Web Appendix to Firm Heterogeneity and Aggregate Welfare (Not for Publication)

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1 Web ppeni to Firm Heterogeneity an ggregate Welfare Not for Publication Marc J. Melitz Harvar University, NBER, an CEPR Stephen J. Reing Princeton University, NBER, an CEPR March 6, 203 Introuction his web appeni contains the technical erivations opressions for each section of the paper an aitional supplementary material. In the interests of clarity, an to ensure that the web appeni is self-containe, we reprouce some material from the paper, but also inclue the intermeiate steps for the erivation opressions. he sections of the web appeni closely follow the sections of the paper with the same name. 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 here is a continuum of firms that are heterogeneous in terms of their prouctivity 0,, which is rawn from a fie istribution g after incurring a sunk entry cost of units of labor. Prouction involves a fie prouction cost an a constant marginal cost that epens on firm prouctivity, so that l f + q / units of labor are require to supply q units of output. Consumers have constant elasticity of substitution CES preferences efine over the ifferentiate varieties supplie by firms, so that the equilibrium revenue for a firm with prouctivity is: r RP p, where R is aggregate revenue; P is the aggregate CES price ine; an p is the price chosen by a firm with prouctivity. Profit maimization implies that equilibrium prices are a constant mark-up over marginal cost: p w. 2 We are grateful to Harvar an Princeton Universities for research support an Davi Krisztian Nagy for research assistance. Responsibility for any results, opinions an errors is the authors alone. 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.

2 ogether the equilibrium revenue function an pricing rule 2 imply that the relative revenues of firms epen only on their relative prouctivities: r r 2 2, 3 an equilibrium profits are a constant fraction of revenue minus the fie prouction cost: π r wf. Fie prouction costs imply a prouctivity cutoff below which firms eit efine by the following zero-profit conition: r R P w wf, 4 where the superscript enotes autarky. 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 to equal the sunk entry cost: G π wfe, 5 which using the relationship between the revenues of firms with ifferent prouctivities 3 an the zero-profit conition 4 can be re-written as follows: G r g wf G w, G r g wf G w, f g, which can be written more compactly as: J f J fe, 6 G G, 7 where is a weighte average of firm prouctivities: g G. 8 Note that lim 0 J, lim J 0, an J is a monotonically ecreasing function. It follows that the free entry conition 6 etermines a unique equilibrium value of the autarkic zero-profit prouctivity inepenently of the other enogenous variables of the moel. he mass ontrants M e equals the mass of proucing firms M ivie by the probability of successful entry G : M e M G R r G 9. 2

3 Using the relationship between average firm revenue r an average firm profits π: r π + wf, an the free entry conition 5 the mass ontrants can be re-epresse as: R M e w + G. 0 f We choose labor as the numeraire w. Using the relationship between the mass ontrants an mass of firms 9 in the free entry conition 5, we obtain: M π M e L e, which implies that total payments to labor use in entry equal total profits. Note that total payments to labor use in prouction equal total revenue minus total profits: L p R M π, which together with labor market clearing implies that aggregate revenue equals total labor payments R L. Using this equality of aggregate revenue an total labor payments in the zero-profit conition 4, welfare can be written solely in terms of the zero-profit prouctivity an parameters: W Het w L P f. herefore the zero-profit prouctivity 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 with eogenous probabilities Ḡ an Ḡ respectively. Fie prouction costs imply that only firms rawing a prouctivity of fin it profitable to prouce. herefore proucing firms are homogeneous an there is a egenerate prouctivity istribution conitional on prouction at. 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. 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 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 + F. 2 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 w. 3

4 while profit maimization an free entry imply that equilibrium output an employment for the representative variety are proportional to the fie prouction cost: q F, l F. Using the common employment for each variety, the mass of firms can be etermine from the labor market clearing conition: M L. 3 F Using the equilibrium pricing rule an the mass of firms, the CES price ine is: P w M, 4 where we again choose labor as the numeraire an hence w. Rearranging the price ine 4, an using the mass of firms 3 an our choice of numeraire, welfare can be written in terms of prouctivity an other parameters: 2.3 ggregate Equilibrium Equivalence w L P F. 5 We now pick the parameters Ḡ an of the egenerate prouctivity istribution with homogeneous firms such that the autarky equilibrium is isomorphic to the heterogeneous firm equilibrium, in the following sense: Proposition Consier a homogeneous firm moel that is a special case of the heterogeneous firm moel with an eogenous probability of successful entry Ḡ G an an eogenous egenerate istribution of prouctivity conitional on successful entry. Given the same value for all remaining parameters {f,, L, }, 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 + / G with the free entry conition 6 in the heterogeneous firm moel, we obtain: F f. 6 Substituting this result into close economy welfare in the homogeneous firm moel 5, we obtain the same welfare in the two moels: W Hom w L P f 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 3, 0 an 6. 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 We assume that the heterogeneous an homogeneous firm moels have the same trae costs, so that there is fie eporting cost of f units of labor an an iceberg variable 4

