Restoing the Poduct Vaiety and Po-competitive Gains fom Tade with Heteogeneous Fims and Bounded Poductivity* by Robet C. Feensta Univesity of Califonia, Davis, and NBER Octobe 203 Abstact The monopolistic competition model in intenational tade offes thee souces of gains fom tade that do not aise in competitive models: expansion in poduct vaiety; a po-competitive eduction in the makups chaged by fims; and the self-selection of moe efficient fims into expoting. Recent liteatue on tade with heteogeneous fims has emphasized the thid of these effects, and the fist two effects ae uled out when using a Paeto distibution fo poductivity with a suppot that is unbounded above. The goal of this pape is to estoe a ole fo poduct vaiety and po-competitive gains fom tade by using a bounded Paeto distibution fo poductivity. * Thanks ae due to Costas Akolakis and Kadee Russ fo helpful discussions. Financial suppot fom the National Science Foundation is gatefully acknowledged.
. Intoduction The monopolistic competition model in intenational tade offes thee souces of gains fom tade that do not aise in competitive models. Fist, opening to tade can lead to expansion in poduct vaiety, as goods not available in autaky becomes impoted. This fist souce is emphasized in the ealiest witings by Kugman (979) and thoughout Helpman and Kugman (985). A second souce of gains emphasized by Kugman (979) is that the po-competitive effect of tade educes the makups chaged by fims, and theefoe lowes consume pices. In ode fo this fall in pices to tanslate in a social gain, and not just a edistibution fom fims to consumes, we need the assumption of zeo pofits due to fee enty. In that case, the educed atio of pice to maginal cost implies a educed atio of aveage to maginal costs, so that fims ae taking geate advantage of economies of scale (as discussed by Helpman and Kugman, 985, p. 34). In this way, the consume gains due to educed makups become social gains because of the accompanying expansion of fim scale. The thid souce of gains aises in the moe ecent models of monopolistic competition and tade with heteogeneous fims, due to Melitz (2003). In this case, tade will lead to the selfselection of moe efficient fims into expoting, while less efficient fims exit the maket, leading to ise in aveage poductivity. This thid souce of gains has been the focus of ecent liteatue. Fo example, if we add the assumption that fim poductivity is unbounded above with a Paeto distibution, as in Chaney (2008), then it can be shown that the gains fom tade in the Melitz (2003) model ae entiely due to the selection of fims: the welfae gains fom new impoted vaieties ae just offset by the loss fom fewe domestic vaieties (Feensta, 200); and of couse, thee is no change in makups due to CES pefeences. Even without the unbounded Paeto assumption, Melitz and Redding (203) have ecently agued that the ise in aveage poductivity
2 due to fim selection and tade in the Melitz model is what distinguishes it most clealy fom the homogeneous fim model of Kugman (980). Even if we allow fo non-ces pefeences with heteogeneous fims, so that in pincipal a po-competitive effect could opeate, Akolakis, Costinot, Donaldson and Rodiguez-Clae (ACDR, 202) have ecently shown that neithe this effect no poduct vaiety leads to any gains; so once again, the key souce of gains fom tade comes fom the selection of fims. That esult in ACDR depends on the assumption of a Paeto distibution of poductivity with a suppot that is unbounded above, which is the stating point fo this pape. The goal of this pape is to estoe a ole fo poduct vaiety and po-competitive gains fom tade with heteogeneous fims, by using a bounded (o tuncated) Paeto distibution fo poductivity. The empiical elevance of this appoach is beyond question: Helpman, Melitz and Rubenstein (2008) have used the bounded Paeto to obtain a gavity equation in tade that is consistent with the many instances of zeo tade flows between counties. 2 It is supising, then, that the bounded Paeto has not eceived moe theoetical attention (though it is consistent with Melitz, 2003, who did not constain the distibution). One eason fo the populaity of the unbounded Paeto is that, like CES pefeences, it leads to highly tactable solutions fo tade and welfae. A seconday goal of this pape is to show that the bounded Paeto distibution still yields tactable solutions, even with a class of pefeences allowing fo non-constant makups. Specifically, we will wok with a class of pefeences intoduced by Diewet (976) known as the quadatic mean of ode (QMOR) expenditue function. This is pehaps the most ACDR futhe show that total gains ae educed by the po-competitive effect when expoting fims only patially pass-though the effect of tade cost eductions to thei pices. But that patial pass-though channel influences gains in thei pape if and only if tastes ae non-homothetic. In contast, we assume homothetic tastes. 2 Anothe motivation fo using bounded poductivity comes fom the theoy of globalization put foth by John Sutton and summaized in his Claendon Lectues (Sutton, 202). Sutton uses thee assumptions to deive the inteaction of fims as globalization poceeds, the thid of which is you can t make something out of nothing (Sutton, 202, p. 55). That assumption is intended to ule out unbounded poductivity.
3 geneal paametic fom fo expenditue that is dual to homothetic pefeences. It diffes fom the class of pefeences used by ACDR because it is homothetic, and moe impotant, because it gives an explicit functional fom fo the expenditue needed fo one unit of utility that is, fo the cost of living. 3 In contast, ACDR ely on an implicit solution fo welfae by integating fom demand, which makes it challenging to deal with non-infinitesimal changes in poduct vaiety. 4 The QMOR expenditue function is intoduced in section 2 whee, because we ae dealing with a monopolistic competition model, we assume that demand is symmetic acoss vaieties and also that it has a finite esevation pice. Given these popeties, we establish the sign patten of the paametes needed to ensue that the QMOR expenditue function is globally well-behaved: a featue that has not been assued in pio applications, mainly empiical. Ou use of the QMOR expenditue function sets this pape apat fom othe ecent, theoetical liteatue dealing with vaiable makups in intenational tade. A moe common choice is to use the additively sepaable utility function intoduced by Kugman (979), possibly with an explicit functional fom fo the sub-utility fom each vaiety. 5 Zhelobodko et al (200, 20), Kichko et al (203) and Dhinga and Moow (202) conside a boade class of additively sepaable functions than Kugman (979) by allowing the elasticity of demand to be inceasing o deceasing in quantity. These authos ague fo a po-competitive effect of tade in the latte case only (as assumed by Kugman and holding hee). But when these authos conside heteogeneous fims, they do not appea to ecognize that a Paeto distibution with unbounded suppot implies that the po-competitive effect vanishes, as we shall explain hee. 3 The class of pefeences used by ACDR includes one homothetic case the tanslog pefeences which ae also included within the QMOR class. This is the only case that is common to both classes, as explained in section 2. 4 In ongoing wok, these authos popose a quantitative method to pefom this integation, theeby obtaining welfae fom estimated demand. 5 Behens and Muata (2007, 202) use exponential functions and the latte pape includes po-competitive effects, while Saue (2009) and Simonovska (200) use a logaithmic function with displaced oigin.
