Measuring the Gains from Trade under Monopolistic Competition

Transcription

1 Preliminary and Incomplee draf Measuring he Gains from Trade under Monopolisic Compeiion by Rober C. Feensra Universiy of California, Davis and NBER April 2009 Absrac Three sources of gains from rade under monopolisic compeiion are: (i) new impor varieies available o consumers; (ii) enhanced efficiency as more producive firms begin eporing and less producive firms ei; (iii) reduced markups charged by firms due o impor compeiion. We show how each of hese gains can be measured. The firs wo sources of gains are analogous o new goods in a CES uiliy funcion for consumers or a consan-elasiciy ransformaion curve for he economy, respecively. Alernaively, he firs and hird sources of gain can be measured using a ranslog ependiure funcion for consumers, which in conras o he CES case, allows for finie reservaion prices for new goods and endogenous markups. Prepared as a Sae of he Ar lecure for he Canadian Economics Associaion meeings, Torono, May 29-30, 2009.

2 . Inroducion One of he grea achievemens of inernaional rade heory in he las hree decades is he incorporaion of he monopolisic compeiion model. The need o include increasing reurns o scale in rade heory was recognized as early as Graham (923; see also Ehier 982), and in he Canadian cone, by Easman and Sykol (967) and Melvin (969). Sill, i was no unil he formalizaion of he monopolisic compeiion model by Dii and Sigliz (977), in parallel wih Spence (976) and Lancaser (979), ha a se of global equilibrium condiions ha avoided he problems of large firms and muliple equilibria could be developed. Tha se of equilibrium condiions was firs wrien down by Krugman (979, 980, 98). 2 There is no doub ha hese developmens have had imporan policy implicaions. For eample, he simulaion resuls of Harris (984a,b) demonsraed large gains o Canada from free rade wih he U.S., and were very influenial in convincing policy makers o proceed wih he Canada-U.S. free rade agreemen in 989; ha agreemen in urn paved he way for he Norh American free rade agreemen in 994. Subsequen empirical work for Canada by Trefler (2004), as well as Head and Ries (999, 200), confirmed he efficiency gains for Canada due o opening rade, hough no in he manner prediced by Krugman s work. Bu a comprehensive empirical assessmen of he gains from rade under monopolisic compeiion has no ye been made. The goal of his paper is o describe how hese gains can be measured, using mehods ha draw heavily on dualiy heory from Diewer (974, 976). The monopolisic compeiion model predics hree sources of gains from rade ha are The muliple equilibria problem due o increasing reurns was described by Chipman (965). Ehier (979) poined ou ha his is avoided in a model wih differeniaed inermediae inpus, leading o a form of eernal (or inernaional ) economies of scale. More recenly, Grossman and Rossi-Hansberg (2008) argue ha a careful specificaion of he marke srucure allows muliple equilibria o be avoided even wih eernal economies. 2 See also he early conribuions of Dii and Norman (980, chaper 9), Lancaser (980) and Helpman (98); hese various approaches were inegraed by Helpman and Krugman (985).

3 2 no presen in radiional models. Firs, here are he consumer gains from having access o new impor varieies of differeniaed producs. Those gains have recenly been measured for he Unied Saes by Broda and Weinsein (2006), using he mehods from Feensra (994), as described in secion 2. Their approach assumes a consan elasiciy of subsiuion (CES) uiliy funcion for consumers, in which case he impor varieies are analogous o new goods in he uiliy funcion. We show ha he gains from rade depend on he impor share and he elasiciy of subsiuion. The eension of he monopolisic compeiion model o allow for heerogeneous firms, due o Meliz (2003), leads o a second source of gains from he self-selecion of more efficien firms ino epor markes. This aciviy drives ou less efficien firms and herefore raises overall produciviy. We argue ha his self-selecion can sill be inerpreed as a gain from produc variey, bu now on he epor side of he economy raher han for impors. Surprisingly, he consumer gains from new impor varieies do no appear in his case, because hey cancel ou wih disappearing domesic varieies. This finding, demonsraed in secion 3, helps o eplain he heoreical resuls of Arkolakis e al (2008a), where he gains from rade depend on he impor share bu are oherwise independen of he elasiciy of subsiuion in consumpion. Raher, he gains come from he producion side of he economy, where he self-selecion of firms leads o a consan-elasiciy ransformaion curve beween domesic and epor varieies, wih an elasiciy depending on he Pareo parameer of produciviy draws. Third, he monopolisic compeiion model also allows for gains from a reducion in firm markups due o impor compeiion. This hird source of gains was sressed in Krugman (979), bu has been absen from much of he laer lieraure due o he assumpion of CES preferences, leading o consan markups. In secion 4, I summarize my curren research wih David