5 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 RP p, r τ r, 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 because of the variable costs of trae: p w, p τp, 7 his equilibrium pricing rule implies that profits in each market are a constant proportion of revenues minus the fie costs: π r wf, π r wf, where 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 an eport market are efine by the following zero-profit conitions: r R P w wf. 8 r R P τw wf, 9 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: τ f f. 20 For sufficiently high fie an variable trae costs τ f /f >, only the most prouctive firms eport, consistent with an etensive empirical literature see for eample the review in Bernar, Jensen, Reing an Schott he free entry conition again equates the epecte value ontry to the sunk entry cost: G π wfe. 2 Noting that the relative revenues of firms within the same market epen solely on their relative prouctivities, an using the omestic cutoff conition 8 an the eport cutoff conition 9, the 5

6 free entry conition can be re-written as follows: or equivalently: G G + G G r wf g + G G r r G wf g G g wf G r g wf G f g + f g w, w,. f J + f J fe, 22 where J is efine in 7 an weighte average prouctivity in the eport market is efine in an analogous way to weighte average prouctivity in the omestic market in 8: g G. Using the relationship between the prouctivity cutoffs 20, the free entry conition can be written solely in terms of the omestic prouctivity cutoff: f g + τf /f /. 23 f g τf /f / Noting that the left-han sie converges towars infinity as converges towars zero; the left-han sie converges towars zero as converges towars infinity; an the left-han sie is monotonically ecreasing in. It follows that the free-entry conition 23 etermines a unique equilibrium value of inepenently of the other enogenous variables of the moel. he unique equilibrium value of follows immeiately from the relationship between the cutoffs 20. Since the left-han sies of the close an open economy free entry conitions 6 an 23 respectively are monotonically ecreasing in, it also follows that the omestic cutoff in the open economy is greater than the omestic cutoff in the close economy >. s in the close economy, the mass ontrants M e equals the mass of firms M ivie by the probability of successful entry G : M e M G R r G 24. Using the relationship between average firm revenue r an average firm profits π: r π + wf + G G wf, 6

7 an the free entry conition 2 the mass ontrants can be re-epresse as: M e R w + G. f + G f an aggregate revenue equals total labor payments R wl. Using the equilibrium pricing rule an the mass of firms, the CES price ine in the open economy can be written as: P M + χτ w, 25 where χ G / G is the proportion oporting firms. We choose labor in one country as the numeraire, which with symmetric countries implies that the wage in each country is equal to one w. Rearranging the price ine 25, an using the mass of firms 24 an our choice of numeraire, welfare can be epresse in terms of prouctivity an parameters: W Het w P L G + f + χf + χτ. 26 In an open economy equilibrium with selection into eport markets >, the zero-profit conition for the omestic market 8 implies that open economy welfare can be written equivalently in terms of the omestic prouctivity cutoff an parameters: W Het w L P f. 27 Comparing an 27, an noting that the omestic cutoff is higher in the open economy than in the close economy >, 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, 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 again can be written equivalently in terms of the omestic prouctivity cutoff an parameters: W Het w + τ P L f + f. 28 Comparing an 28, an noting that f /f τ 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 respectively. For sufficiently high fie an variable trae costs τ 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 close economy welfare, there are welfare gains from trae, an trae only occurs in the heterogeneous 7

