New Trade Models, New Welfare Implications

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1 New rae Moels, New Welfare Implications he Harvar community has mae this article openly available. Please share how this access benefits you. Your story matters Citation Melitz, Marc J., an Stephen J. Reing New rae Moels, New Welfare Implications. merican Economic Review 05 3 March: oi:0.257/aer Publishe Version oi:0.257/aer Citable link erms of Use his article was ownloae from Harvar University s DSH repository, an is mae available uner the terms an conitions applicable to Open ccess Policy rticles, as set forth at nrs.harvar.eu/urn-3:hul.instrepos:ash.current.terms-ofuse#op

2 Web ppeni to New rae Moels, New Welfare Implications Not for Publication Marc J. Melitz Harvar University, NBER, an CEPR Stephen J. Reing Princeton University, NBER, an CEPR ugust 3, 204 Introuction his web appeni contains the technical erivations of epressions 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 of epressions. Sections 2-7 of the web appeni closely follow the sections of the paper with the same name. Section 8 provies a formal treatment of the reveale preference intuition iscusse in the paper. he heterogeneous firm moel has an aitional ajustment margin that is absent in the homogeneous firm moel namely enogenous firm selection. With constant elasticity of substitution CES preferences, ajustment along this aitional margin is efficient. s a result, if the egenerate prouctivity istribution in the homogeneous firm moel is chosen so that the two moels have the same welfare for an initial value of trae costs, the heterogeneous firm moel has higher welfare for all other values of trae costs. s part of this analysis, we show that the social planner s problem can be reuce to a choice of the prouctivity cutoffs in an unconstraine maimization problem. In Section 9, we show that the social planner s problem also has an equivalent representation in terms of a constraine maimization problem. In Section 0, we use this equivalent representation as a constraine maimization problem to consier the case of variable elasticity substitution VES preferences. In this case, the market equilibrium nee not be efficient. Nonetheless, enogenous firm selection has the potential to generate higher welfare in the heterogeneous firm moel than in the homogeneous firm moel, as long as ajustment along this aitional margin is similar in the market equilibrium an social optimum. 2 Heterogeneous an Homogeneous Firm Moels We compare the canonical heterogeneous firm moel of Melitz 2003 to a homogeneous firm moel that is a special case with a egenerate prouctivity istribution as in Krugman 980. We hol all other parameters incluing the traing technology constant across the two moels. 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.

3 2. Close Economy 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, ma, which is rawn from a fie istribution g after incurring a sunk entry cost of f e units of labor. We allow the upper boun of the support of the prouctivity istribution to be either finite ma < or infinite ma =. 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 σ p σ, 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 = σ w σ. 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 σ, 3 an equilibrium profits are a constant fraction of revenue minus the fie prouction cost: π = r σ wf. Fie prouction costs imply a prouctivity cutoff below which firms eit efine by the following zero-profit conition: σ σ r = R σ P w σ = σwf, 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: ma G r σ wf G G = wf e, ma G σ r G wf σ G = wf e, ma σ f G = f e, 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

4 which can be written more compactly as: f J = fe, 6 J ma σ = G = G σ, 7 where is a weighte average of firm prouctivities that correspons to a harmonic mean weighte by output shares: ma σ = σ 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 of firms M equals the mass of entrants M e times the probability of successful entry G : M = G M e =. 9 R r Using the relationship between average firm revenue r an average firm profits π: r = σ π + wf, an the free entry conition 5, the mass of firms can be re-epresse as: M = R 0 f σw e G + f We choose labor as the numeraire w =. Using the relationship between the mass of entrants an mass of firms 9 in the free entry conition 5, we obtain: M π = M e f 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 = L M π, which together with labor market clearing implies that aggregate revenue equals total labor payments R = wl = L. Using this result, the mass of firms 0 can be epresse as: M = L = L f σ e G + f σf, where F summarizes average fie costs per firm in the close economy. he CES price ine in the close economy can be written as: P = Mp σ σ. Using the mass of firms an the pricing rule 2, the CES price ine becomes: P = σ { } L σ σ σ σf. 3

5 herefore, using our choice of numeraire, close economy welfare can be written in terms of the mass of firms L/σF an the weighte average prouctivity of these firms : W Het = w P = σ σ { } L σ σ σf. 2 From the zero-profit conition 4 an the equality of aggregate revenue an total labor payments, welfare can be equivalently written solely in terms of the zero-profit prouctivity an parameters: W Het = w L P = σ σ σf σ. 3 herefore the zero-profit prouctivity cutoff is a sufficient statistic for welfare. 2.2 Open Economy Heterogeneous Firm Moel We consier trae between two symmetric countries. We assume 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. Equilibrium firm revenues in the omestic an eport markets are: r = RP σ p σ, r = τ σ 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 = σ w σ, p = τp, 4 his equilibrium pricing rule implies that profits in each market are a constant proportion of revenues minus the fie costs: π = r σ wf, π = r σ 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: σ σ r = R σ P w σ = σwf. 5 σ σ r = R σ P τw σ = σwf, 6 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: f σ = τ. 7 f 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

