutoffs following the opening of trae, even though it is feasible to leave them unchange an replicate the open economy equilibrium of the homogeneous firm moel. herefore, by reveale preference, open economy welfare must be at least as high in the heterogeneous firm moel as in the homogeneous firm moel, an we show that it is in general higher. 4. Close Economy In the close economy of the heterogeneous firm moel, the real consumption ine an aggregate labor constraint for the social planner can be written as: Q = {[ Gϕ )] M e } σ/σ ) q, L = [ Gϕ )] M e [ q + f + ] Gϕ ), where q = ) q is the output of a firm with a prouctivity equal to omestic weighte average prouctivity ; an we assume that the social planner faces the same prouctivity istribution Gϕ) an entry cost per firm as in the market allocation. Using the aggregate labor constraint in the real consumption ine, the social planner chooses the omestic cutoff ϕ firm with weighte average prouctivity q an hence the prouctivity range of proucing firms) an the output of a an hence the mass ontrants) to solve the following 4 o highlight the efficiency properties of the market equilibrium, we assume a worl planner, which abstracts from the incentives of national planners to manipulate the terms of trae between countries. 5 For an analysis of how the efficiency of the monopolistically competitive equilibrium epens on the etent to which the elasticity of substitution between varieties is constant or variable, see Diit an Stiglitz 977) for homogeneous firm moels an Dhingra an Morrow 202) for heterogeneous firm moels. 2

14 unconstraine maimization problem: 6 ma ϕ, q { L σ/σ ) [ q + f + ] σ/σ ) Gϕ ) q }. 26) he trae-off face by the social planner is as follows. On the one han, a lower prouctivity cutoff reuces epecte entry costs conitional on successful entry, which releases more labor for prouction. On the other han, the lower prouctivity cutoff involves prouction by lower prouctivity firms, which reuces epecte output conitional on successful entry. From the first-orer conitions for q ϕ, we obtain: [ [ Gϕ )] ϕ ] f =, 27) which correspons to the free entry conition in the market economy 2). herefore, the social planner chooses the same prouctivity cutoff ϕ an the same values of all other enogenous variables) as in the market equilibrium of the heterogeneous firm moel. his confirms that the market equilibrium is efficient. In the heterogeneous firm moel, changes in fie costs an other parameters inuce the social planner to change both the prouctivity cutoff ϕ an the output of a firm with weighte average prouctivity q an hence the mass ontrants M e). In contrast, in the homogeneous firm moel, prouctivity is constant by assumption, which implies that changes in fie costs an other parameters only inuce changes in the mass of proucing firms an output per firm. 4.2 Open Economy In the open economy of the heterogeneous firm moel, the real consumption ine an the aggregate labor constraint for the social planner can be written as: Q = M σ/σ ) t q t, L = M t [ q t t + + χ f + χf + )] Gϕ ), where M t is the mass of firms serving each market an q t q t ) is the omestic output of a firm with a prouctivity equal to a weighte average across all non-eporting an eporting firms t. 7 he mass of firms serving each market an weighte average prouctivity are efine as follows: an M t = [ + χ] M = [ + χ] [ G ϕ )] Me, 28) { [ ) t = σ + χ + χτ σ ] } /σ ). 29) Using the aggregate labor constraint in the real consumption ine, the social planner chooses the omestic cutoff ϕ an hence the prouctivity range for proucing firms), the eport cutoff ϕ 6 In the web appeni, we also erive the same results from an equivalent representation of the planner s problem as a Lagrangian. 7 Recall that χ = [ G )] [ )] ϕ / G ϕ is the proportion oporting firms. 3

15 an hence the proportion oporting firms) an the output of a firm with the weighte average prouctivity an hence the mass of firms) as the solution to the following unconstraine maimization problem: ma ϕ,ϕ, q t { L σ/σ ) [ q t t + )] σ/σ ) } f + χf + + χ Gϕ ) q t. 30) he trae-offs face by the social planner are as follows. lower omestic cutoff again reuces epecte entry costs conitional on successful entry an thereby releases more labor for prouction. But this lower cutoff involves prouction lower prouctivity firms, which reuces epecte output conitional on successful entry. lower eport cutoff for a given omestic cutoff increases the proportion oporting firms. his uses more labor to cover fie eporting costs an reuces epecte output conitional on eporting, but increases the fraction of foreign varieties available to omestic consumers. he first-orer conition for q t is: [ Gϕ ) ] [ + χ) q t σ ) t f χf ] =. 