4 Anothe line of liteatue elated to this pape assumes a finite numbe of fims, in which case makups ae endogenous even with nested-ces pefeences. 6 Initiated by Atkeson and Bustein (2008), this famewok is used by Edmond, Midigan and Xu (202) to compute the po-competitive gains fom tade between the United States and Taiwan. Specializing to the case of Betand competition between fims, De Blas and Russ (202) contast the esults obtained by Benad, Eaton, Jensen, and Kotum (2003) using an infinite numbe of ivals to those obtained instead with a finite numbe of ivals; only in the latte case does a po-competitive effect of tade opeate. Ou pape is most closely elated to Holmes, Hsu and Lee (203), who also use Betand competition and show that if and only if the distibution of poductivities is unbounded Paeto, then tade leads to gains only though selection and not though makups. In these papes, Betand competition occus between fims poducing pefect substitutes, so thee ae no gains fom poduct vaiety. Befoe poceeding, we should give a bief intuition as to why the po-competitive effect of tade vanishes with heteogeneous fims and the unbounded Paeto distibution. Suppose that we measue makups by the atio (not the diffeence) of pice and maginal cost. The most poductive fim has zeo cost, but a non-zeo pice, so its makup is infinite. The least poductive suviving fim will have its maginal cost equal to the esevation pice, so its makup is zeo. This ange of [0,+) fo makups applies equally well to domestic and foeign fims, even if the latte face vaiable tade costs. Futhemoe, the distibution of makups within this ange is detemined by the Paeto distibution of poductivity. So changes in tade costs have no impact at all on the distibution of makups, fom eithe domestic o foeign fims, but still affect the mass (o extensive magin) of expotes. The fixed distibution of makups no longe holds, howeve, 6 Eaton, Kotum and Sotelo (202) also conside a model with a finite but stochastic numbe of fims.
5 when poductivity and makups ae bounded above, since then the highest foeign makup depends on tade costs (so tade costs also affect the intensive magin). Ou pape poceeds as follows. We show in sections 2 and 3 that the QMOR expenditue function allows us to decompose the cost of living and theefoe welfae into components that coespond to poduct vaiety, the po-competitive effect, and the selection effect which is captued by aveage fim poductivity. In the tade envionments we shall conside, we ae able to establish how these components change individually and jointly due to libealization. This allows us to establish the gains compaing autaky to fictionless tade (section 4), and fo small changes in tade costs aound the fictionless equilibium (section 5). Impotantly, we contast the souce of gains with unbounded vesus bounded Paeto, and show that it is only in the bounded case whee the poduct vaiety and po-competitive gains apply. Finally, we ae able to compae the magnitude of total gains fom tade using unbounded vesus bounded Paeto. Measued in elation to initial utility, we find that the popotionate ise in welfae due to tade libealization is lagest in the unbounded Paeto case, despite the fact that neithe the poduct vaiety no the po-competitive channels opeate in this case. Constaining the Paeto distibution to be bounded allows those exta souces of gains to opeate, but educes the gains due to fim selection, so that the total popotionate gains ae lowe. This esult is elated to Melitz and Redding (203), who compae a heteogeneous fim model (with any poductivity distibution) to a homogeneous fim model (i.e. with a degeneate distibution), both with CES pefeences. They find highe popotionate gains when poductivity is dispese acoss fims. We ae using non-ces pefeences, and find highe popotionate gains when poductivity is the most dispese acoss fims, i.e. unbounded above. Conclusions ae given in section 6, and the poofs of popositions ae in the Appendix.