6 he free entry conition again equates the epecte value of entry to the sunk entry cost: G π = wfe. 8 Noting that the relative revenues of firms within the same market epen solely on their relative prouctivities, an using the omestic cutoff conition 5 an the eport cutoff conition 6, the free entry conition can be re-written as follows: or equivalently: ma r G σ wf G G + G ma G ma σ r G G σ wf G + G ma σ r G σ wf f ma +f ma r σ wf G G G G σ G σ G = wf e, = wf e, = f e. f J + f J = fe, 9 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: = ma σ σ G G. Using the relationship between the prouctivity cutoffs 7, the free entry conition can be written solely in terms of the omestic prouctivity cutoff: ma σ f G σ ma +f = f e. 20 G τf /f /σ τf /f /σ 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 20 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 7. Since the left-han sies of the close an open economy free entry conitions 6 an 20 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 of firms M equals the mass of entrants M e times the probability of successful entry G : M = G M e = R r. 2 5

7 Using the relationship between average firm revenue r an average firm profits π: r = σ π + wf + G G wf, an the free entry conition 8 the mass of firms can be re-epresse as: M = R. f σw e G + f + χf where χ = G / G is the proportion of eporting firms. s in the close economy, aggregate revenue equals total labor payments R = wl. 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 =. herefore the mass of firms can be written as: M = σ L = f e G + f + χf L σf, 22 where F summarizes average fie costs per firm in the open economy. Using the equilibrium pricing rule 4 an our choice of numeraire, the CES price ine in the open economy can be written as: P = σ M σ σ + χτ σ σ σ. 23 Using the price ine 23, the mass of firms 22 an our choice of numeraire, open economy welfare can be written in terms of the mass of varieties available for consumption L + χ/σf an the weighte average prouctivity of these varieties t : W Het = w P = σ σ { L + χ σf } σ σ t, 24 where this weighte average prouctivity t epens on weighte average prouctivity in the omestic an eport markets an respectively an the proportion of eporting firms χ: σ t = σ + χ + χ τ σ. 25 In an open economy equilibrium with selection into eport markets >, the zero-profit conition for the omestic market 5 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. 26 Comparing 3 an 26, 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 σ P = L σ σ σ f + f σ. 27 Comparing 3 an 27, 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. 6

8 2.3 Close Economy Homogeneous Firm Moel We construct a homogeneous firm moel that is a special case of the heterogeneous firm moel with a egenerate prouctivity istribution. Firms pay a sunk entry cost of f e units of labor an raw a prouctivity of either 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. he close economy equilibrium of this homogeneous firm moel is isomorphic to that in 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 of entry costs F = f + f e / Ḡ. hese values for the representative firm s prouctivity an the fie prouction cost are eogenous an hel constant. o simplify the eposition, we aopt this Krugman 980 interpretation. he representative firm s prouction technology is: l = q + F. 28 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 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 σ, l = σ F. Using the common employment for each variety, the mass of firms can be etermine from the labor market clearing conition: M = L σ F. 29 Using the equilibrium pricing rule an the mass of firms, the CES price ine is: P σ = M σ w σ, 30 σ where we again choose labor as the numeraire an hence w =. Rearranging the price ine 30, an using the mass of firms 29 an our choice of numeraire, close economy welfare can be written in terms of the mass of firms L/σ F an prouctivity : W Hom = w P = σ σ 2.4 Open Economy Homogeneous Firm Moel { } L σ F σ σ. 3 We again consier trae between two symmetric countries an assume the same traing technology as in the heterogeneous firm moel, so that there is a fie eporting cost of f units of labor an an iceberg variable trae cost of τ > units of each variety. 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 σ f / F >, the representative firm oes not fin it profitable to eport. In contrast, for sufficiently low fie an variable trae costs τ σ f / F <, the representative firm fins it profitable to eport, an there is trae in both moels. he open 7