4 3 Weinsein (Feensra and Weinsein, 2009), ha shows how a ranslog ependiure funcion leads o racable formulas for he gains from produc variey and he pro-compeiive effec of impors on reducing markups. Conclusions are given in secion Consumer Benefis from Impor Variey We sar wih he consumer gains from impor variey. From a echnical poin of view, measuring he benefis of new impor varieies is equivalen o he so-called new goods problem in inde number heory. Tha has always been a favorie problem of Erwin Diewer s, and arises because he price for a produc before i is available is no observed, so we don know wha price o ener in an inde number formula. The answer given many years ago by Hicks (940) was ha he relevan price of a produc before i is available is he reservaion price for consumers, namely, a price so high ha demand is zero. Once he produc appears on he marke hen i has a lower price, deermined by supply and demand. The fall in he price from is reservaion level o he acual price can be used in an inde number formula o obain he consumer gains from he appearance of ha new good. For he consan elasiciy of subsiuion (CES) uiliy funcion, we immediaely run ino a problem wih implemening his suggesion because he reservaion price for any good is infinie: he demand curve approaches he verical ais as he price approaches infiniy. 3 Bu provided ha he elasiciy of subsiuion is greaer han uniy, hen he area under he demand curve is bounded above, as shown in Figure, where he raio of areas A/B = /( ) is easily calculaed for a demand curve wih elasiciy. The second problem we run ino is how o epress hese consumer gains when here is no jus one bu many new goods available. 3 In Feensra (2006), we show ha an infinie reservaion price leads o a well-behaved limi for he quadraic mean of order r inde number formula of Diewer (976), providing an alernaive proof of Theorem below.

5 4 CES Uiliy Funcion To address his problem, we will work wih he non-symmeric CES funcion, U /( ) ( ) / U(q, I ) aiqi,, () ii where a i > 0 are ases parameers ha can change over ime, and I denoes he se of goods available in period a he prices p i. The minimum ependiure o obain one uni of uiliy is, /() i e(p, I ) b i p,, b i ai. (2) ii For simpliciy, firs consider he case where I - = I = I, so here is no change in he se of goods, and also b i- = b i, so here is no change in ases. We assume ha he observed purchases q i are opimal for he prices and uiliy, ha is, q U ( e / p ). Then he inde number due o Sao (976) and Varia (976) shows us how o measure he raio of uni-ependiures: i i Theorem (Sao, 976; Varia, 976) If he se of goods available is fied a I - = I = I, ase parameers are consan, b i- = b i, and observed quaniies are opimal, hen: w (I) i e(p, I) pi PSV(p, p, q,q, I), e(p, I) p (3) i ii ii where he weighs w i (I) are consruced from he ependiure shares s i (I) p q p q as, si (I) si(i) si (I) si(i) wi(i) ln si (I) ln si (I) ii ln si (I) ln si(i). (4) i i i i

6 5 The numeraor in (4) is he logarihmic mean of he shares s i (I) and s (I) i, and lies in-beween hese wo shares, while he denominaor ensures ha he weighs w i (I) sum o uniy. The special formula for hese weighs in (4) is needed o precisely measure he raio of uniependiures in (3), bu in pracice he Sao-Varia formula will give very similar resuls o using w i 2 i i oher weighs, such as (I) [s (I) s (I)], as used for he Törnqvis price inde. In boh cases, he geomeric mean formula in (3) applies. The imporan poin from Theorem is ha goods wih high ase parameers a i will also end o have high weighs, so even wihou knowing he rue values of a i, he eac raio of uni-ependiures is obained. Now consider he case where he se of goods is changing over ime, bu some of he goods are available in boh periods, so ha I I. We again le e(p,i) denoe he uniependiure funcion defined over he goods wihin he se I, which is a non-empy subse of hose goods available boh periods, I I I. We someimes refer o he se I as he common se of goods. Then he raio e(p, I) / e(p, I) is sill measured by he Sao-Varia inde in he above heorem. Our ineres is in he raio (p, I ) / e(p, I ), which can be measured as follows: e Theorem 2 (Feensra, 994) Assume ha b i- = b i for i I I I, and ha he observed quaniies are opimal. Then for > : /( ) e(p, I ) (I) PSV(p, p,q, q, I), e(p, I) (I) (5) ii where he weighs w i (I) are consruced from he ependiure shares s i (I) p q p q as in (4), and he values (I) and - (I) are consruced as: i i i i