6 2. Consume Pefeences Expenditue Function We shall adopt the quadatic mean of ode (QMOR) expenditue function, which is defined by Diewet (976, p. 30) ove a discete numbe of goods as: / ij i j, 0 i j b p p, whee b ij ae paametes. We will conside the symmetic case whee b ii =, b ij = fo i j, and the QMOR function is expessed ove a continuum of goods indexed by : 2 / e ( p ) pd p d, 0. () This function is the expenditue needed to obtain one unit of utility, o the cost of living. Fo specific values of the paametes, and, this expenditue function takes on familia foms. Fo > 0, = 0 and =( ), the expenditue function is CES, so that < 0 fo >. Fo = 2, we obtain a quadatic expenditue function, but without the additively sepaable outside good used by Melitz and Ottaviano (2008). Fo =, we obtain what Diewet (97) calls a Genealized Leontief function (since the dual to a Leontief poduction function is linea in pices like the fist tem of () fo =, while the second tem adds geneality). And as we show below, as 0 then () appoaches a tanslog function. So the quadatic mean of ode function nests the commonly used homothetic cases. While the special cases of the quadatic mean of ode function have been applied empiically, it has not been applied in a monopolistic competition setting. To do so, we need to ecognize that demand is positive if and only if pices ae less than a esevation pice p*, equal acoss goods since the expenditue function is symmetic. In the CES case the esevation pice is infinite, but we will focus hee on finite esevation pices, while obtaining CES as a limiting
7 case. Goods that ae not available should have thei pices in () eplaced by p*, because that is the economically elevant pice to evaluate expenditue, demand and welfae. To fomalize this, without eplacing any pices define p p* as the set of available goods, with mass N d 0. Denote the mass of all possible goods by N d N. Then eplacing the pices in () by p* fo, we ewite the expenditue function as: e ( p) pd ( N N )( p*) p d 2 / 2 2 ( N N )( p*) p d ( N N ) ( p*). Diffeentiating this expession with espect to p*, dividing by ( N N) and multiplying by utility u, we obtain the demand fo a good with pice p*. Setting this demand equal to zeo we solve fo the esevation pice: 2/ N p* p d N [ N ( / )] N. (2) The second tem in (2) is a mean of ode of the pices p (also called a powe mean), and fo all values of this mean lies between the minimum and maximum values of p. The esevation pice is above this mean pice if and only if the fist tem in (2) is geate than unity. To ensue this and also ule out the CES case of an infinite esevation pice, we assume: 2/ Assumption (a) If < 0 then > 0, < 0 and [ N ( / )] < 0 ; (b) If > 0 then < 0, > 0 and 0 <[ N ( / )] < N ; 2 (c) As 0 then N and 2 fo any > 0. N
8 It is eadily confimed that pats (a) and (b) of Assumption ensue that the fist tem on the ight of (2) exceeds unity, so the esevation pice exceeds the mean pice. Pat (c) is consistent with (a) and (b) in the sense that eithe set of inequalities hold fo small so the fist tem of (2) is again geate than unity. Futhemoe, in this limit it is shown by Diewet (980, p. 45) and in the Appendix that the expenditue function in () appoaches the tanslog fom, ln e 0 ( p ) ln p d ln (ln ln ') ' N 2N p p p d d. While we have motivated Assumption by the equiement that the esevation pice in (2) exceeds the mean pice, we should be moe igoous in checking that the QMOR expenditue function satisfies the necessay conditions fo an expenditue function: that it is positive and nondeceasing, homogeneous of degee one and concave in pices. Concavity implies that demand cuves slope downwads, which will ensue that the esevation pice in (2) exceeds all pices p fo goods with positive demand, and not just the mean pice. Confiming that these egulaity conditions hold ensues that the expenditue function can be deived fom a well-behaved homothetic utility function. To check these conditions, substitute the esevation pice (2) back into () to obtain expenditue defined ove the available goods p p* : 2 / e ( p ) pd p d N [ N ( / )]. (3) We can compute demand by setting p ' p fo pices in a small inteval ' [, ], diffeentiate (3) with espect to p, divide by, multiply by utility u and use (2) to obtain: p p* q ( p) u e ( p) p. (4)
9 Note that this expession equals CES demand if we specify > 0, = 0 and = ( ) < 0 fo >, in which case p* and the final backeted tem above vanishes. Fo p < p* this final tem has the same sign as unde Assumption, so demand is positive except when the pice is geate than o equal to the esevation pice p*. This guaantees that the expenditue function is non-deceasing in pices. We can eliminate utility u in (4) by using total expenditue pq( p) d e ( p ) u. Evaluating the integal using (4), we solve fo educed-fom expenditue: p p* e ( p ) p * d. (5) p* p Again, fo p < p* the tem in squae backets in (5) has the same sign as unde Assumption, so expenditue is positive povided that a non-empty set of goods ae puchased. Anothe condition that we need to confim is that the expenditue function is concave in pices. Concavity implies that demand is downwad sloping, and fo the symmetic QMOR expenditue function, the evese is also tue as we show in the Appendix. We diectly evaluate the elasticity of demand by diffeentiating (4) with espect to p p*, holding utility u and expenditue e ( p ) constant, obtaining: ln q p* p*. (6) ln p 2 p p 0 The final tem above appoaches zeo in the CES case when 0 and =( ) < 0, so p* and =. But with 0 unde Assumption, this tem is positive fo p p* and so. With 0 we then obtain. Fo 0 2, the tem in culy backets in (6) exceeds unity, so that 0. Summaizing, we have: 2 2 2 /
0 Poposition Unde Assumption, fo N > 0 and < 2 the QMOR expenditue function (3) is globally positive, non-deceasing, homogeneous of degee one and concave in pices, with a finite esevation pice. Fo values of > 2, the demand cuves in (4) ae still downwad sloping in a neighbohood of the esevation pice, but we cannot guaantee this popety globally. Poposition is that fist time that the QMOR expenditue function has been shown to be globally well-behaved, and the fact that we can establish these popeties is made possible by the assumed symmety acoss goods. A final popety is obtained by diffeentiating (6) and simplifying, to obtain: 0, ln( p / p*) 2 so the elasticity is inceasing in pice, using the inequalities discussed just befoe Poposition. Welfae Gains Having confimed that the expenditue function is well-behaved fo < 2, we should explain how ou demand system elates to that used by ACDR and also deive conditions to ensue welfae gains. Assume fo convenience that labo is the only facto of poduction and each consume has one unit, so that income equals the wage, w. Then w ue ( p ), which we can use in (4) and (5) to obtain the demand shaes: p q ( p) d( p / p*) s ( p), with w D( p) p p* d( p / p*) p* p, (7) and D( p ) d( p / p*) d. The tem D( p) in the denominato of the shae expession ensues that the shaes integate to unity. In compaison, ACDR assume an expession fo the demand shaes that depend on the wage, the pice p, and also the esevation pice p*, but does not
involve any tem like D( p ). Nevetheless, the shaes integate to unity because of the pesence of the esevation pice. It tuns out that the ACDR demand shaes and those in (7) have only one case in common: the tanslog case, as 0. The tanslog demand shaes ae given by: s ( p ) ln p ln p d N. N Setting the demand shae equal to zeo, we see that the esevation pice is: ln p* ln p d N, (8) N so that the demand shae is also expessed as s ( p ) ln p ln p *. Integating ove the set of poducts, we immediately see that the shaes integate to unity without the pesence of a tem like D( p ). Indeed, using Assumption (c) it can be shown that lim 0 D( p ) in the tanslog case. 7 ACDR conside a whole family of demand functions with the convenient popety that the shaes ae defined using the pice and the esevation pice. All those demand functions except fo the tanslog coespond to non-homothetic utility functions, while the tanslog is the only case in common with ou homothetic expenditue function. Combining (5) with the definition of D( p ), we see that: / ( ) * ( ) e p p D p. (9) With labo as the only facto of poduction, and fim pofits equal to zeo unde monopolistic competition, welfae is u w / e ( p ), so a dop in the expenditue function will indicate welfae gains. Poposition guaantees that as the pice of any good falls fom the esevation pice then welfae ises, so inceased vaiety is beneficial fo the consume. Ou goal hee is to develop 7 This is shown fom (7), (8), Assumption (c), and 2 /2 lim 0 ( p * / p) ln( p * / p); see the Appendix.