3) It equates the epecte profits from entry to the sunk entry cost, since +χ) q t σ ) t profits conitional on successful entry. he first-orer conition for ϕ yiels: is epecte variable ϕ t q t σ t = f. 32) It equates the omestic variable profits of the least prouctive firm with the fie prouction cost. he first-orer conition for ϕ yiels: ) ϕ σ τ σ σ t q t t = f. 33) It equates the eport variable profits of the least prouctive eporter with the fie eporting costs. Combining these three first-orer conitions, we obtain the same relationship between the omestic an eport cutoffs 5) an free entry conition 6) as in the open economy market equilibrium. s a result, the social planner chooses the same prouctivity cutoffs ϕ an ϕ an the same value of all other enogenous variables) as in the open economy market equilibrium of the heterogeneous firm moel. Hence the open economy market equilibrium of the heterogeneous firm moel is efficient. 4.3 Open Versus Close Economy Recall that the heterogeneous an homogeneous firm moels generate the same values of all aggregate variables incluing welfare) in the close economy. Furthermore, it is technologically feasible for the social planner to choose the same omestic prouctivity cutoff in the open economy as in the close economy ϕ = ϕ ) an to let all firms eport ϕ = ϕ ). In this hypothetical allocation, the heterogeneous an homogeneous firm moels woul also generate the same values of all aggregate variables incluing welfare) in the open economy. 4

16 However, comparing the close an open economy free entry conitions 2) an 6), the social planner in general chooses a ifferent omestic prouctivity cutoff in the open economy than in the close economy ϕ ϕ ) an in general restricts eporting to a proper subset of more prouctive firms ϕ < ϕ < ). Since the social planner chooses ifferent prouctivity cutoffs in the open economy when it is technological feasible to choose the same prouctivity cutoffs as in the close economy, reveale preference implies that these ifferent prouctivity cutoffs must yiel welfare levels at least as high as a hypothetical allocation with the same prouctivity cutoffs. Furthermore, the social planner s objective 30) is globally concave in {ϕ, ϕ, q t }, which implies that the ifferent prouctivity cutoffs must yiel strictly higher welfare than the hypothetical allocation with the same prouctivity cutoffs. itionally, the social planner s choice correspons to the open economy equilibrium of the heterogeneous firm moel, while the hypothetical allocation correspons to the open economy equilibrium of the homogeneous firm moel. herefore it follows that open economy welfare is strictly higher in the heterogeneous firm moel than in the homogeneous firm moel whenever the prouctivity cutoffs iffer in the open an close economies of the heterogeneous firm moel. 8 Comparing the social planner s objectives in the close an open economies 26) an 30), she faces ifferent combinations of fie versus variable costs in the close versus open economy. he social planner s optimal response to these ifferent cost combinations is to ajust the prouctivity range for both proucing an eporting firms as well as average output per firm). hus the greater welfare gains in the heterogeneous firm moel reflect the presence of an aitional ajustment margin the firm prouctivity ranges) relative to the homogeneous firm moel. In the heterogeneous firm moel, the social planner can allocate low-prouctivity firms to serve only the omestic market an reallocate resources to higher-prouctivity eporting firms. herefore the welfare that the social planner can achieve in a moel with this aitional ajustment margin must be at least as high an in general higher) than in a moel without it. While we have focuse on opening the close economy to trae, an analogous analysis can be unertaken for the reverse comparative static of closing the open economy. Choosing the egenerate prouctivity istribution in the homogeneous firm moel such that the two moels to have the same welfare in an initial open economy equilibrium, the welfare costs of closing the open economy are smaller in the heterogeneous firm moel than in the homogeneous firm. gain the welfare that the social planner can achieve in the heterogeneous firm moel with its aitional ajustment margin must be at least as high an in general higher) than in the homogeneous moel. s a result, whether we consier increases or reuctions in trae costs, welfare in the two moels is the same for the calibrate value of trae costs, but is strictly higher in the heterogeneous firm moel than in the homogeneous firm moel for all other values of trae costs. 8 While to make our argument as clearly as possible we focus on symmetric countries, this reveale preference argument applies more generally for asymmetric countries. 5