7 6 p ii iqi p ii,ii iqi ( I), = -,. (6) p ii iqi p ii iqi Each of he erms (I) < can be inerpreed as he period ependiure on he good in he common se I, relaive o he period oal ependiure. Alernaively, his can be inerpreed as one minus he period ependiure on new goods (no in he se I), relaive o he period oal ependiure. When here is a greaer number of new goods in period, his will end o lower he value of (I), which leads o a greaer fall in he raio of uni coss in (5), by an amoun ha depends on he elasiciy of subsiuion. The imporance of he elasiciy of subsiuion can be seen from Figure 2, where we suppose ha he consumer minimizes he ependiure needed o obain uiliy along he indifference curve AD. If iniially only good is available, hen he consumer chooses poin A wih he budge line AB. When good 2 becomes available, he same level of uiliy can be obained wih consumpion a poin C. Then he drop in he cos of living is measured by he inward movemen of he budge line from AB o he line hrough C, and his shif depends on he conveiy of he indifference curve, or he elasiciy of subsiuion. Krugman (980) Model Turning o he inernaional rade applicaion, we will suppose ha he uiliy funcion in () applies o he purchases of a good from various source counries i I. Tha is, he elasiciy of subsiuion we are ineresed in is he Armingon (969) elasiciy beween he source counries for impors. We refer o he source counries as providing varieies of he differeniaed

8 7 good, so he gains being measured in (5) are he gains from impor variey. In his case, we can compare he formula in (5) wih he gain from rade obained in he model of Krugman (980), as analyzed by Arkolakis e al (2008a). In paricular, suppose here are any number of counries, where he represenaive consumer in each has a CES uiliy funcion wih elasiciy >. Labor is he only facor of producion and here is a single monopolisically compeiive secor, wih no oher goods. 4 Firms face a fied cos of f o manufacure any good, and an iceberg ranspor cos o sell i abroad, bu no oher fied cos for epors. Then i is well known ha wih profi-maimizaion and zero profis hrough free enry, he oupu of each firm is fied a he amoun: 5 q ( ) f, (7) where is he produciviy of he firm, i.e. he number of unis of oupu per uni of labor. Wih he populaion of L, he full-employmen condiion is hen: L N[(q / ) f ] Nf, (8) which deermines he number of produc varieies produced in equilibrium as N L / f. This condiion holds under auarky or rade, so opening a counry o rade has no impac on he number of varieies produced wihin a counry. The gains from opening rade can be measured by he raio of real wages under free rade and auarky. Wih labor as he only facor of producion we can normalize wages a uniy, so he gains from rade are simply measured by he drop in he cos of living, which is he inverse of (5). The common se of goods are hose domesic varieies ha are available boh in auarky and under rade. Then he Sao-Varia inde P SV is jus he change in he price of he domesic 4 In paricular, we are ruling ou he addiively-separable numeraire good someimes inroduced ino his model o obain a home marke effec; see Krugman (980). 5 See Arkolakis e al (2008a, p. 3).

9 8 varieies, and wih consan markups ha is he change in home wages, which we have normalized o uniy. So he gains from rade are measured by /( ) / ) ( in (5). The denominaor of ha raio reflecs he disappearance of domesic varieies, i.e. hose varieies available in period - bu no in period. As we have shown above, here are no disappearing domesic varieies in his model, so - =. The numeraor measures he ependiure on he domesic varieies relaive o oal ependiure wih rade, or one minus he impor share. The gains from rade are herefore /( ), which is precisely he formula obained by Arkolakis e al (2008b). While his formula is no oo surprising, i will ake on greaer significance when we compare i o he resuls from he Meliz (2003) model, in he ne secion. Broda and Weinsein (2006) measure hese gains from rade for he U.S. They define a good as a 0-digi Harmonized Sysem (HS) caegory, or before 989, as a 7-digi Tariff Schedule of he Unied Saes (TSUSA) caegory. The impors from various source counries are he varieies available for each good. The raio / ) is consruced for each good, using he ( ependiure on new and disappearing source counries. In addiion, hey esimae for each good, using he GMM mehod from Feensra (994), which eplois heeroskedasiciy across counries o idenify his elasiciy. Puing hese ogeher, hey measure /( ) / ) ( for 30,000 goods available in he HS and TSUSA daa. For he TSUSA daa hey used 972 as he base year and measured he gains from new supply counries up o 988, and hen for he HS daa hey used 990 as he base year and measured he gains from new supplying counries up o Aggregaing over goods, hey obain an esimae of he gains from rade for he US due o he epansion of impor varieies, which amoun o 2.6% of GDP in is omied because Wes and Eas Germany unified hen, making comparisons wih laer years difficul.