2 moe geneal sufficient conditions fo welfae to ise fom one equilibia to anothe. To achieve this, we need a chaacteization of the tem D. Such an chaacteization is easiest in the tanslog case, fo which: 2 lim / p 2 D p d s 0 2 p * 2 p ln ( ) ln ( ). The fist equality can be shown by taking the limit of / ln[ D( p ) ] (see the Appendix), while the second follows fom the tanslog shae s ( p ) ln p ln p * the tanslog esevation pice in (8), we find that log expenditue is:. Combining this esult with 2 ln e ( p ) ln p d s ( p ). 0 N N 2 Po-competitive Poduct and Selection Hefindahl Vaiety Thus, expenditue is decomposed into thee tems: the fist eflects the po-competitive and selection effects in loweing aveage pices; the second eflects the benefits of poduct vaiety; while the thid is the Hefindahl index. Having moe dispese expenditue shaes will lowe the Hefindahl index and aise expenditue, theeby loweing welfae. That counte-intuitive esult is intepeted by Feensta and Weinstein (200) as eflecting cowding in poduct space. Fo othe values of, we can still obtain a type of Hefindahl index by defining the adjusted demand shaes: z ( p) s( p)( p * / p ). s ( p)( p * / p ) d ' ' ' Fo the tanslog case, = 0, these adjusted shaes equal the conventional shaes, while fo the quadatic case, = 2, these adjusted demand shaes equal the quantity shae of each poduct. Then defining the Hefindahl index, 2 H z ( p ) d, it is shown in the Appendix that,
3 / / / D( ) N N p H, constant>0 in H (0) whee the final tem is deceasing in the Hefindahl index H (since [ N ( / )] has the same sign as, fom Assumption ). Recall that expenditue is / ( ) * ( ) e p p D p, fom (9). As the esevation pice falls, then so does expenditue and welfae ises. But that gain is offset if the Hefindahl index also falls. In the tade envionments we shall conside, that esult will be likely wheneve vaiety inceases: while thee is not a one-to-one coespondence between changes in the mass of poducts N and the Hefindahl, in all cases that we examine an incease in N implies a lowe Hefindahl, which in tun implies an incease in / ( ) D p. It follows that if expenditue falls, then it falls by less than the eduction in the esevation pice. The question is whethe we achieve some bound to this offsetting effect on welfae due to cowding in poduct space. That question is answeed in the affimative, as shown by the following decomposition of expenditue: Lemma Unde Assumption, the cost of living can be decomposed altenatively as: / / e ( ) p * N N p H constant>0 in H / / () p * N s( ) p d p constant>0 in pices p Sufficient conditions fo a fall in the cost of living and ise in welfae ae that: (i) the esevation pice falls; and (ii) the Hefindahl index does not fall o the weighted-aveage pice tem on the second line does not ise.
4 The fist line of (), obtained by substituting (0) into (9), has aleady been discussed. It shows that if the esevation pice is falling, but the Hefindahl index is also falling due to an incease in vaiety, then the decline in the cost of living (and incease in welfae) will be less than that in p*. We have efeed to this outcome as a cowding effect, though it also can be given anothe intepetation. Conside, fo example, the CES case with =( ) < 0 and 0, fo which thee is no cowding in poduct space. The esevation pice appoaches in this case, but a limiting value of () still exists because / lim 0[ N ( / )] 0. Using the fomula fo p* in (2), it can be shown that the fist line of () appoaches: 2/ lim 0 p* N lim 2/ / e ( ) 0 p d lim 0 H / p d p, fo < 0. 2/ p d We see that lim 0 H equals the atio of two means: the mean of ode in the numeato, which is the exact index fo the CES function; and the mean of ode in the denominato, which uses an elasticity that is too low fo the CES case, since = ( )/2 < ( ). That numeato just cancels with the same tem appeaing in the esevation pice in (2). A moe geneal intepetation of the Hefindahl index, then, is that it is needed in () to coect the esevation pice to obtain an exact measue of the cost of living. Now suppose that the Hefindahl falls when compaing two equilibia, which will tend to incease the cost of living, but that the esevation pice also falls. Can we easily detemine whethe welfae ises o falls? The second line of () gives an affimative answe. Regadless of the change in the Hefindahl, the cost of living falls and welfae ises if along with the fall in the esevation pice the shae-weighted mean of ode shown on the second line does not ise. If the shae-weighted mean of pices is falling, then it follows that the decline in the cost of
5 living and incease in welfae exceeds the fall in p *. So unde these conditions, the changes in p* and p * effectively become bounds fo the change in the cost of living. In the next section we shall futhe decompose the esevation pice into tems eflecting (i) poduct vaiety, (ii) the makups chaged by fims, and (iii) an aveage of fim costs. The latte will eflect selection acoss fims. Using this decomposition in (), we will obtain a decomposition of the cost of living into the thee potential souces of gains fom tade, togethe with an additional tem (eithe the Hefindahl index o the shae-weighted aveage of pices) that essentially combines all of these effects. 8 3. Autaky Equilibium We have aleady assumed that labo is the only facto of poduction, and now we nomalize the wage at unity. As in Melitz (2003), we assume that fims eceive a andom daw of poductivity denoted by, so maginal costs ae a /, whee a is the labo need pe unit of output fo a fim with the lowest poductivity of. We will allow the Paeto distibution of poductivity to have eithe an uppe-bound in its suppot, as in Helpman, Melitz and Rubenstein (2008), o to be unbounded above: Assumption 2 (a) The poductivity distibution is Paeto, G( ) ( ) / ( b ), b, whee the uppe bound is b(, ] and > max{0, }; (b) Thee is a sunk cost F of obtaining a poductivity daw, but no fixed cost of poduction. In pat (a), we allow the Paeto distibution to be unbounded (b = ) o bounded (b < ). The 8 Substituting the esevation pice fom (2) into (), we obtain a futhe decomposition of expenditue into tems eflecting an exact pice index simila to that in Diewet (976), and changes in vaiety. See the Appendix.