9 economy equilibrium of this homogeneous firm moel is isomorphic to a version of Krugman 980 with the same traing technology as in Melitz 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 = σ w, p = τp. 32 σ Free entry implies that the representative firm s operating profits equal its fie costs. In an equilibrium in which the representative firm oes not fin it profitable to eport, we have: r σ = F. 33 In contrast, in an equilibrium in which the representative firm fins it profitable to eport, we have: + τ σ r = σ F + f. 34 Using these two free entry conitions 33 an 34, we can confirm that the representative firm eports if: τ σ f <, F = f + f e. 35 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 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 of eporting. 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. 36 Using the equilibrium pricing rule an the mass of firms, the CES price ine in the open economy is: P σ = + τ σ M σ w σ, 37 σ where we again choose labor as the numeraire an hence w =. Using the price ine 37, the mass of firms 36 an our choice of numeraire, open economy welfare can be written in terms of the mass of varieties available for consumption 2L/σ F + f an average prouctivity t : W Hom = w P = σ σ { 2L σ F + f t σ } σ. 38 where this average prouctivity t epens on the prouctivity of the representative firm an variable trae costs τ: t σ = σ + τ σ

10 3 heoretical Comparative Static o eamine the aggregate welfare properties of the two moels, we first pick the parameters Ḡ an of the egenerate prouctivity istribution with homogeneous firms such that welfare in an initial equilibrium is the same as with heterogeneous firms. We net eamine the effects of changes in trae costs. his calibration enables us to unertake a theoretical comparative static in which we eamine the effect of the prouctivity istribution on the moel s welfare properties keeping all other structural parameters the same same f, f e, f, τ, L, σ. We eamine both the opening of the close economy to trae an changes in trae costs in the open economy equilibrium. 3. Opening the Close Economy to rae We begin by picking the parameters Ḡ an of the egenerate prouctivity istribution with homogeneous firms such that the autarky equilibrium is isomorphic to that with heterogeneous firms, an eamine the effect of opening the close economy to trae. 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, f e, 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. Comparing 3 an 3, equal welfare follows immeiately from = an Ḡ = G, which implies F = F. 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 immeiately from equal price inices an =. Equal aggregate revenue follows from R = wl = L in both moels. his first proposition reflects the aggregation properties of the heterogeneous firm moel. ll aggregate variables in this moel take the same value as if there were a representative firm with prouctivity an fie costs F. But the key ifference between the heterogeneous firm moel an such a representative firm moel is that aggregate prouctivity in the heterogeneous firm moel is enogenous an respons to changes in trae costs. From the epressions for open economy welfare in an equilibrium with trae in both moels 24 an 38, open economy welfare is higher in the heterogeneous firm moel than in the homogeneous firm moel if the following inequality is satisfie: σ + χτ σ σ + τ σ σ f e G + f >. + χf F + f From the free entry conition in the open economy equilibrium of the heterogeneous firm moel, an noting that F = f + f e / G an =, this inequality is necessarily satisfie in any open economy equilibrium in the heterogeneous firm moel in which the prouctivity cutoffs iffer in the open an close economies an ma. Since the two moels have the same close economy welfare, it follows that the welfare gains from opening the close economy to trae are larger in the heterogeneous firm moel than in the homogeneous firm moel. Proposition 2 Choosing the egenerate prouctivity istribution in the homogeneous firm moel so that the two moels have the same close economy welfare an the same structural parameters f, f e, f, τ, L, σ, the proportional welfare gains from opening the close economy to 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 opening the close economy to trae are the same in the two moels. 9

11 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 /σ >. 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 /σ < < τ f /f /σ. From 24 an 38, open economy welfare is higher in the heterogeneous firm moel than in the homogeneous firm moel if the following inequality is satisfie: σ + χτ σ σ f e G + f > + χf + τ σ σ F + f. 40 o show that this inequality must be satisfie, we use the open economy free entry conition in the heterogeneous firm moel, which implies: ma σ ma σ f G + f G = f e, f G σ + f G σ = f e, σ G σ f e + f = + f + χf. f G Using σ = σ τ σ f /f, we obtain: f σ σ + χτ σ σ = G f e G + f + χf. 4 Note that the open economy free entry conition in the heterogeneous firm moel also implies: ma σ ma σ f G + f G < f e, 42 since < < an σ < 0, for <, σ < 0 for <. Rewriting 42, we have: f G σ + f G σ < f e, σ f σ + f < f e G + f + f. 0

12 Using σ = σ τ σ f /f, we obtain: From 4 an 43, we have: f σ + τ σ σ f e < G + f + f. 43 f σ σ + χτ σ σ f + τ σ σ σ f e G + f = + f f e G + f =, 44 + χf f + τ σ σ σ <, F + f which establishes that inequality 40 is satisfie. From 26 an 38, the conition for open economy welfare to be higher in the heterogeneous firm moel with eport market selection than in the homogeneous firm moel can be also written as: σ + τ σ σ > F + f. f Using 40 an 44, 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 /σ < τ f /f /σ. 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 40 is satisfie an hence open economy welfare is higher in the heterogeneous firm moel than in 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: since < an Rewriting 45, we obtain: f + f ma σ G = f e, ma σ f + f G < f e, 45 σ < 0, for <. f + f σ f e < G + f + f = F + f. 46 From 27 an 38, the conition for open economy welfare to be higher in the heterogeneous firm moel without eport market selection than in the homogeneous firm moel can be also written as: σ σ f + f > F + f.