10 9 Two feaures of Broda and Weinsein s mehods deserve special menion. Firs, by measuring he ependiure on new supplying counries relaive o a base year, hey are following he hypohesis of Theorem 2 ha he common se of counries should be hose wih consan ase parameers. In conras, when counries firs sar eporing goods, i is reasonable o epec ha he demand curve in he imporing counry shifs ou over some number of years, as consumers become informed abou he produc. Broda and Weinsein are allowing for such shifs for new and disappearing counries afer he base year, and all such changes in demand for hese counries are incorporaed ino he erms in Theorem 2. Tha is he correc way o measure he gains from new impor varieies. 7 Second, Broda and Weinsein (2006) did no incorporae any changes in he number of U.S. varieies ino heir esimaion, nor include he U.S. as a source counry in he esimaion of he elasiciy of subsiuion for each good. Tha is he correc approach only under he limied case where he number of U.S. varieies is consan. While ha is rue under our assumpions in he model of Krugman (980), i is cerainly no he case in more general models: we could epec ha increases in impor variey would resul in some reducion in domesic varieies. In ha case, he gains from impor varieies would be offse by he welfare loss from reduced domesic varieies. Tha poenial loss was no addressed by Broda and Weinsein (2006), and we shall begin o address i in he remainder of he paper. 3. Producer Benefis from Oupu Variey While we have so far resriced ou aenion o > in he uiliy and ependiure funcions () and (2), a wider range of values for his elasiciy can be considered. In paricular, if 7 In addiion, counries ha are suspeced of selling a changing range of produc varieies wihin each HS good should be ecluded from he se I, and insead included in he erms.

11 0 0 hen insead of obaining conve indifference curves from () for a fied level of U, we obain a concave ransformaion curve as shown in Figure 3. 8 The parameer U in his case measures he resources devoed o producion of he goods q, i I, and he elasiciy of he i ransformaion curve (measured as a posiive number) equals. This reinerpreaion of () comes from Diewer (976), who uses he general erm aggregaor funcion o refer o uiliy funcions, producion funcion, or ransformaion funcions for an economy. To make his reinerpreaion eplici, when 0 we will denoe is posiive value by, which is he elasiciy of ransformaion. Then we will rewrie () using labor resources L o replace uiliy U, obaining: L /( ) ( ) / a q i i, ai 0, 0. (9) ii The maimum revenue obained using one uni of labor resources, dual o (9), is hen: e(p, I ) i I b i p i /( ), i b a, 0. (0) i Wih his reinerpreaion, Theorem 2 coninues o hold as: /( ) e(p, I ) (I) PSV(p, p,q, q, I), e(p, I) (I) () where he eponen appearing on ( / - ) is now negaive. In oher words, he appearance of new oupus, so ha <, will raise revenue on he producer side of he economy. To undersand where his increase in revenue is coming from, consider he ransformaion 8 Noice ha he range 0 canno be considered, since hen all goods are essenial in (), wih a zero quaniy for any single good resuling in zero for he enire CES aggregae. In ha case he welfare gain from a new good is infinie.

12 curve in Figure 3. If only good is available, hen he economy would be producing a he corner A, wih revenue shown by he line AB. Then if good 2 becomes available o producers, he new equilibrium will be a poin C, wih an increase in revenue. This illusraes he benefis of oupu variey. In Figure 4 we illusrae he same idea in a parial equilibrium diagram, for a supply curve wih consan elasiciy. When he good becomes available for producion, here is an effecive price increase from he reservaion price for producers (which is zero wih a consan-elasiciy supply curve) o he acual price. The gain in producer surplus is area C, and measured relaive o oal sales C+D, we can readily compue ha C/(C+D) = /(+). While his reinerpreaion of our earlier consumer model is mahemaically valid, here is a problem in is applicaion o inernaional rade: he ransformaion curve beween wo oupus is ofen aken o be linear raher han sricly concave. Tha is he case in he Ricardian model, for eample, or in he ransformaion curve (8) in Krugman s (980) model. In ha case, he gains from oupu variey vanish. So he quesion arises as o wheher he sricly concave case we illusrae in Figure 3 has any pracical applicaion? We will now argue ha he case of a sricly concave ransformaion curve is indeed relevan, and in fac, arises in he generalizaion of he monopolisic compeiion model due o Meliz (2003). Meliz assumes ha labor is he only facor of producion, bu he allows firms o differ in heir produciviies. In he equilibrium wih zero epeced profis, only firms above some cuoff produciviy * survive; and of hese, only firms wih produciviies above * * acually epor. We will argue ha he endogenous deerminaion of hese cuoff produciviies leads o a sricly concave consan-elasiciy ransformaion curve beween domesic and epor varieies, adjused for he quaniy produced of each.