6 estiction that > {0, } becomes > ( ) > 0 in the CES case, which is needed fo cetain fist moments to convege in that case; this estiction is needed hee fo much the same eason. The assumption that thee is no fixed cost of poduction in (b) is made fo convenience, and follows Melitz and Ottaviano (2008). The optimal pice fo a fim with poductivity is p( a/ ) / ( ). We follow ACDR and let p/ ( a/ ) denote the atio of pice to maginal cost, while v p * /( a / ) denotes the atio of the esevation pice to maginal cost. Fom (6), the elasticity ( p/ p*) is a function of the pice elative to the esevation pice, so ( / v) and using this notation the makup of the fim is witten as: ( / v) ( ), ( / v) v 2 2 whee the second expession follows fom (6) and is used to solve fo ( v). 9 Diffeentiating this expession, it is shown in the Appendix that elasticity of the makup is 0 v'( v) /, so that changes in maginal cost ae only patially passed-though to pices. We can now wite the equilibium conditions in autaky. A fim paying the sunk cost of F eceives a daw of poductivity with pobability g( ) G'( ). We make a change of vaiables fom to v. Since v p * /( a / ) then av / p *, so using the Paeto distibution: v p * p * g( ) d d dv g( v) dv b b a a. (2) This change of vaiables suggested by ACDR will consideably simplify ou expessions. 9 The left side of the second expession can be evaluated at = and = v to show that it is above and below ½, so that a solution (, v) whee it equals ½ always exists.
7 Because thee ae no fixed costs of poduction, the lowest-poductivity fim that will continue poduction will have maginal costs equal to the esevation pice, so v = is the lowe bound. The uppe bound fo v, denoted by v*, is obtained when poductivity is b av / p *, so that: v* bp * / a. Stating with the demand shae d / D( p ) fom (7), we multiply that by expenditue L to obtain total demand, and then by ( )/ to obtain pofits. Using the bounds v [, v*], the expected pofit fom enteing the maket must equal the sunk costs of F in equilibium, so that: v* ( v) ( v) L p * F d g( v) dv ( v) v D( p) a v* ( v) ( v) p * L d g( v) dv ( v) v a, v* ( v) p* N d g( v) dv v a e (3) whee in the second line we substitute the expession fo D( p ) d( p / p*) d. Rathe than using the geneal notation fo the set of available poducts, with the change in vaiables in (2) we ae now defining that set by the bounds fo v and the mass of enteing fims fims emaining afte those with lowest poductivity exit will be: v* e N e. The mass of p* p* N N g( v) dv Ne G( v*) a a, (4) which also equals N e [ G(a/p*)], whee (a/p*) is the poductivity of the fim with maginal cost (a/) just equal to the esevation pice. So as usual in the Melitz model, the mass of suviving fims N equals the mass of enteing fims times the pobability of suvival, which is equivalently witten as [ G(a/p*)] = ( p * / a) G( v*).
8 We have aleady used the condition that expenditue equals the wokfoce L, so that fullemployment holds. The emaining equilibium condition is obtained fom the esevation pice in (2), e-witten slightly by dividing by p* and using the Paeto distibution: v* ( v) p* N [ N ( / )] Ne g( v) dv v a. (5) The solution to the equilibium conditions (3)-(5) is summaized in the following esult: Poposition 2 Unde Assumptions and 2: (a) the autaky equilibium conditions (3)-(5) have a positive solution fo p*, N e and N; (b) if and only if b =, the solution fo N e is popotional to the county size L, while the solution fo N is independent of county size L. The existence esult in (a) elies on > max{0, } in Assumption 2, so that the integals in (3) and (5) emain bounded even fo v*. The esults in pat (b), when poductivity is unbounded, ae obtained by inspection of the equilibium conditions. In that case the uppe-limit of integation in (3) is infinite, so the two integals ae constant and (3) becomes F L / N. e It is immediate that the mass of entants is popotional to county size in this case. This esult is also obtained by ACDR, and follows fom the popotionality elation between expected pofits fom enteing the maket and expected evenue. While these two vaiables ae popotional fo evey fim in the CES case (i.e. egadless of poductivity), they ae popotional in expected tems fo the geneal demand system that we o ACDR adopt, povided that the Paeto distibution is unbounded above and thee ae no fixed costs of poduction. Those two assumptions ensue that the uppe (v = ) and lowe (v =) limits of integation in (3) ae exogenous.