13 From 46, 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 /σ = τ f /f /σ. 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: ma σ f + f G = f e, σ f + f = f e G + f + f = F + f, where we have use =. From 27 an 38, it follows immeiately that open economy welfare is the same in the two moels when fie eporting costs are equal to zero. 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 chieving the same proportional welfare gains from trae in the two moels 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 38. 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 3 an 38. It follows that both of the above propositions hol for any value of the representative firm s prouctivity. Both these propositions also hol for general continuous prouctivity istributions. 3.2 Changes in rae Costs in the Open Economy Equilibrium he role of the aitional ajustment margin of firm entry an eit ecisions for generating ifferent aggregate welfare implications is not limite to the opening of the close economy to trae an also hols for reuctions in trae costs in the open economy equilibrium. o show this, we recast our heterogeneous an homogeneous firm moels so that they have the same welfare in an initial open economy equilibrium. In orer to ensure that the two moels have the same initial welfare an only iffer in their prouctivity istribution keeping the same structural parameters f, f e, f, τ, L, σ, we eten the homogeneous firm moel to allow for two types of firms: eporters an non-eporters. In this etension, firms again pay a sunk entry cost of f e units of labor before observing their prouctivity. With probability Ḡ a firm raws a prouctivity of an can eport; with probability Ḡ the firm raws a prouctivity of an cannot eport; with probability Ḡ Ḡ the firm raws a prouctivity of zero an oes not fin it profitable to prouce. We pick the parameters of this etene homogeneous firm moel,, Ḡ, Ḡ such that the open economy equilibrium features the same aggregate variables as the initial open economy 2

14 equilibrium with heterogeneous firms same welfare, price ine, mass of firms, aggregate revenue, an omestic trae share. his requires equating those parameters with their corresponing averages uner firm heterogeneity: Ḡ = G, Ḡ = G G, { =, = G G σ G } σ where the last term represents the average prouctivity of non-eporters in the heterogeneous firm moel. We keep the values of all the other structural parameters f, f e, f, τ, L, σ constant across the two moels. his ensures that the aggregate statistics line-up across the two moels in the initial open economy equilibrium. Nevertheless these two moels respon ifferently to changes in trae costs from this common initial equilibrium along a key imension. In the heterogeneous firm moel, the enogenous selection responses to trae costs lea to changes in the average prouctivity of eporting an non-eporting firms an in the proportion of eporting firms. In contrast, in the etene homogeneous firm moel, the average prouctivity levels of eporters an non-eporters an the proportion of eporting firms remain constant. 2 he presence of this aitional ajustment margin in the heterogeneous firm moel implies that welfare following the change in trae costs must be strictly higher than in the homogeneous from moel. his argument hols irrespective of whether trae costs ecrease or increase. herefore, welfare gains are larger in the heterogeneous firm moel whenever trae costs fall, an welfare losses are smaller in the heterogeneous firm moel whenever trae costs increase. Proposition 4 Starting from an initial open economy equilibrium with the same welfare an the same structural parameters in the two moels f, f e, f, τ, L, σ, a common ecrease increase in variable or fie trae costs generates larger welfare gains smaller welfare losses in the heterogeneous firm moel than in the etene homogeneous firm moel. Proof. In the initial open economy equilibrium before the change in trae costs, 24 implies that welfare in both the heterogeneous firm moel an in the etene homogeneous firm moel can be written as: σ L σ W σ Het = σ σ f e G σ + χ τ σ σ. + f + χ f In the new open economy equilibrium after the change in trae costs, 24 implies that welfare in the heterogeneous firm moel is: σ L σ W σ 2 Het = σ 2 σ f e G 2 σ + χ2 τ σ 2 2 σ. + f + χ 2 f 2 In contrast, in the new open economy equilibrium after the change in trae costs, welfare in the 2 Unless trae costs become sufficiently high that firms with prouctivity no longer fin it profitable to eport or firms with prouctivity no longer fin it profitable to prouce. In both cases, the average prouctivity of the two types of firms remains constant at an, 3