13 2 Meliz (2003) model We ouline here a wo counry version of he Meliz (2003) model ha does no assume symmery across he counries. We focus on he home counry H, while denoing foreign variables wih he superscrip F. A home here is a mass of M firms operaing in equilibrium. Each period, a fracion of hese firms go bankrup and are replaced by new enrans. Each new enran pays a fied cos of f e o receive a draw of produciviy from a cumulaive disribuion G( ), which gives rise o he marginal cos of w /, where w is he wage and labor is he only facor of producion. Only hose firms wih produciviy above a cuoff level * find i profiable o acually produce (he cuoff level will be deermined below). Leing M e denoe he mass of new enrans, hen [ G( *)]M e firms successfully produce. In a saionary equilibrium, hese should replace he firms going bankrup, so ha: [ G( *)]Me M. (2) Condiional on successful enry, he disribuion of produciviies for home firms is hen: where g( ) if *, [ G( *)] ( ) (3) 0 oherwise, g ( ) G( ) / is he densiy funcion. Home and foreign consumers boh have CES preferences ha are symmeric over produc varieies. Given home ependiure of wl, he revenue earned by a home firm from selling a he price p( ) is: p( ) r( ) p( )q( ) wl, >, (4) H P

14 3 where q( ) is he quaniy sold and H P is he home CES price inde. The profi-maimizing price from selling in he domesic marke is he usual consan markup over marginal coss: w p ( ). (5) Using his, we can calculae variable profis from domesic sales as r ( ) (w / )q( ) r( ) /. The lowes produciviy firm ha jus breaks even in he domesic marke here saisfies he zero- cuoff-profi (ZCP) condiion: r( *) / wf q( *) ( )f *, (6) where f is he fied labor cos. Noe ha his cuoff condiion for he marginal firm is idenical o wha is obained in Krugman s (980) model, in (7), for all firms. While firms wih produciviies * find i profiable o produce for he domesic * marke, only hose wih higher produciviies * find i profiable o epor. A home eporing firm faces he iceberg ranspor coss of meaning ha unis mus be sen in order for one uni o arrive in he foreign counry. Leing p ( ) and q ( ) denoe he price received and quaniy shipped a he facory-gae, he revenue earned by he eporer is: where p ( ) r ( ) p ( )q ( ) w * L *, (7) F P F P is he aggregae CES price in he foreign counry, and w*l* is foreign ependiure. Again, he opimal epor price is a consan markup over marginal coss: w p ( ). (8)

15 4 The variable profis from epor sales are herefore r ( ) (w / )q ( ) r ( ) /, so he ZCP condiion for he eporing firm is: * r ( ) / wf q * * ( ) ( ) f, (9) where f is he addiional fied labor cos for eporing. Provided ha r ( ) / f r( ) / f, which we assume is he case, hen he cuoff produciviy for he eporing firm will eceed ha for he * domesic firm, *. Then he mass of eporing firms is compued as: M * M( )d M. (20) To close he model, we use he full employmen condiion and also zero epeced profis for any enran. The labor needed for domesic sales for a firm wih produciviy is [ q( ) / f ], and for epor sales is q ( ) / f ], so he full employmen condiion is: * [ [q ( ) / f ] ( ) d * L Me fe M [q( ) / f ] ( )d M, (2) where he disribuion of produciviies condiional on eporing is ( ) g( ) /[ G( )] if * *, and zero oherwise. We can rewrie (2) by muliplying by w, and using he fac ha ( w / )q( ) r( )( ) /, and likewise for eporers, o obain: w wl w M e f Mf M M f Mf M f wl, e e e f M * r( ) ( )d M * r ( ) ( )d where he second line is obained using he definiion of GDP, wih zero epeced profis. I follows immediaely ha here is a linear ransformaion curve beween he mass of enering, domesic and eporing firms, ha is:

16 5 L M f Mf M f. (22) e e To obain furher resuls, we assume a Pareo disribuion for produciviies: G ( ), wih 0. (23) In ha case, i can be shown (see he Appendi) ha he number of enering firms is proporional o he labor force, M e L( ) / f e, which was assumed by Chaney (2008), for eample. So he ransformaion curve beween domesic and epor varieies is furher simplified as: L Mf M f. (24) ( ) The fac ha his ransformaion curve is linear beween he mass of domesic and epored varieies is similar o ha found in he Krugman (980) model, in (7). Bu his fac does no ell us abou he ransformaion curve beween he economy s oupus, because we also need o ake ino accoun he quaniy produced of each variey. In Krugman s model, he quaniy produced by each firm is fied, as in (6), so he ransformaion is also linear in he quaniy produced by any groups of firms. Bu in he Meliz (2003) model, only he zero-profi-cuoff firm has oupu idenical o ha in Krugman s model, and he cuoff produciviy * iself is endogenously deermined. So o deermine he ransformaion curve for he economy, we firs need o deermine he correc measure of oupu used o adjus he varieies M and M. To deermine he appropriae measure of quaniy, i is convenien o inver he demand curve and rea revenue as a funcion of quaniy, so from (4) we obain: r ( ) A d q( ) H, where wl A d P. (25) H P