8 firm moel. In contrast, for sufficiently low fie an variable trae costs τ 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 because of the variable costs of trae: p w, p τp. 29 Free entry implies that total revenue equals total costs. In an equilibrium in which the representative firm oes not fin it profitable to eport, we have: r F. 30 In contrast, in an equilibrium in which the representative firm fins it profitable to eport, we have: + τ r F + f. 3 Using these two free entry conitions 30 an 3, we can confirm that the representative firm eports if: τ f <, F f F Ḡ In an equilibrium in which the representative firm eports, profit maimization an free entry imply that equilibrium output an employment for the representative variety are proportional to fie costs: q F + f, l 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 F + f. 33 Using the equilibrium pricing rule an the mass of firms, the CES price ine in the open economy is: P + τ M w, 34 where we again choose labor as the numeraire an hence w. Rearranging the price ine 34, an using the mass of firms 33 an our choice of numeraire, welfare can be again epresse in terms of prouctivity an parameters: W Hom w + τ P L F + f. 35 8

9 3.3 Relative Welfare in the Open Economy In the homogeneous firm moel, aggregate prouctivity is eogenous an hence constant by assumption. In contrast, in the heterogeneous firm moel, aggregate prouctivity is the result of the enogenous entry an eit ecisions of heterogeneous firms, which 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 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,. 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 such that the autarky equilibrium is isomorphic to the heterogeneous firm one as outline in Proposition : Ḡ G an. 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. We establish the proposition for the various possible types of open economy equilibria epening on parameter values. I First, we consier parameter values for which the representative firm oes not fin it profitable to eport in the homogeneous firm moel τ f /F / >. For these parameter values, the proposition follows immeiately from the fact that the two moels have the same close economy welfare, there are welfare gains from trae, an trae only occurs in the heterogeneous firm moel. II Secon, we consier parameter values for which 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 / < < τ f /f /. From 26 an 35, open economy welfare is higher in the heterogeneous firm moel than in the homogeneous firm moel if the following inequality is satisfie: + χτ + τ G + f > + χf F + f. 36 o show that this inequality must be satisfie, we use the open economy free entry conition in the heterogeneous firm moel, which implies: f G + f G, f G + f G, G + f + f + χf. f G Using τ f /f, we obtain: f + χτ 9 G G + f + χf. 37

10 Note that the open economy free entry conition in the heterogeneous firm moel also implies: f G + f G <, 38 since < < an < 0, for <, < 0 for <. Rewriting 38, we have: f G + f G <, f + f < Using τ f /f, we obtain: From 37 an 39, we have: G + f + f. f + τ < G + f + f. 39 f + χτ f + τ G + f + f G + f, 40 + χf f + τ <, F + f which establishes that inequality 36 is satisfie. From 27 an 35, the conition for open economy welfare to be higher in the heterogeneous firm moel than in the homogeneous firm moel can be also written as: + τ > F + f. f Using 36 an 40, this equivalent inequality is necessarily satisfie. Since close economy welfare is the same in the two moels, an open economy welfare is higher in the heterogeneous firm moel than in the homogeneous firm moel, it follows that the proportional welfare gains from trae are larger in the heterogeneous firm moel W Het /W Het > W Hom /W Hom. III hir, we consier parameter values for which the representative firm eports in the homogeneous firm moel an all firms eport in the heterogeneous firm moel, but fie eporting costs are still positive 0 < τ f /F / < τ f /f /. his is simply a special case of II in which, an G G. herefore the same line of reasoning as in II can be use to show that the inequality 36 is satisfie an hence open economy welfare is higher in the heterogeneous firm moel than in 0

11 the homogeneous firm moel. In this special case in which all firms eport, the free entry conition in the open economy equilibrium of the heterogeneous firm moel implies: f + f G, f + f G <, 4 since < an Rewriting 4, we obtain: < 0, for <. f + f < G + f + f F + f. 42 From 28 an 35, the conition for open economy welfare to be higher in the heterogeneous firm moel than in the homogeneous firm moel can be also written as: f + f > F + f. From 42, this inequality is necessarily satisfie. Since close economy welfare is the same in the two moels, an open economy welfare is higher in the heterogeneous firm moel than in the homogeneous firm moel, it follows that the proportional welfare gains from trae are larger in the heterogeneous firm moel W Het /W Het > W Hom /W Hom. IV Finally, when fie eporting costs are zero, we have 0 τ f /F / τ f /f /. his is a special case of III in which, an G. In this special case of zero fie eporting costs, the free entry G conition in the open economy equilibrium of the heterogeneous firm moel implies: f + f G, f + f G + f + f F + f, where we have use. From 28 an 35, it follows immeiately that open economy welfare is the same in the two moels when fie eporting costs are equal to zero. 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. 2 2 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.