15 etene homogeneous firm moel is: σ L σ W σ 2 Hom = σ σ f e G σ + χ τ2 σ + f + χ f 2 σ o show that welfare in the new open economy equilibrium is higher in the heterogeneous firm moel than in the homogeneous firm moel, we nee to show that: σ 2 + χ2 τ2 σ 2 σ σ + χ τ σ 2 σ f e > f + f + χ 2 f e f + χ f 2 G 2 o establish this inequality, we use the free entry conition in the new open economy equilibrium of the heterogeneous firm moel, which implies: σ ma f ma σ G + f 2 G = f e, 2 f G 2 2 σ 2 2 σ 2 G 2 f + f 2 2 G Using 2 σ = 2 f G 2 + f 2 G 2 2 σ = 2 σ τ σ 2 f 2 /f, we obtain: σ 2 σ + χ2 τ σ 2 f e G 2 σ = 2 σ = f 2 e, 2 + f + G 2 f 2. G f e G f + χ 2 f Note that the free entry conition in the new open economy equilibrium of the heterogeneous firm moel also implies: σ ma f ma σ G + f 2 G < f e, 49 2 since < 2 an > 2 an Rewriting 49, we have: f G 2 2 σ σ σ 2 2 < 0, for < 2, > 0 for 2 < <. + f 2 G 4 σ < f 2 e,

16 σ G f + f 2 2 G Using 2 σ = 2 f 2 σ < 2 σ τ σ 2 f 2 /f, we obtain: σ From 48 an 50, we have: f 2 f 2 which establishes the inequality 47. σ + χ τ σ 2 σ 2 f e G 2 σ f e G f e G σ < σ + χ2 τ σ 2 + f + G f 2. G f e G 2 σ + f + χ 2 f 2 =, σ + χ τ σ 2 σ + f + χ f 2 <, + f + χ f 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 an initial open economy equilibrium. Put another way, consier a planner who is constraine to keep the same set of firms operating in both the omestic an eport markets i.e. the enogenous selection margin is inoperative. Uner this constraint, the welfare-maimizing allocation coincies with the market equilibrium of the etene homogeneous firm moel. In contrast, in the absence of this constraint, the welfare-maimizing allocation coincies with the market equilibrium of the heterogeneous firm moel. herefore, the welfare ifferential between the two moels provies a irect measure of the impact of selection on aggregate welfare. In other wors, it isolates the aitional contribution to aggregate welfare of the new enogenous selection/prouctivity channel highlighte by the heterogeneous firm moel of trae this represents the new welfare implications that we refer to in the title of this paper. Later in Section 6, we show that this aitional welfare channel is quantitatively substantial for a moel calibrate to U.S. aggregate an firm statistics. tkeson an Burstein 200 consiers this welfare ifferential from enogenous firm selection in a moel with prouct an process innovation. hey fin that this welfare ifferential is of seconorer. Proposition 4 is consistent with this result. s iscusse above an shown formally in the web appeni, 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. But, as we show later, these secon-orer welfare effects can be quite substantial for larger changes in trae costs. 3.3 Untruncate Pareto Distribution Since the homogeneous firm moel is a special case of the heterogeneous firm moel, our above comparisons of the two moels are equivalent to a iscrete comparative static of moving from a nonegenerate to a egenerate prouctivity istribution within the heterogeneous firm moel. In the special case of an untruncate Pareto prouctivity istribution, the egree of firm heterogeneity is summarize by a single parameter: the shape parameter k. Lower values of k correspon to greater firm heterogeneity an the homogeneous firm moel correspons to the limiting case in which k. 5

17 herefore we can complement the above iscrete comparative static with a continuous comparative static in the egree of firm heterogeneity. Proposition 5 ssuming that prouctivity in the heterogeneous firm moel has an untruncate Pareto istribution g = k k min k+, where min > 0 an k > σ an fie eporting costs are positive, the greater the ispersion of firm prouctivity smaller k, a the larger the welfare gains from opening the close economy to trae larger W Het /W Het, b the larger smaller the welfare gains losses from a ecrease increase in variable trae costs in the open economy equilibrium. 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 /σ < < τ f /f /σ. From 3 an 26, we have: W Het W Het =. 5 In the special case of an untruncate Pareto prouctivity istribution an for these parameter values for which there is selection into eport markets in the heterogeneous firm moel, we have: which can be written as: Note that ln / k ln = k 2 ln + = + = k ln + τ f /f /σ τf /f /σ k f f k f f τ f /f /σ /k, k f f. k lnτf /f /σ + τf/f /σ k f τf/f /σ f k f f < 0, 52 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 /σ < τ f /f /σ. From 3 an 27, we have: W Het W Het = + τ σ f f + f σ. In the special case of an untruncate Pareto prouctivity istribution an for these parameter values for which all firms eport in the heterogeneous firm moel, we have: = + f /k, f which can be written as: ln = k ln + f. f 6