17 5 rae Shares an Welfare We now eamine the relationship between the heterogeneous an homogeneous firm moels preictions for welfare an their preictions for omestic trae shares. In an open economy equilibrium of the heterogeneous firm moel, the omestic trae share is: λ Het = p ) σ p ) σ + χτ σ p ) σ = + τ σ Λ, 34) where Λ = ϕ ϕ ϕσ G ϕ) ϕ σ G ϕ). In contrast, in an open economy equilibrium of the homogeneous firm moel in which the representative firm eports, the omestic trae share is: λ Hom = p ) σ + τ σ ) p ) σ =. 35) + τ σ Comparing the omestic trae shares in the heterogeneous an homogeneous firm moels 34) an 35) respectively) for the same trae costs {f, τ}, we obtain the following result: Proposition 4 Whenever the representative firm eports in the homogeneous firm moel an there is selection into eport markets in the heterogeneous firm moel 0 < τ f /F ) /σ ) < < τ f /f ) /σ ) ), the omestic trae share λ is strictly higher in the heterogeneous firm moel than in the homogeneous firm moel. If all firms eport in both moels 0 τ f /F ) /σ ) < τ f /f ) /σ ) ), the omestic trae shares are the same. Proof. For 0 < τ f /F ) /σ ) < < τ f /f ) /σ ), we have ϕ > ϕ, which implies 0 < Λ < an hence λ Het > λ Hom in the omestic trae shares 34) an 35). For 0 τ f /F ) /σ ) < τ f /f ) /σ ), we have ϕ = ϕ, Λ = an λ Het = λ Hom. Proposition 4 implies a non-monotonic relationship between relative omestic trae shares in the two moels an trae costs. We illustrate this non-monotonic relationship for the special case of a Pareto istribution in our simulations of the moel in Section 8 below see in particular Panel D of Figures an 2). For sufficiently high trae costs, there is trae in the heterogeneous firm moel, but the representative firm oes not fin it profitable to eport in the homogeneous firm moel. s a result, the omestic trae share is lower in the heterogeneous firm moel than in the homogeneous firm moel. s trae costs fall, the representative firm starts to eport in homogeneous firm moel, at which point the omestic trae share in the homogeneous firm moel falls iscretely from one to a value below that in the heterogeneous firm moel. For all values of trae costs with selection into eport markets in the heterogeneous firm moel an positive trae in the homogeneous firm moel, the omestic trae share is higher in the heterogeneous firm moel than in the homogeneous firm moel. s trae costs fall further, ϕ ϕ an the omestic trae share in the heterogeneous firm 6

18 moel converges towars the omestic trae share in the homogeneous firm moel. Once trae costs reach the point at which all firms eport in the heterogeneous firm moel, the omestic trae share is the same in the two moels. Proposition 4 highlights the comple relationship between omestic trae shares an welfare in the heterogeneous an homogeneous firm moels. he omestic trae share is an enogenous variable an in general iffers between the heterogeneous an homogeneous firm moels given the same structural parameters an trae costs. Furthermore, the omestic trae share can be strictly higher in the heterogeneous firm moel than in the homogeneous firm moel even though welfare is strictly higher in the heterogeneous firm moel Propositions 2 an 4). Heterogeneous firms enogenous entry an eit ecisions also have implications for the elasticity of trae flows with respect to trae costs. In the homogeneous firm moel, the elasticity of the omestic trae share with respect to variable trae costs epens solely on the elasticity of substitution between varieties an the omestic trae share: λ Hom τ τ λ Hom = σ ) λ), an the elasticity of the omestic trae share with respect to fie eporting costs is equal to zero: λ Hom τ f λ Hom = 0. In contrast, in the heterogeneous firm moel, the elasticities of the omestic trae share with respect to variable trae costs an fie eporting costs are enogenous variables that epen on the entire prouctivity istribution: Λ τ Λ τ λ Het τ [ τ λ = σ ) Λ τ τ λ Het ] λ), λ Het f = Λ f λ), τ λ Het τ λ Het [ ) τ ϕ σ ) = g ϕ ) ϕ σ ] λ Het ϕ ϕσ G ϕ) g ϕ ) ϕ ϕ σ τ, G ϕ) [ ) f ϕ σ ) = g ϕ ) ϕ σ ] λ Het ϕ ϕσ G ϕ) g ϕ ) ϕ ϕ σ f, G ϕ) where ϕ > ϕ an ϕ ϕσ G ϕ) < ϕ ϕ σ G ϕ). herefore the elasticity of the omestic trae share with respect to trae costs varies with the functional form of the prouctivity istribution, the level of trae costs, an the relative importance of fie versus variable trae costs. It follows that there is no single elasticity of trae flows with respect to trae costs in the heterogeneous firm moel with a general continuous prouctivity istribution. Rather empirical measures of this elasticity capture an enogenous variable that epens on the full structure of the moel. 7