12 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 of zero fie eporting costs. 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 35. 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 5 an 35. It 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, which with symmetric countries is equivalent to maimizing the welfare of the representative consumer in each country. 3 We show that the social 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. 4 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 he real consumption ine for the representative consumer is: / Q M e q / G. 43 he social planner chooses the prouctivity cutoff, the output levels q for all proucing firms, an the mass ontrants M e to maimize Q subject to the aggregate labor constraint: L M e { q G + G f + }, 44 3 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. 4 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

13 where the social planner faces the same prouctivity istribution G an entry cost per firm as in the market allocation. he planner chooses the output levels q to equate the marginal rates of transformations an marginal rates of substitution for firms with ifferent prouctivities. he marginal rate of substitution between varieties for any two firms with prouctivities an 2 is: q / MRS. q 2 he marginal rate of transformation between varieties for any two firms with prouctivities an 2 is: MR 2. Efficiency requires: MRS MR q q 2 which yiels the same relationship between relative quantities an relative prouctivities as in the market equilibrium. Using this relationship, we can rewrite the consumption ine Q an the aggregate labor constraint as a function of the output level q q of a firm with a weighte average prouctivity : Q { G M e } / q, L G M e q + f + 2, G, where we have use q / q an the efinition of in 8. Using the aggregate labor constraint we can rewrite the real consumption ine as: Q L / q + f + / G q. 45 he social planner chooses the cutoff an quantity q to maimize this consumption ine. he trae-off face by the social planner is that a lower prouctivity cutoff reuces epecte entry costs conitional on successful entry, an thereby releases more labor for prouction. But this lower prouctivity cutoff involves firms of lower prouctivities proucing, which reuces epecte output conitional on successful entry. he first-orer conition for q yiels: q f + G. 46 he first-orer conition for is: q 2 fe g G 2. Noting that: g G, the first-orer conition for can be re-written as: q 3 G.

14 Substituting the optimal quantity q from 46 elivers the planner s solution for the optimal cutoff : G f, 47 which correspons to the free entry conition in the market economy 6. herefore, the social planner chooses the same prouctivity cutoff an the same values of all other enogenous variables as in the market equilibrium of the heterogeneous firm moel, which implies that the market equilibrium is efficient. In the heterogeneous firm moel, changes in fie costs an other parameters inuce the social planner to change the prouctivity cutoff in aition to 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, an hence changes in fie costs an other parameters only inuce changes in the mass of proucing firms an output per firm. 4.2 Open Economy he open economy real consumption ine for the representative consumer is: Q M e q / q / / G + M e G. 48 τ he social planner chooses the prouctivity cutoffs an, the output levels q for the omestic market for all proucing firms, the output levels q for the eport market for all eporting firms, an the mass ontrants M e to maimize Q subject to the aggregate labor constraint: { q L M e G + q G + G f + G f + where the social planner faces the same prouctivity istribution G an entry cost per firm as in the market allocation. he planner chooses the output levels q an q to equate the marginal rates of transformations an marginal rates of substitution for firms with ifferent prouctivities: q q 2 q q 2 q q 2 τ which are the same relationships between relative quantities an relative prouctivities as in the market equilibrium. Using these relationships, we can rewrite the consumption ine Q an the aggregate labor constraint as a function of the mass of firms serving each market M t an the output level q t q t of a firm with a weighte average prouctivity t : 2 2 Q M / t q t,,, }, L M t q t t M t q t t + G f + G f + G + G, + f + χf + + χ G. 4

15 he mass of firms serving each market M t an weighte average prouctivity t are efine as follows: M t + χ M + χ G Me, 49 t { G + G τ } /, 50 G + G { + χ + χτ } /. where χ is the proportion oporting firms: χ G G. Using the aggregate labor constraint, we can rewrite the real consumption ine as: Q L / q t t + Q L / q t t G f + G / f + G + G q t + / f + χf + + χ G q t. 5 he social planner chooses the prouctivity cutoffs an an quantity q t to maimize this consumption ine. 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 omestic cutoff involves firms of lower prouctivities proucing, which reuces epecte output conitional on successful entry. lower eport cutoff for a given omestic cutoff increases the probability oporting, which uses more labor in the fie costs oporting an reuces epecte output conitional on eporting. But this lower eport cutoff also increases the fraction of foreign varieties available to omestic consumers. he first-orer conition for q t yiels: q t G f + G f +, G + G t q t t f f + χf + e, + χ G G + χ q t t f χf, 52 which equates the epecte profits from entry to the sunk entry cost, because +χ q t t epecte variable profits conitional on successful entry. he first-orer conition for requires: +χ rt is q t t t 2 G g f { G + G } 2 G + g f { G + G } 2 + g { G + G } 2 0. fe 5