18 Note that ln / = k 2 ln + f < k f Since a smaller value of k correspons to greater prouctivity ispersion, it follows that greater prouctivity ispersion again implies larger /. aking 52 an 53 together an using 5, 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 /σ >. In the special case of an untruncate Pareto prouctivity istribution, we have: k /k σ /k f = + f τf /f /σ min. k σ f e herefore: τ τ k f τf /f /σ τ = k τ f + f τf /f /σ = ξτ. Hence: which implies: τ τ τ k = ln τ f /f /σ τ τ f + τ τ k τ τ k τf /f /σ k f τf /f /σ 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 26 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 26 implies smaller proportional welfare costs from increases in variable trae costs. In this special case of an untruncate Pareto istribution, the heterogeneous firm moel falls within the class consiere by CR. herefore Proposition 5 confirms that the egree of firm heterogeneity affects the aggregate welfare implications of trae, in the sense of our theoretical comparative static, even within the class of moels consiere by CR. In the special case of an untruncate Pareto istribution, we obtain the following close-form solutions for the omestic an eport prouctivity cutoffs in the heterogeneous firm moel in terms of the moel s parameters: k σ f = k k σ f min, 54 e 7

19 k = σ k σ k f + τ k f σ f f e f k min. 55 Using these omestic an eport prouctivity cutoffs, we can obtain close-form solutions for the relative welfare gains from trae in heterogeneous an homogeneous firm moels. We begin by consiering the opening of the close economy to trae, in which case the egenerate prouctivity istribution in the homogeneous firm moel is chosen so that the two moels have the same close economy welfare. herefore the relative welfare gains from trae in the two moels equal relative open economy welfare. From 40, relative open economy welfare in the two moels is: σ + χτ σ σ f e + τ σ G + f + f σ f e G + f > χf In the special case of an untruncate Pareto istribution, this epression for relative welfare becomes: k k k σ + τ σ k σ f e + min f + min f + τ σ k σ k k >, 57 f e + min f + min f where = τf /f /σ an we have close-form solutions for {, } from 54 an 55. We net compare two open economy equilibria with ifferent values of trae costs, in which case the prouctivity istribution in the etene homogeneous firm moel is chosen so that the two moels have the same welfare in the initial open economy equilibrium inee by. herefore the relative welfare gains from trae liberalization in the two moels equal relative welfare in the new open economy equilibrium following the reuction in variable trae costs inee by 2. From 47, relative open economy welfare in the two moels following trae liberalization is: σ 2 + χ2 τ σ 2 2 σ + χ τ2 σ σ σ f e G f e G 2 + f + χ f 2 > 58 + f + χ 2 f 2 In the special case of an untruncate Pareto istribution, this epression for relative welfare becomes: k k 2 k σ + τ2 σ 2 k σ f e + min f + min f2 k σ + τ2 σ k σ k >, 59 k f e + f + f2 min 2 min 2 where = τf /f /σ in an open economy equilibrium with eport market selection an we have close-form solutions for from 55 above. 4 Welfare an rae Policy Evaluation Our theoretical comparative static in the previous section eamines the impact of changes in the istribution of prouctivity holing other eogenous variables fie across moels. his eercise oes not restrict the equilibrium values of the enogenous variables in particular the omestic trae share λ an the trae elasticity θ to be the same in the two moels. Instea the equilibrium values for these enogenous variables iffer systematically across the two moels. We now compare trae shares an trae elasticities in the two moels given the same values of the eogenous variables. 8

20 rae Shares: In an open economy equilibrium of the homogeneous firm moel in which the representative firm eports, the omestic trae share is: σ λ Hom = + τ σ σ = τ σ In contrast, in an open economy equilibrium of the heterogeneous firm moel, the omestic trae share is: σ λ Het = σ + χτ σ σ = + τ σ Λ, 6 where Λ = δ δ = ma ma σ G σ G. In an open economy equilibrium of the heterogeneous firm moel in which only some firms eport 0 < <, the eport market selection term Λ is strictly less than one. Proposition 6 Given the same structural parameters f, f e, f, τ, L, σ, the omestic trae share is strictly greater in the heterogeneous firm moel than in the homogeneous firm moel λ Het > λ Hom for parameter values for which there is trae in both moels an selection into eport markets in the heterogeneous firm moel 0 < τ f / F /σ < < τ f /f /σ. he omestic trae share is only the same in the two moels λ Het = λ Hom for parameter values for which all firms eport in the heterogeneous firm moel 0 τ f / F /σ < τ f /f /σ. Proof. For 0 < τ f / F /σ < < τ f /f /σ, we have >, which implies 0 < Λ < an hence λ,het > λ,hom in the omestic trae shares 6 an 60. For 0 τ f / F /σ < τ f /f /σ, we have =, Λ = an λ Het = λ Hom. In the special case of an untruncate Pareto prouctivity istribution from Proposition 5, we can solve in close form for the eport market selection term Λ in the heterogeneous firm moel as a function of the prouctivity cutoffs {, }: Λ = ma ma σ G k σ σ G =. 62 In an open economy equilibrium with selection into eport markets <, we have: λ Het = + τ σ k σ = k σ + τ k f σ f > λ Hom = τ σ generalization of this functional form for the prouctivity istribution is the case of a truncate Pareto istribution, in which the support of the prouctivity istribution is boune from above: g = kk min k+ k. min ma with the corresponing cumulative istribution function: G = k min k, min ma 9