17 6 We inroduce he noaion A d as shif parameer in he demand curve facing home firms for heir domesic sales. I depends on he CES price inde H P, and also on domesic ependiure wl. Likewise, epor revenue can be wrien as: ( ) Aq ( ) r, where F P iw * L * A. (26) F P Inegraing domesic and epor revenue over firms, we obain GDP: wl A M d * q( ) ( )d A M q ( ) ( ) d. (27) * Thus, in order o measure GDP he mass of domesic and epor varieies are muliplied by he quaniies shown above. Feensra and Kee (2008) demonsrae ha he firs-order condiions for maimizing GDP subjec o he resource consrain for he economy, aking A and A as given, are precisely he monopolisic compeiion equilibrium condiions. So he quaniies appearing in his epression are he righ way o adjus he mass of domesic and epor varieies. We can simplify hese quaniies by noing ha CES demand, combined wih consanmarkup prices in (5), imply ha he quaniy sold equals ~ ) q( ) ( / q( ~ ) for any choice of reference produciviy ~. We follow Meliz (2003) in specifying ~ as average produciviy: and likewise for he average produciviy /( ) ~ ( ) ( )d *, (28) ~ for eporers, compued using * and. I follows ha GDP simply equals ( A M ~ A M ~ d ), using he adjused mass of varieies: ( ) / ( ) / M ~ Mq( ~ ) and M ~ M q ( ~ ). (29)

18 7 To simplify his epression for GDP furher, we noe ha a propery of he Pareo disribuion is ha an inegral like (28) is always a consan muliple of he lower bound of inegraion. Tha is: ~ ( ) /( ) *, (30) as obained by evaluaing he inegral in (28), which is finie provided ha. The cuoff produciviy * is in urn relaed o he mass of firms by [ G( *)]Me M, and using he mass of enering firms M e L( ) / f and he Pareo disribuion, i follows ha: e fe ( *) M. (3) L( ) Gahering ogeher hese resuls, we can use q( ~ ) ( ~ / *) q( *) o compue ha he adjused mass of domesic varieies is: ( ) ~ M f M ~ e M q( *) [( )f*] kf M, * ( ) L where he second equaliy uses (30) and he ZCP condiion q( *) ( )f *, and he hird follows from (3), where k > 0 depends on he parameers, and. Thus, he adjused mass of domesic varieies is an increasing bu nonlinear funcion of he mass M. A similar epression holds for epors, bu replacing f, M, and M ~ wih f, M, and M ~. Solving for M and M and subsiuing hese ino he linear ransformaion curve (24), we obain a concave ransformaion curve beween M ~ and M ~, wih elasiciy 0 ( ) : /( ) ( )( ) ( )( ) /( ) 2 e M ~ L k f f M ~ f, (32)

19 8 where k 2 > 0 again depends on he parameers, and. Summing up, from he Meliz (2003) model we have obained a consan-elasiciy ransformaion curve, wih elasiciy 0, jus like in (9) as we iniially assered. ( ) Our earlier resuls in Theorems and 2 coninue o apply o his ransformaion curve. In paricular, consider he problem of maimizing ( A M ~ A M ~ d ) subjec o his ransformaion curve. This Lagrangian problem leads o he following soluion, analogous o (0): Theorem 3 (Feensra and Kee, 2008) Assume ha he disribuion of firm produciviy in Pareo, as in (23). Then maimizing GDP subjec o he ransformaion curve (32) resuls in e(a d, A ) L, where: w e(a d, A ) k /( ) 2fe A d f ( ) A ( f ) ( ). (33) The funcion e(ad, A ) is he revenue earned wih L = on he ransformaion curve, and equals wages. Noe ha he eponens appearing on he fied coss f and f in (33) are obained ( ) as ( ) ] 0. This epression also appears as he eponen on fied coss [ ( ) in he graviy equaion of Chaney (2008). We can now apply Theorem 2 o compue he gain from rade. Denoing auarky by, he economy is a he corner of he ransformaion curve wih M ~ 0, as illusraed A by poin A in Figure 5. Using o denoe he rade siuaion, under free rade we have A 0 and M ~ 0, as a poin C. We can herefore evaluae he gain from rade as he raio of real wages in rade and under auarky:

20 9 w w / P H / P H A A w w e(a e(a d d d / P d R d wl H / P, A ) P,0) P H H H R d w L P P H H (34) where he firs line follows from wages in Theorem 3; he second line follows from Theorem 2, using he domesic price A d as he common good available boh periods, wih spending on domesic goods in period of R ; and he hird line follows direcly from he d AdM ~ definiion of A d in (25). We use his equaion o solve for he raio of real wages, obaining he resul: Theorem 4 (Arkolakis, e al, 2008a) The gains from rade in he Meliz (2003) model are: w w / P H / P H R d wl R d w L, (35) where he final equaliy is obained because, so ( ). ( ) Noe ha he raio of domesic ependiure R d o oal income w L is equal o one minus he impor share, so his formula is idenical o he gains from rade in he Krugman (980) model, ecep ha we replace he eponen ( ) in ha case wih in (35). This resul is precisely he resul derived by Arkolakis e al (2008a), and remarkably, he elasiciy of subsiuion does no ener he formula a all (ecep insofar as i affecs he impor share). Our

21 20 derivaion gives some inuiion as o where his simple formula comes from. Namely, he movemen from a corner of he ransformaion curve A in Figure 5, wih epors equal o zero, o an inerior posiion like C, gives rise o gains equal o one minus he impor (or epor) share wih he eponen ( ), which is a sraighforward applicaion of Theorem 2 on he producion side of he economy. We migh inerpre hese gains as due o epor variey. These gains are shown in he second line of (34), and reflec he increase in wages due o he produciviy improvemen as he eporing firms drive ou less producive domesic firms. Bu in addiion, his produciviy improvemen drives down prices, and herefore furher increase real wages: ha is shown as we subsiue for he endogenous value of A d, and hereby solve for real wages in (35). Through hese wo channels, he gains equal one minus he impor (or epor) share wih he eponen, which eceeds ( ) ( ) in absolue value. Bu wha abou any furher gain due o impor variey? Now we mus be careful, because he Meliz model leads o he ei of domesic firms and herefore a reducion in domesic varieies, which mus be weighed agains he increase in impor variey. Baldwin and Forslid (2004) argue ha he oal number of produc varieies falls wih rade liberalizaion, whereas Arkolakis e al (2008) show ha i can rise or fall. Bu simply couning he oal number of varieies is no he righ way o evaluae he welfare gains: insead, we need o ake he raio /( ) / ) ( on he consumpion side of he economy, as in Theorem 2. As we now show, his raio urns ou o be uniy: he gains due o new impor varieies are eacly offse for reduced domesic varieies. Therefore, he producion-side gains we have already idenified in Theorem 4 are all ha is available. To obain his resul, we use he CES price inde for he Meliz model:

22 2 P H * p( ) M( )d F* p F ( ) M F F ( )d, (36) where F* denoes he zero-profi-cuoff for he foreign eporers, wih prices p F ( ). This CES price inde is concepually idenical o wha we referred o as he uni-ependiure funcion in (2). The average prices of domesic goods appearing in (36) are: w p( ) M ( )d M ~, (37) * which uses he prices (5) ogeher wih he definiion of average produciviy in (28). When comparing auarky (denoed by ) wih free rade (denoed by ), we need o ake ino accoun he changing price of domesic goods and heir changing variey, as in (37), along wih he fac he all impored goods are new. Applying Theorem 2 gives rise o he following raio of uni-ependiures: P P H H w / ~ R d / w L w ~ / M / M. (38) The firs erm appearing on he righ of (38) is jus he change in he average price of domesic goods, reflecing he change in wages and in average produciviy. The aggregae domesic good is available in boh periods, so he firs erm reflecs he Sao-Varia inde P SV over he common good in Theorem 2. The numeraor of he second erm on he righ is he spending on domesic goods relaive o oal spending in period ; his equals in Theorem 2, or one minus he share of spending on new impored varieies. The denominaor of he second erm is - in Theorem 2, and reflecs he reducion in he number of domesic varieies, M < M -.