9 Thee is a second implication of unbounded poductivity that is less well known than the linea elationship between enty and county size, and that concens the mass N of suviving fims. Substituting (4) into (5), the equilibium condition fo the esevation pice becomes: v* ( v) N N g( v) dv, (6) v which equals a positive constant on the left, fom Assumption. Fo b = and v* bp * / a, it is immediate that (6) solves uniquely fo N, independent of county size L. This supising esult is also found by Akolakis, Costinot and Rodiguez-Clae (200) fo the tanslog case, and ACDR fo thei moe geneal demand function. The finding that poduct vaiety is independent of county size holds only fo b = and will have stong implications fo the souces of gains fom tade, examined in the following section. Befoe tuning to that discussion, we use the fim-level stuctue intoduced in this section to futhe decompose the cost of living. The esevation pice in Lemma can be witten as the poduct of tems that eflect the aveage of fim makups and costs, as follows: Lemma 2 The esevation pice in the closed economy is: 2/ * 2/ * /2 2/ v /2 ( ) v * ( ) N g v p g v p* ( v) dv N [ N ( / )] Gv ( *) v G( v*), (7) in vaiety N Aveage makup Aveage of costs whee g( v) g( v) / v is an adjusted density function with distibution v* G( v*) g( v) dv. The fist tem appeaing on the ight of (7) is the same vaiety tem appeaing in (2). The second tem is a mean of ode of the makups ( v). To intepet the last tem, ecall that
20 v p * /( a / ) is the atio of the esevation pice to maginal cost, so p * / v / a is the maginal cost of a fim with poductivity. The last tem in (7) is theefoe a mean of ode of the maginal cost of fims. If all thee tems in Lemma 2 fall and also the shae-weighted pice tem in Lemma does not ise, then we ae assued of a welfae gain. We now examine tade envionments allowing fo such welfae gains. 4. Fictionless Tade We initially conside fictionless tade, whee in addition to the assumption of no fixed costs of poduction o expot, we also ignoe vaiable costs of tade (while intoducing such tade costs in the next section). We suppose that the expenditue function in () along with Assumptions and 2 holds acoss counties. In this envionment, moving fom autaky to fictionless tade is equivalent to gowth in the labo foce L. We have aleady shown in Poposition 2 that with an unbounded Paeto distibution, poduct vaiety N does not change but N e ises in popotion to L. It follows that the pobability of suvival is falling, so thee is a positive selection effect: only fims with poductivity above a highe cutoff level poduce in the lage maket, while smalle fims ae cowded out. Futhemoe, this selection effect is the only souce of welfae gain in the lage maket: vaiety N is independent of L and it will follow that the Hefindahl index in () does not change; the aveage makup in (7) does not change because the uppe-limit of integation is v * p*b/a as b ; and it follows fom Lemmas and 2 that only the fall in fims costs changes the esevation pice and welfae. When poductivity is bounded, howeve, then we shall find that all thee souces of gains fom county gowth opeate: vaiety inceases, the aveage makup falls, and thee is a positive selection effect. To show this, we pefom the compaative statics on (3)-(5). Diffeentiating (6) and simplifying, we obtain:
2 N e b ( v*) d ln Ne ( A) d ln p*, A [ N ( / )], b v* whee v * p*b/a. Re-expess (3) by moving the denominato D( p ) to the left, obtaining: v* ( v) ( v) p * 0 L FNe d g( v) dv. ( v) v a Totally diffeentiating this condition, and substituting fo dln N e fom above, we obtain: A d ln Ne d ln L AB and dln L dln p* AB, (8) whee L ( v *) d [ ( v *) / v *] B N b e F ( v*) D( p)( b ). To give the intuition fo these esults, conside the fee enty condition (3). The makups appeaing in the numeato of this expession ae inceasing as the esevation pice ises o maginal cost falls, '( v) 0, and likewise fo the Lene index [ ( v) ] / ( v). The ising makup follows fom the fact that the demand elasticity in inceasing in pice, as noted ealie. So as the esevation pice falls, the expected makup in the numeato of (3) falls elative to the integal in the denominato. It follows that N e ises less than popotionately with L. That is shown in the fist esult in (8), whee it is immediate that A > 0 fo b <, while B > 0 fo b < because the Lene index takes on its highest value at the uppe bound v*, so that: v* ( v) ( *) ( ) d g( v) L v L v v dv N F ( v*) F ( v) v* ( v') d g( v') dv' v' The inequality states that the highest Lene index exceeds its aveage, and then the equality follows diectly fom the fee enty condition (3) and ensues that B > 0. e.