16 Using the first-orer conition for q t an noting that: t G g g + { G + G }2 G + G + G g τ { G + G }2 t, the first-orer conition for can we written as follows: t q t t r f, 53 which correspons to the omestic cutoff conition for the open economy market equilibrium 8. he first-orer conition for requires: t q t 2 + t G g f { G + G } 2 G g f { G + G } 2 + g fe { G + G } 2 0. Using the first-orer conition for q t an noting that: G t g G g { G + G }2 g + τ G + G τ { G + G }2 t, the first-orer conition for can be epresse as: τ t q t t r f, 54 which correspons to the eport cutoff conition for the open economy market equilibrium 9. From the two cutoff conitions 53 an 54, the relationship between the omestic an eport cutoffs is the same as in the open economy market equilibrium: τ f f, an the free entry conition 52 can be written in the same form as in the open economy market equilibrium 22: G f + G f. 55 Since the free entry, omestic cutoff an eport cutoff conitions for the social planner are the same as those for the open economy market equilibrium, the social planner chooses the same prouctivity cutoffs an 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. 6

17 4.3 Open Versus Close Economy Recall that the heterogeneous an homogeneous firm moels are specifie to yiel 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 an to choose all firms to eport. 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. However, comparing the close an open economy free entry conitions 47 an 55, the social planner in general chooses a ifferent omestic prouctivity cutoff in the open economy than in the close economy an in general restricts eporting to a proper subset of more prouctive firms < <. 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 at least as high level of welfare as the hypothetical allocation with the same prouctivity cutoffs. Furthermore, the social planner s objective 5 is globally concave in {,, q t }, which implies that the ifferent prouctivity cutoffs must yiel strictly higher welfare than the hypothetical allocation with the same prouctivity cutoffs. Since the social planner s choice correspons to the open economy equilibrium of the heterogeneous firm moel, an the hypothetical allocation correspons to the open economy equilibrium of the homogeneous firm moel, 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. his reveale preference argument implies larger welfare gains in the heterogeneous firm moel both when there is selection into eport markets τ f /f / > an > an when all firms eport τ f /f / an, as long as fie eporting costs are positive f > 0. Comparing the free entry conitions in the open an close economies 47 an 55, the social planner chooses ifferent omestic prouctivity cutoffs in the open an close economies for positive fie eporting costs. he reason is that the social planner s objective in the open economy 5 features a ifferent combination of fie an variable costs from her objective in the close economy 45. he social planner s optimal response to these ifferent fie an variable costs is to change the range of prouctivities for which firms serve each market as well as the mass of firms an 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. 7

18 5 lternative Reveale Preference Derivation In this section, we provie a complementary alternative erivation of the result that the market equilibrium of the heterogeneous firm moel is efficient, which is use as part of our reveale preference argument above. his alternative erivation formulates the social planner s problem as a Lagrangian an uses the first-orer conitions for this Lagrangian to show that the socially optimal allocation replicates the allocation in the market equilibrium. 5. Close Economy 5.. Social Planner s Problem We first show that in the close economy the social planner chooses the same values of {h,, M e, Q} as in the market equilibrium, where h enotes variable labor input. he social planner s problem is to maimize the welfare of the representative consumer subject to the constraints of the entry an prouction technologies. he welfare of the representative consumer is etermine by the real consumption ine: Q M e h ρ G where ρ. herefore the social planner maimizes the following Lagrangian: ma {h,,me} M e h ρ G where λ is the multiplier on the social planner s constraint First-orer Conitions ρ ρ. 56 λ M e h G + M e G f + M e L, he first-orer conition for M e is : Q λ M e h G + M e G ρm e M f + M e e he first-orer conition for h is: M e ρ h ρ g Q ρ λm e g he first-orer conition for is: M e ρ h ρ g Q ρ + λm e h g + λg Me f he first-orer conition for λ is: M e h G + M e G f + M e L