21 which implies: G = k k min min ma k. min ma where the untruncate Pareto istribution is the special case in which ma. With this more general functional form, the eport market selection term Λ continues to be a close-form epression of the prouctivity cutoffs {, }: Λ = ma ma σ G σ G = k σ ma k σ ma k σ. 64 In an open economy equilibrium with selection into eport markets <, we have: λ Het = + τ σ k σ k σ ma k σ ma > λ Hom = τ σ rae Elasticities: Given the same structural parameters f, f e, f, τ, L, σ, the heterogeneous an homogeneous firm moels also have ifferent implications for the elasticity of trae flows with respect to trae costs. In the homogeneous firm moel, the elasticity of trae with respect to trae costs is zero for parameter values for which the representative firm oes not fin it profitable to eport. For parameter values for which the representative firm eports, there is a constant elasticity of trae with respect to variable trae costs an a zero elasticity of trae with respect to fie trae costs: ln λhom σ 0 < τ f / F /σ < θhom τ λ Hom = = ln τ 0 0 < < τ, 66 f / F /σ ln λhom Hom = θ f λ Hom = ln f 0 0 < τ f / F /σ < 0 0 < < τ f / F /σ, 67 where trae increases iscontinuously from zero to a positive value as trae costs fall below the value at which the representative firm begins to eport: τ f / F /σ =. In the heterogeneous firm moel, the elasticities of trae with respect to variable an fie trae costs are in general enogenous variables. For parameter values for which only some firms eport <, these enogenous variables epen on the functional form of the prouctivity istribution an the level of trae costs. For parameter values for which all firms eport =, the elasticities of trae with respect to variable an fie trae costs are the same as in the homogeneous firm moel: θhet τ = θ f Het = σ ln Λ ln τ > 0 τ f /f /σ > σ τ f /f /σ < ln Λ ln f > 0 τ f /f /σ > 0 τ f /f /σ <, 68, 69 where the trae elasticity can change iscontinuously as trae costs fall below the value at which only some firms eport: τ f /f /σ =. 20

22 Proposition 7 Given the same structural parameters f, f e, f, τ, L, σ, the elasticities of trae with respect to variable an fie trae costs are strictly larger in absolute value in the heterogeneous firm moel than in the homogeneous firm moel for parameter values for which there is trae in both moels an selection into eport markets in the heterogeneous firm moel 0 < τ f / F /σ < < τ f /f /σ. he trae elasticities are the same in the two moels for parameter values for which all firms eport in the heterogeneous firm moel 0 τ f / F /σ < τ f /f /σ. Proof. he proposition follows immeiately from the trae elasticities in the homogeneous firm moel 66 an 67 an the trae elasticities in the heterogeneous firm moel 68 an 69, since Λ τ τ Λ < 0 an Λ f Λ f < 0. rae Elasticities with an Untruncate Pareto Prouctivity Distribution: In the special case of an untruncate Pareto istribution, the elasticities of trae with respect to variable an fie trae costs are constants that epen only on whether or not there is selection into eport markets, as can be seen from the omestic trae share 63: { θhet τ k τ f = /f /σ > σ τ f /f /σ <, 70 θ f Het = { k σ σ τ f /f /σ > 0 τ f /f /σ <, 7 where the trae elasticity changes iscontinuously as trae costs fall below the value at which only some firms eport: τ f /f /σ =. rae Elasticities with a runcate Pareto Prouctivity Distribution: Even a slight generalization of an untruncate Pareto istribution to introuce a finite upper boun to the support of the prouctivity istribution ma < implies that the elasticities of trae with respect to variable an fie trae costs become variable rather than constant. We now report the close form solutions for these variable trae elasticities for a truncate Pareto prouctivity istribution. he elasticity of trae with respect to variable trae costs is: ln λhetr θhetr τ λ HetR =. ln τ Using the omestic trae share for a truncate Pareto istribution 65, the elasticity of trae with respect to variable trae costs can be epresse as: θhetr τ = σ +k σ min 4. CR Welfare Derivation No further erivations require. k σ min ln ln τ k σ min ma k σ 4.2 Gains from rae in the Homogeneous Firm Moel No further erivations require. min k σ min ln ln τ k σ min ma k σ 72 2