19 6 rae Liberalization in the Open Economy We now eamine the aggregate welfare implications of the two moels starting from an open economy equilibrium. In Proposition 4, we showe that whenever the heterogeneous firm moel features eport market selection, the homogeneous firm counterpart moel cannot be calibrate to generate the same initial omestic trae share given the same trae costs {f, τ}. 9 hus, we cannot construct an open economy homogeneous firm moel that features a) the same structural parameters as our heterogeneous firm moel with eport market selection an b) the same initial omestic trae share an welfare. We therefore consier an etension of the homogeneous firm moel to allow for two types of firms: eporting an non-eporting firms. In this etension, firms again pay a sunk entry cost of units of labor before observing their prouctivity. n entering firms raws a prouctivity level ϕ with probability [ Ḡ], an raws a prouctivity level ϕ with probability Ḡ. he remaining firms raw a prouctivity level of zero which occurs with probability ] [Ḡ Ḡ ). We pick the parameters of the etene homogeneous firm moel ϕ, ϕ, Ḡ, Ḡ) such that the open economy equilibrium features the same aggregate variables as in the initial open economy equilibrium with heterogenous firms same welfare, wage, price ine, mass of firms, aggregate revenue, an omestic trae share). his requires equating these parameters with their corresponing averages uner firm heterogeneity: Ḡ = G ϕ ), Ḡ = G ϕ ) ) G ϕ, { ϕ =, ϕ = G ϕ ) G ) ϕ ϕ ϕ ϕ σ G ϕ) } σ since this last term represents the average prouctivity of non-eporters in the heterogeneous firm moel. s with all our previous comparative statics, we keep the values of all the other structural parameters f,, f, τ, L, σ) constant across the two moels. hese assumptions ensure that, in our etene-homogeneous-firm setup, the firms with prouctivity ϕ will fin it profitable to prouce for both the omestic an eport markets, while the firms with prouctivity ϕ will only fin it profitable to prouce for the omestic market leaing to the property that the aggregate statistics line-up across the two moels in the initial open economy equilibrium). hese two moels will nevertheless feature a key ifference in response to a reuction in trae cots from this common initial equilibrium: In the heterogeneous firm moel, the enogenous selection responses to trae costs lea to changes in the average prouctivity oporting an non-eporting firms also inucing a response in the proportion oporting firms)., In contrast, those average prouctivity levels for eporters an non-eporters remain fie in our etene-homogeneous-firm 9 he calibration of the heterogeneous an homogeneous firm moels to achieve the same omestic trae share an welfare uner a Pareto prouctivity istribution as in rkolakis, Costinot an Roriguez Clare 202) involves assuming ifferent fie an variable trae costs in the two moels, as iscusse further in the net section. 8

20 moel. 0 Once again, this enogenous response of firm selection an average prouctivity leas to larger welfare gains from a given reuction in trae costs uner firm heterogeneity: Proposition 5 Starting from an initial open economy equilibrium with the same welfare an omestic trae share, a common reuction in variable an/or fie trae costs generates greater welfare gains in the heterogeneous firm moel than in our etene-homogeneous moel. chieving the same welfare gains in both moels requires a larger reuction in trae costs in the etene homogeneous firm moel than in the heterogeneous firm moel. Proof. See the ppeni. Note that the etene homogeneous firm moel is equivalent to a version of the heterogeneous firm moel in which the omestic an eport prouctivity cutoffs are hel constant at their values in the initial equilibrium. herefore the comparison of the heterogeneous an etene homogeneous firm moels in Proposition 5 is equivalent to the following thought eperiment. Start from an initial open economy equilibrium of the heterogeneous firm moel with