19 5..3 Lagrange Multiplier Using the first-orer conition for λ 60, the first-orer conition for M e 57 becomes: ρm e Q λ M e L 0, λ Q ρl Employment Using the first-orer conition for λ 60, the first-orer conition for h 58 can be written: h ρ ρ Q ρ ρ ρl ρ. 62 or equivalently: h Q L, Labor Market Clearing Using the solution for h from 62 in the first-orer conition for λ 60, the labor market clearing conition can be written as follows: L M e h G + M e G f + M e. L M e Q ρl G + Me G f + M e Real Consumption Ine an Mass of Firms G G. Using the solution for h from 62 in the real consumption ine 56 we get: Q M e ρl G. 65 Substituting this epression for Q into the labor market clearing conition 64, we obtain the following epression for the mass of firms as a function of an parameters: M e ρ L G f Using this result in 65, the real consumption ine can also be written in terms of an parameters: Q ρ ρ L G. 67 G f Prouctivity Cutoff Conition Using the first-orer conition for λ 60 an the solution for h in 62, the first-orer conition for can be written as: Q ρl f. 68 9

20 5..8 Free Entry o erive the analogue of the free entry conition for the social planner, note that the labor market clearing conition 64 gives us one epression for L/M e : L Q ρl G M + G f +, e while the mass of firms 66 gives us another epression for L/M e : L M e G f + ρ G f +. Equating these two epressions, we obtain the analogue of the free entry conition for the social planner: G Q ρl G f. 69 Using the prouctivity cutoff conition 68 to substitute for Q ρl, the analogue of the free entry conition becomes: f G. 70 he social planner s choices of {h,, M e, Q} are etermine by 63, 70, 66 an 67. his is the same system oquations that characterizes the open economy market equilibrium. It follows that the welfare-maimizing social planner chooses the same allocation {h,, M e, Q} as the market equilibrium an that the market equilibrium is efficient. 5.2 Open Economy 5.2. Social Planner s Problem We now show that in the open economy the social planner chooses the same values of {h, h,,, M e, Q} as in the market equilibrium, where h enotes variable labor input for the omestic market an h enotes variable labor input for the eport market. he social planner s problem is to maimize the joint welfare of the two countries subject to the constraints of the entry an prouction technologies. Since the two countries are symmetric, the social planner maimizes the welfare of the representative consumer in each country, which is etermine by the real consumption ine: Q M e h ρ G + M e τ ρ h ρ G herefore the social planner maimizes the following Lagrangian: ma {h,h,,,me} M e h ρ G + M e τ ρ h ρ G λ where λ is the multiplier on the constraint. ρ ρ. 7 Me h G + M e h G + M e G f +M e G f + M e L, 20

21 5.2.2 First-orer Conitions he first-orer conition for M e is : ρm e Q λ M e Me h G + M e h G + M e G f +M e G f + M e he first-orer conition for h is: he first-orer conition for h is: he first-orer conition for is: M e ρ he first-orer conition for is: M e ρ h ρ g Q ρ λm e g τ ρ M e ρ h ρ g Q ρ λm e g h ρ g Q ρ + λm e h g + λg Me f M e ρ τ ρ h ρ g Q ρ + λm e h g + λg Me f he first-orer conition for λ is: Me h G + M e h G + M e G f M e G f + M e L Lagrange Multiplier Using the first-orer conition for λ 77, the first-orer conition for M E 72 becomes: ρm e Q λ M e L 0, λ Q ρl Employment Using the first-orer conition for λ 77, the first-orer conition for h 73 can be written: or equivalently: h ρ ρ Q ρ ρ ρl ρ. 79 h Q L, 80 Using the first-orer conition for λ 77, the first-orer conition for h 74 can be written: h τ ρ ρ ρ ρ Q ρ ρ ρl ρ. 8 or equivalently: h τ Q L, 82 2

22 5.2.5 Labor Market Clearing Using the solutions for h an h from 79 an 8 in the first-orer conition for λ 77, the labor market clearing conition can be written as follows: L M e h G + M e h G + M e G f + M e G f + M e. L M e Q ρl G + τ G 83 + M e G f + M e G f + M e, where G G, G G Real Consumption Ine an Mass of Firms Using the solutions for h an h from 79 an 8 in the real consumption ine 7 we get: Q Me ρl G + τ G. 84 Substituting this epression for Q into the labor market clearing conition 83, we obtain the following epression for the mass of firms as a function of, an parameters: M e ρ L G f + G f Using this result in 84, the real consumption ine can also be written in terms of an parameters: Q ρ ρ L G + τ G G f + G. 86 f Prouctivity Cutoff Conitions Using the first-orer conition for λ 77 an the solution for h in 79, the first-orer conition for can be written as: Q ρl f. 87 Using the first-orer conition for λ 77 an the solution for h in 8, the first-orer conition for can be written as: τ Q ρl f. 88 ogether these cutoff conitions imply that the relative prouctivity cutoffs satisfy: τ f f