22 As poductivity is unbounded and b, then A, B 0, and so in that case we have d ln N d ln L, as asseted in Poposition 2(b). Fo unbounded poductivity we also have that e the esevation pice changes by d ln p* d ln L / in (8) and, as discussed above, this change is puely due to the dop in the aveage of fim costs, i.e. the self-selection of moe efficient fims in the lage maket. When poductivity is bounded, the esevation pice falls by less than dln L/, as shown by (8) with A, B > 0. That means that the inceased selection of fims is offset. We conclude that county gowth leads to two opposing effects on poduct vaiety N: enty of fims N e ises less than popotionately with L; but also the esevation pice falls by less, so the inceased selection is offset. It tuns out that this second effect dominates so that poduct vaiety N ises with L. These vaious esults ae summaized as: Poposition 3 Unde Assumptions and 2, an incease in county size L unde fictionless tade leads to: (a) when b =, then p* falls only due to the dop in the aveage of fim costs, with the Hefindahl index H fixed; (b) when b <, then vaiety N ises, the Hefindahl falls, and the aveage of fim costs, makups and the weighted-aveage pice tem in () all fall; (c) the popotional welfae gain when b < is less than that with b =. Pat (a), with an unbounded Paeto distibution, has aleady been discussed above and the constant Hefindahl is shown in the Appendix. The ise in poduct vaiety with bounded Paeto, in pat (b), has also been motivated by the compaative statics above and the falling Hefindahl is shown in the Appendix. The fact that the aveage of fim costs and makups both fall follows quite easily fom those tems in (7): as the esevation pice falls then so does v* p*b/a, and
23 so we ae excluding the highest makup tem ( v*) in (7); but because p* appeas explicitly within the integal of costs, we ae also educing the aveage of fim costs, as confimed in the Appendix. The hadest piece to pove is that the shae-weighted mean pice tem appeaing in () also falls as county size inceases, which togethe with the dop in the esevation pice, ensues that the epesentative consume gains in the lage county. Pat (c) shows that despite the fact that all thee souces of gains fom tade opeate in the bounded Paeto case, the total popotional gains fom tade ae smalle with bounded than with unbounded Paeto. This esult follows fom ou welfae decomposition in Lemma and the compaative statics above. With b =, we found that d ln p* d ln L / and the Hefindahl index is fixed, so it follows immediately fom the fist line of () that the incease in welfae is dln L/. But with b <, we found above that d ln p* d ln L /, and we also confim in the Appendix that the Hefindahl index is falling as vaiety inceases. Fo both easons, if follows that the fictionless tade leads to an incease in welfae that is less than dln L/, obtained in the unbounded case. The esult in pat (c) is elated to that in Melitz and Redding (203), who focus on the CES case only. They show that, povided thee ae fixed costs of expoting, then the gains fom tade with heteogeneous fims exceed that with homogeneous fims as in the Kugman (980) model. Homogeneous fims ae an exteme case of bounded Paeto whee thee is a mass point at a single poductivity. So Melitz and Redding (203) ae compaing any poductivity distibution fo heteogeneous fims (including bounded o unbounded Paeto) with a degeneate distibution with a single mass point. In compaison, we find that even without fixed costs of tade, the gains fom fictionless tade with unbounded Paeto exceed those with bounded Paeto fo the QMOR
24 class of pefeences. So we ae compaing the unbounded Paeto case to any bounded Paeto (but not including the degeneate case b =, uled out in Assumption 2). With this diffeence in ou compaisons undestood, the spiit of ou esults ae simila: having a geate spead of poductivities leads to highe popotional gains fom tade. That is an especially supising esult in ou context because by esticting the ange of poductivities we give scope fo additional souces of gains fom tade due to poduct vaiety and educed makups that do not opeate with the unbounded Paeto distibution. We have found with unbounded Paeto that these additional souces of gains necessaily educes the self-selection of moe efficient fims, so that the total popotional gains ae lowe. This esult can be usefully compaed to the fomula fo welfae gains found by ACDR, which emphasizes the shae of total consumption puchased fom the domestic maket. Denoting that faction by, the domestic labo foce by L, and the wold labo foce by L L, then = in autaky and = L / Lwith fictionless tade. With gowth it follows that d ln d ln L 0. Applying ou esult above that the welfae gain is d ln p* d ln L / with unbounded Paeto, but smalle with bounded Paeto, we have theefoe poved: Coollay The gain fom fictionless tade equals d ln / 0 with an unbounded Paeto distibution, but is stictly less than this amount with a bounded Paeto distibution fo poductivity. Ou calculation of the gains fom tade goes beyond ACDR by allowing fo changes in poduct vaiety and makups unde a bounded Paeto distibution, even though it is thei own fomula (obtained with unbounded Paeto) that becomes the uppe bound.
25 5. Vaiable Tade Costs We now allow fo vaiable costs of tade, but fo simplicity, will suppose that the tading counties ae symmetic. We shall let C 2 denote the numbe of (identical) counties in the wold, but due to tade costs, each county does not necessaily tade with all othes. We numbe counties by thei poximity to an expote, so c = denotes the local maket, c = 2 denotes the next closest maket, etc. In equilibium we allow fo tade with whole counties o a faction of a county, as explained below. We shall assume the following stuctue of tade costs: Assumption 3 Numbeing counties by thei poximity to an expote, deliveing one unit to county c means that ( c) 0 units must be sent, with 0, 0 and c C. c These costs apply onto to coss-bode tade, while local sales (c = ) have. Notice that numbe of counties c that a nation is tading with plays the same ole in Assumption 3 as distance does in an empiical specification of vaiable tanspot costs, while 0 plays the same ole as a bode effect, i.e. the exta amount that must be sent egadless of distance. We can biefly povide a mico-stuctue that justifies the tade costs descibed in Assumption 3. Suppose that counties ae located evenly on a cicle of cicumfeence C 2, with the capital city at the cente of each county. By constuction, the capitals ae distance ½ fom each bode, as shown in Figue fo a cicle of cicumfeence C 4. We assume that impoted goods much each the capital city (e.g. an aipot) befoe being costlessly dispesed thoughout the county. Then letting dist denote the distance a good tavels to each the bode of an impoting county, the good tavels (½+dist) to each the capital city. We assume that the
26 Cicumfeence = C = 4 indicates a bode Tading edge in county 3 distance /2 Capital in county /2 distance No tade with county 4 Tading edge in county 2 Figue : Geogaphy of Tade with Fou Counties vaiable tade costs ae 0 ½ dist. Since goods can ente fom the bode on eithe side, if dist = then the impote is tading with ( c ) 2 counties (not counting itself), while if dist = 2 then the impote is tading with ( c ) 4 counties (not counting itself). Tade with a faction of a county is also allowed, as illustated Figue whee county is tading with a faction of the counties on eithe side up the tading edge. In geneal we have ( c ) 2dist so that dist ( c ) / 2. Substituting this into the fomula fo tade costs we obtain: 0 ½ dist 0 ½ c 0 c ( c) [ ( ) / 2] 2. Absobing 2 into the bode costs 0, we obtain Assumption 3.