symmetric countries an selection into eport markets. Now reuce trae costs an consier two cases. In the first case, let the omestic an eport cutoffs ajust enogenously. In the secon case, hol the threshols constant at their initial values before the reuction in trae costs. s shown in the proof of the proposition, welfare is higher in the first case in which the omestic an eport cutoffs ajust enogenously. tkeson an Burstein 200) consier this thought eperiment an show that enogenous changes in the omestic an eport cutoffs have only secon-orer effects on welfare. Proposition 5 is consistent with this result. s shown in the previous section, the initial equilibrium of the heterogeneous firm moel is efficient. herefore the envelope theorem implies that the changes in the prouctivity cutoffs in the heterogeneous firm moel have only secon-orer effects on welfare 30). But for substantial reuctions in trae costs, these changes in the prouctivity cutoffs can have substantial effects on welfare, because the secon-orer terms can be large. In Section 8, we eamine the quantitative magnitue of these effects using a calibration of the heterogeneous firm moel to key U.S. aggregate an firm statistics. While Proposition 5 focuses on reuctions in trae costs, an analogous analysis can be unertaken for the reverse comparative static of increases in trae costs, an the same arguments apply as alreay iscusse above. Proposition 5 makes clear that the new aggregate welfare implications of the heterogeneous firm moel o not arise from firm heterogeneity per se, since eogenous ifferences between eporters an non-eporters eist in the etene homogeneous firm moel. Instea, they stem from enogenous firm heterogeneity: the enogenous selection of firms into the omestic an eport markets. his introuces a new ajustment margin through which the economy can respon to trae liberalization; a margin that is absent in both the homogeneous an etene-homogeneous firm moels. 0 Unless the reuction in trae costs is so large that the non-eporting firms in the initial equilibrium fin it profitable to start eporting in the new equilibrium. In this case, the prouctivity levels for the two types of firms remain constant, but the proportion oporting firms changes. 9

21 7 Pareto Prouctivity Distribution While our results in the previous sections are evelope for general continuous prouctivity istributions, we now consier the special case of a Pareto prouctivity istribution. Uner this functional form assumption, our heterogeneous firm moel falls within the class consiere by rkolakis, Costinot an Roriguez-Clare 202), where a country s omestic trae share is a sufficient statistic for the welfare gains of trae. ssuming that prouctivity has the following Pareto istribution: g ϕ) = k ϕ min ) k ϕ k+), k > σ, ϕ min > 0, we can solve in close form for the omestic an eport prouctivity cutoffs. Using these solutions, welfare in the heterogeneous firm moel 20) can be epresse solely in terms of a country s omestic trae share an parameters: W Het = w P = λ /k Het L /.σ ) ϕk min f k σ σ σ ) k σ k σ σ k σ ) k. 36) herefore, uner this assumption of a Pareto prouctivity istribution, knowing a country s omestic trae share λ) an the shape parameter of the prouctivity istribution k) is sufficient to etermine the welfare gains from trae in the heterogeneous firm moel: W Het W Het [ = λ Het ] k. 37) With a Pareto prouctivity istribution, the elasticity of the omestic trae share with respect to variable trae costs is: λ Het τ { τ = λ Het k λ Het ) τ f /f ) /σ ) σ ) λ Het ) τ f /f ) /σ ) <, 38) In contrast, in the homogeneous firm moel, the omestic trae share 35) an welfare in the close an open economies 0) an 25) respectively) imply that the welfare gains from trae can be epresse as: W Hom W Hom [ = F λ Hom F + f ) ] σ. 39) In this homogeneous firm moel, knowing a country s omestic trae share λ) an the elasticity of substitution σ) is sufficient to etermine the welfare gains from trae only in the absence of fie eport costs. Otherwise, the ratio of those fie eport costs to the remaining fie costs is also neee to compute those welfare gains. In the homogeneous firm moel, the elasticity of the omestic trae share with respect to variable trae costs remains as specifie in Section 5. s in rkolakis, Costinot an Roriguez-Clare 202), the heterogeneous an homogeneous firm moels can be calibrate to have the same omestic trae share λ ), the same elasticity of trae flows with respect to variable trae costs k Het = σ Hom ), an the same welfare gains from trae if there 20