23 5.2.8 Free Entry o erive the analogue of the free entry conition for the social planner, note that the labor market clearing conition 83 gives us one epression for L/M e : L Q ρl G M + τ G e + G f + G f +, while the mass of firms 85 gives us another epression for L/M e : L G f + G f + M e ρ G f + G f +. Equating these two epressions, we obtain the analogue of the free entry conition for the social planner: Q ρl G + τ G G f G f. Using the prouctivity cutoff conitions 87 an 88 to substitute for Q ρl, the analogue of the free entry conition can be written as: f G + f G. 90 he social planner s choices of {h, h,,, M e, Q} are etermine by 80, 82, 90, 89, 85 an 86. his is the same system oquations that characterizes the open economy market equilibrium. It follows that the welfare-maimizing social planner chooses the same allocation {h, h,,, M e, Q} as the market equilibrium an that the market equilibrium is efficient. 6 rae Shares an Welfare No aitional erivations require. 7 rae Liberalization in the Open Economy No aitional erivations require. 8 Pareto Prouctivity Distribution In this section, we eamine the implications of assuming a Pareto prouctivity istribution. We first eamine the eterminants of the omestic trae share an the mass ontrants with a general continuous prouctivity istribution. We net eamine these relationships uner the assumption of a Pareto prouctivity istribution. 23

24 8. General Prouctivity Distribution 8.. Domestic rae Share Each country s trae share with itself in the heterogeneous firm moel is: 8..2 Mass Entrants λ X X M e L M e L M e L r g, 9 f g, G f. Free entry implies a relationship between average revenue, the prouctivity cutoffs an fie costs in the heterogeneous firm moel. We erive this relationship using the free entry conition: π G, 92 an the relationship between average profits an average revenue conitional upon entry: r G + G f + G f, 93 which together imply: G f + G f. 94 itionally, the mass ontrants is relate to the omestic prouctivity cutoff an average revenue as follows: G Me. 95 R r L r 8.2 Pareto Prouctivity Distribution ssume that prouctivity in the heterogeneous firm moel is rawn from a Pareto istribution: 8.2. Prouctivity Cutoffs g k min k k+, k >, min > Uner this istributional assumption, we erive close form solutions for the prouctivity cutoffs an weighte average prouctivity in the close an open economies of the heterogeneous firm moel. he close economy omestic prouctivity cutoff is: while the open economy omestic prouctivity cutoff is: k k k f k k f min, 97 e k f + τ k f f f k min

25 Weighte average prouctivity in the omestic market is relate to the omestic prouctivity cutoff as follows: k k. 99 he eporting prouctivity cutoff in the open economy equilibrium is relate to the omestic prouctivity cutoff through 20: τ f f. Weighte average prouctivity in the eport market is relate to the eporting prouctivity cutoff as follows: Domestic rae Share k k. 00 Using these close form solutions for the Pareto prouctivity istribution, each country s trae share with itself 9 in the heterogeneous firm moel becomes: Mass of Entrants λ X X M e L f min k k k, 0 With a Pareto prouctivity istribution the relationship between average revenue, the prouctivity cutoffs an fie costs 94 in the heterogeneous firm moel simplifies to: G f + G f k. 02 Using this result in average revenue 93, we obtain: r G k, 03 which implies that the mass ontrants 95 is given by: which epens solely on labor supply an other parameters Welfare M e L k, 04 Rearranging the omestic trae share uner a Pareto prouctivity istribution 0, we obtain the following relationship between the omestic prouctivity cutoff an each country s trae share with itself in the heterogeneous firm moel: Me k f k λl k k k min. 05 Using this result in welfare in the open economy equilibrium of the heterogeneous firm moel 27, we can relate welfare to each country s trae share with itself: W Het w k L P k f k k k min M /k e λ /k