27 Equilibium Conditions With Assumption 3, we can eadily solve fo the numbe C of counties that each nation tades with in the symmetic equilibium. The most efficient fim in any county has maginal labo costs of a/b to poduce one unit of output. Nomalizing the wage at unity in evey county, in equilibium the maginal cost of poducing enough to delive one unit to the most distant county C will just equal the esevation pice in that county: a 0 C p*, fo C C. (9) b This equilibium condition povides a vey simple elation between the bode effect 0 and the equilibium numbe of tading patnes. Of couse, changes in the tade costs 0 will also affect the esevation pice in (9), so we will need to specify all the equilibium conditions to account fo the endogenous esponse of both C and p*. Note that if the tade cost 0 ae sufficiently close to unity and is close enough to zeo to have 0 C ( a / b) p*, then the most efficient fim fom each county sells to evey maket, so that C C. To wite the othe equilibium conditions with tade, we evisit the change in vaiables intoduced fo the autaky economy. When a fim is selling to a foeign county, we let v p * /( a / ) denote the atio of the esevation pice to the maginal costs inclusive of the vaiable tade costs. It follows that av / p *, so that fom (2): v p * p * g( ) d d dv g( v) dv b b a a. (20) Fom the final expession in (20), we see that highe tade costs implies a lowe density of fims in any inteval dv, which shows how the tade costs affect the extensive magin of expoting fims. But in contast to the unbounded Paeto case, tade costs now also affect the intensive magin of expotes, and of the highest-poductivity expote in paticula.
28 The uppe bound fo v when selling to the domestic maket is still denoted by v* bp * / a, and the uppe bound when selling to a foeign county c is: v * / ( c) bp * / a( c). (2) With unbounded poductivity, b, the atio of esevation pice to maginal costs fo foeign fims inclusive of the vaiable tade costs is in the ange [,+), the same as fo home fims. So thee is no diffeence in the distibution of maginal costs and pices chaged by home and foeign fims: both counties have fims with essentially zeo costs, chaging an infinite makup, and fims with maginal costs equal to the esevation pice, with zeo makup. But with bounded poductivity, we see fom (2) that the atio of the esevation pice to maginal costs is in the ange v/ [, bp */ a ), which depends on the esevation pice and tade costs. Now the pice of the highest poductivity fim is affected by tade costs, and we efe to this as an impact on the intensive magin of the highest poductivity fim. We continue to let N denote the total mass of poducts available to the epesentative consume in each county, so this notation fom section 2 stands. But in section 3, dealing with the autaky economy, we peviously let N e denote the mass of enteing fims, while N was the mass of suviving fims. With tade we need to intoduce a new notation fo the mass of fims in a single county, so we now let M e denote the mass of enteing fims in a single county, and M denote the mass of suviving fims. These ae elated by the equilibium condition (4), ewitten using this new notation as, v* p* p* M M g( v) dv Me G( v*). a a (22) e Conditional on selling at home, the pobability of fims in the inteval dv selling to county c is then obtained by dividing (20) by the final tems in (22):
29 [ p * / a( c)] g( v) ( c) g( v) dv dv. ( p * / a) G( v*) Gv ( *) The total mass of poducts N available within a county is obtained by stating with the mass M available in each county, and then integating ove the conditional density above: C v*/ ( c) ( c) g( v) N M dvdc Gv ( *) ( ) C ( C )( v*) M 0, ( v*) ( v*) (23) whee C ( ) ( C ) / ( ) is the Box-Cox tansfomation of C. 0 We see fom (23) that tade costs have a diect negative effect on the mass of poducts available in a county though 0, and also an indiect effect though the esevation pice; both of these channels eflect the extensive magin of expoting fims, using the conditional density above. In addition, changes in tade costs have two futhe effects: though changing the ange of counties C that ae expoting to each destination; and though changing the mass of domestic poducts M. While we will need to take into account all these effects, fo the moment just concentate on the diect impact of tade costs on the mass of expoting fims and counties, holding fixed the mass of domestic poducts M and also p*. The effect of changing tade costs on the numbe of tading patnes C can be obtained quite easily fom the equilibium condition (9). Using that condition to solve fo C, substituting the esult into (23), and diffeentiating with espect to tade costs while holding M and p* fixed, we can obtain: ( ) d ln N MC C ( v*). dln N [ ( v*) ] 0 M, p* 0 The esult in (23) is obtained by fist integating ove v, obtaining ( c) G[ v*/ ( c)]/ G( v*) ; then using the Paeto distibution and tade costs in Assumptions 2 and 3; and then integating ove tading patnes c.
30 This expession shows the patial effect of declining tade costs on expanding the ange of available poducts, though the extensive magin of expoting fims and counties. The esult is inceasing in the numbe of tading patnes C, because C ( ) in inceasing in C egadless of the sign of. In othe wods, the geatest impact of educing tade costs on poduct vaiety comes when a county is aleady tading with the most patnes ( C C), so the gain in vaiety comes exclusively fom expanding the extensive magin of expoting fims athe than by expanding the ange of expoting counties. Retuning to the full equilibium conditions, they ae (9) fo the numbe of tading patnes, (22) fo the mass of domestically poduced goods, and (23) fo the mass of poducts available to consumes. We still need the fee enty condition analogous to (3) in the closed economy, which is now witten as: F LJ ( v) ( v) d ( v) v M J e d ( v) v, (24) with, v* C v*/ ( c) p* p* a a ( c) J [ f ] f ( v) g( v) dv f ( v) g( v) dvdc. (25) domestic maket expot makets To move fom the fee enty condition in the closed economy to the open economy in (24), we have added the integals ove tading patnes so that expected pofits ae computed ove domestic sales plus expots. We have intoduced the notation J [ f] as a functional depending on, fom the tade costs, that integates any function f(v) ove the densities of fims selling in the domestic and all expot makets. In each case, we use the density of these fims given by (2) and (20), including the tems (p*/a) and (p*/a(c)) appeaing in these densities, eflecting the pobability of selling domestically o expoting. Because these pobabilities detemine the mass