23 4.3 Gains from rae in the Heterogeneous Firm Moel We now seek to epress the welfare gains from trae liberalization in terms of observable empirical moments for the general case of our heterogeneous firm moel. Since trae continuously rops to zero when trae costs increase, we can start from an open economy trae regime without loss of generality. o simplify notation, we rop the superscript. For now, we also assume that there is eport market selection in this trae regime so that >. Full an partial trae elasticities: From the omestic trae share 6, the full elasticity of trae with respect to variable trae costs θ can be epresse as follows:: θ = ln λ λ = σ ln Λ ln τ ln τ, = σ ln δ + ln δ, ln τ ln τ = σ ln δ ln ln ln τ + ln δ ln ln ln τ, = σ + γ ln ln τ γ ln ln τ, 73 where δ j = ma j σ G is proportional to the cumulative market share in any given market of firms above any prouctivity cutoff j ; γ j = ln δ/ ln j is the elasticity of δ j for market j {, }; hence γ j represents the hazar function for the istribution of log firm size within a market. s argue by CR, only the partial trae elasticity capturing the irect effect of τ is observe empirically, since it is estimate from a gravity equation with eporter an importer fie effects. In the contet of our symmetric country moel, this partial elasticity can be erive from 6, which relates the omestic trae share to variable trae costs an the two prouctivity cutoffs λ = λτ,,, an from 7, which relates the two prouctivity cutoffs to one another = τ,. 3 aking the partial erivative of the omestic trae share with respect to τ holing constant, we have: ϑ = ln λ λ = σ ln Λ ln ln τ ln ln τ, where the relationship between the prouctivity cutoffs 7 implies ln / ln τ =. herefore the partial elasticity can be further written as: ϑ = σ ln Λ ln, = σ + γ, 74 where γ j = ln δ/ ln j is the elasticity of δ j for market j {, }. Welfare with a general prouctivity istribution: Using the omestic trae share 6, welfare 24 can be re-epresse as: W Het = σ σ M σ δ σ e, 75 λ where δ /σ is the prouctivity of a firm that has omestic market revenue equal to epecte omestic market revenue per entering firm; the eponent on the omestic trae share /σ 3 In the web appeni, we show how a multi-country version of our moel yiels an epression for changes in log bilateral trae that is linear in eporter an importer fie effects an ϑ ln τ. 22

24 epens on the elasticity of substitution. his relationship hols in both the close economy λ = an an open economy equilibrium with selection into eport markets λ <. otally ifferentiating this epression for welfare 75, the change in welfare epens on the change in the mass of entrants, the change in the omestic trae share, an the change in epecte omestic market prouctivity per entering firm: ln W Het = σ ln M e σ ln λ + σ ln δ. 76 otally ifferentiating our earlier epression for welfare 26, the change in welfare is also equal to the change in the omestic prouctivity cutoff: ln W Het = ln. 77 Combining these two relationships an the efinition of γ, the change in welfare following trae liberalization can be epresse in terms of the change in the omestic trae share, the change in the mass of entrants, a variable partial trae elasticity ϑ, an the ifference in the hazar function γ γ between the omestic an eport markets: ln W Het = ϑ + γ γ ln M e ln λ, ϑ = σ + γ, 78 Welfare with an untruncate Pareto prouctivity istribution: In the special case of an untruncate Pareto prouctivity istribution, the relationship between welfare an the omestic trae share can be simplifie further using the following three results: i the mass of entrants is a constant that epens on parameters alone: M e = σ L ; 79 σk f e ii epecte prouctivity in each market per entering firm δ /σ j for market j {, } is a constant elasticity function of the prouctivity cutoff for that market: δ j = k k σ k min j k σ ; 80 iii the trae share for each market is a constant elasticity function of the prouctivity cutoff for that market: λ j = k Me k j σf L k σ k min. 8 ogether these three results in turn imply: a a common constant hazar function in the omestic an eport markets: γ = γ = γ = k σ ; 82 b a constant partial trae elasticity: ϑ = σ + γ = k; 83 c a constant full trae elasticity that equals the partial trae elasticity: θ = ϑ = k. 84 he property that the partial an full trae elasticities are both constant an equal to one another is specific to the untruncate Pareto istribution. In this special case, the omestic trae share 6 uner eport market selection < can be written as: λ = + τ σ k σ = 23 k σ + τ k f σ f, 85