26 Noting that uner a Pareto prouctivity istribution the mass ontrants 04 is proportional to labor supply, we obtain: W Het w P λ /k L k min f k k k k k. 07 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. 08 With a Pareto prouctivity istribution, the elasticity of the omestic trae share with respect to variable trae costs is: { λ Het τ k λ Het τ f /f / τ λ Het λ Het τ f /f / <, 09 In contrast, in the homogeneous firm moel, the omestic trae share an welfare in the close an open economies 5 an 35 respectively imply that the welfare gains from trae can be epresse as: W Hom W Hom F λ Hom F + f. 0 In this homogeneous firm moel, knowing a country s omestic trae share λ an the elasticity of substitution 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 of the paper. Proposition 6 ssuming a Pareto prouctivity istribution with shape parameter k > an positive fie eporting costs, the greater the ispersion of firm prouctivity smaller k, a the larger the proportional welfare gains from opening the close economy to trae in the heterogeneous firm moel larger W Het /W Het, b the larger smaller the proportional welfare gains costs from a reuction increase in variable trae costs. Proof. a First, consier parameter values for which 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 / < < τ f /f /. From an 27, we have: W Het W Het. In the special case of a Pareto prouctivity istribution an for these parameter values for which there is selection into eport markets in the heterogeneous firm moel, we have: + τ f /f / k f f /k, 26

27 which can be written as: Note that ln / k ln k 2 ln + k ln + τf /f / k f f τ f /f / k f f. k lnτf /f / + τf/f / k f τf/f / f k f f < 0, 2 where we have use a / ln a a. Since a smaller value of k correspons to greater prouctivity ispersion, it follows that greater prouctivity ispersion implies larger /. Secon, consier parameter values for which the representative firm eports in the homogeneous firm moel an all firms eport in the heterogeneous firm moel, but fie eporting costs are still positive 0 < τ f /F / < τ f /f /. From an 28, we have: W Het W Het + τ f f + f. In the special case of a Pareto prouctivity istribution an for these parameter values for which all firms eport in the heterogeneous firm moel, we have: which can be written as: Note that + f /k, f ln k ln + f. f ln / k 2 ln + f < 0. 3 k f Since a smaller value of k correspons to greater prouctivity ispersion, it follows that greater prouctivity ispersion again implies larger /. aking 2 an 3 together an using, it follows that greater ispersion of firm prouctivity smaller k implies larger proportional welfare gains from opening the close economy to trae. b Consier parameter values for which there is selection into eport markets in the open economy equilibrium of the heterogeneous firm moel τ f /f / >. In the special case of a Pareto prouctivity istribution, we have: k /k f + τf /f / k f /k min. We have: τ τ k f τf /f / τ k τ f + f τf /f / ξτ. 27

28 Hence: which implies: τ τ τ k ln τ f /f / τ τ f + τ τ k τ τ k τf /f / k f τf /f / k f < 0 for τ < 0, > 0 for τ > 0. ξ τ, herefore greater ispersion of firm prouctivity smaller k implies a larger elasticity of the omestic prouctivity cutoff with respect to reuctions in variable trae costs, which from 27 implies greater proportional welfare gains from reuctions in variable trae costs. By the same reasoning, greater ispersion of firm prouctivity smaller k implies a smaller elasticity of the omestic prouctivity cutoff with respect to increases in variable trae costs, which from 27 implies smaller proportional welfare costs from increases in variable trae costs. 9 Quantitative Relevance In this section, we use the assumption of a Pareto prouctivity istribution to show that the ifferences in the aggregate implications of heterogeneous an homogeneous firms are of quantitative relevance. We choose stanar values for the moel s parameters base on central estimates from the eisting empirical literature an moments of the U.S. ata. We set the elasticity of substitution between varieties 4, which is consistent with the estimates using plant-level U.S. manufacturing ata in Bernar, Eaton, Jensen an Kortum Since prouctivity is Pareto istribute an firm revenue is a power function of firm prouctivity, firm revenue also has a Pareto istribution: G r k r r, where r is the revenue of the least prouctive firm. Eisting empirical estimates suggest that the firm size istribution is well approimate by a Pareto istribution with a shape parameter close to one see for eample tell 200. herefore, we set the Pareto shape parameter for firm prouctivity k 4.25, which ensures a Pareto shape parameter for firm revenue close to one an that log firm revenue has a finite mean k.42 >. choice for the Pareto scale parameter is equivalent to a choice of units in which to measure prouctivity, an hence we normalize min. he general equilibrium of the moel uner the assumption of a Pareto istribution can be summarize by the following system oquations: k f + τ k f f k f k min k. τ f f, M e L k, M min 28 k M e,