23 22 We now show ha M / M = R d / wl in (38), so he reducion in he number of domesic varieies jus cancels wih share of spending on new impored varieies, and here are no furher consumpion gains. This resul is obained from he ZCP condiion for domesic firms, in (6). The second epression appearing in (6) is q( *) ( )f *, which is familiar from he Krugman model see (7). We will combine his wih he firs epression appearing in (6), r( *) / wf, which can be rewrien using he inverse demand curve in (25), o obain: A A d * d q( ) ww * q( ). Using he definiion A H H / d P (wl / P ), we readily simplify his epression as: q( q( ) w ) w * H / P * H / P Now using he ZCP condiion ha q( *) ( )f *, we immediaely obain: w w * H / P * H / P., (39) so ha he increase in real wages reflecs he increase in he ZCP produciviies. From (30) he raio of ZCP produciviies equals he raio of average produciviies, ~ / ~ ), hen comparing ( (38) wih (39) we immediaely see ha M / M = R d / wl, as we inended o show. This finding ha here are no addiional consumpion gains from variey in he Meliz (2003) model, which is implici in Arkolakis e al (2008a), is discussed eplicily by di Giovanni and Levchenko (2009), who argue ha if he disribuion of firm size follows Zipf s Law hen he eensive margin of impors accouns for a vanishing small porion of he oal gains from rade. Their model differs somewha from our discussion above because firms also use differeniaed

24 23 inermediae inpus, bu hey sill assume a Pareo disribuion for produciviies. This assumpion implies ha he disribuion of firms by size follows a power disribuion, which correspond o Zipf s Law as ( ). Tha is he case where hey find ha he eensive margin of impors has a vanishing conribuion o he gains from rade. In comparison, our resuls above are more general because we show ha he eensive margin of impors has a welfare conribuion ha jus cancels wih he reduced eensive margin of domesic goods, and his resul holds for all ( ). These resuls from he Meliz (2003) model obviously challenge he empirical finding of Broda and Weinsein (2008), who reaed domesic varieies as unchanged. In he ne secion, we consider an alernaive framework o CES ha allows for changes in domesic varieies as well as changes in he markups charged by firms. Changing markups have already been inroduced in heory by Meliz and Oaviano (2008), using a quadraic uiliy funcion wih an addiively separable numeraire good, leading o linear demand funcions. As useful as ha framework is, is zero income elasiciies sugges ha in empirical applicaion i is bes suied for parial equilibrium analysis. We will consider insead a ranslog ependiure funcion, which has income elasiciies of uniy and price elasiciies ha are no consan. Before urning o he ranslog case, we conclude by noing ha he gains from rade in he Meliz (2003) model have been esimaed on he producion side of he economy. Inuiively, movemens along he ransformaion curve in Figure 5 due o greaer epor variey will be associaed wih higher GDP and produciviy. Tha hypohesis is srongly confirmed empirically by Feensra and Kee (2008). They analyze 48 counries eporing o he U.S. over , and find ha average epor variey o he Unied Saes increases by 3.3% per year, so i nearly doubles over hese wo decades. Tha oal increase in epor variey is associaed wih a

25 24 cumulaive 3.3% produciviy improvemen for eporing counries, i.e. afer wo decades, GDP is 3.3% higher han oherwise due o growh in epor variey, on average. Tha esimae is greaer han he welfare gains for he U.S. found by Broda and Weinsein (2006), which was ha afer 30 years, real GDP was 2.6% higher han oherwise due o growh in impor variey. Of course, because he U.S. has a low impor share we migh epec o find greaer gains o eporers, bu hese resuls sill demonsrae ha he gains on he producion side of he economy can be subsanial. 4. Translog Ependiure Funcion We urn now o consider a ranslog uni-ependiure funcion. In a monopolisic compeiion model we need o be eplici abou which goods and available and which are no, so le N ~ denoe he maimum number of goods conceivably available, which we rea as fied. The ranslog uni-ependiure funcion (Diewer,976) is defined as: 9 N ~ N ~ N ~ 0 i ln pi 2 ij ln pi ln p j i i j ln e, wih ij = ji and i > 0. (40) Noe ha he resricion ha ij = ji is made wihou loss of generaliy. To ensure ha he ependiure funcion is homogenous of degree one, we add he condiions ha: N ~ i i N ~, and ij 0 i. (4) The share of each good in ependiure is obained by differeniaing (40) wih respec o ln pi, obaining: N ~ s ln p. (42) i i j ij j 9 The ranslog direc and indirec uiliy funcions were inroduced by Chrisensen, Jorgenson and Lau (975), and he ependiure funcion in (40) was proposed by Diewer (976, p. 22).

26 25 These shares mus be non-negaive, of course, bu we will allow for a subse of goods o have zero shares because hey are no available for purchase. To be precise, suppose ha s i > 0 for i=,,n, while s j = 0 for j=n+,, N ~. Then for he laer goods, we se s j = 0 wihin he share equaions (42), and use hese ( N ~ N) equaions o solve for he reservaion prices ~ p j, j=n+,, N ~, in erms of he observed prices p i, i=,,n. Solving for he reservaion prices inroduces a level of compleiy ha did no arise in he CES case, where reservaion prices are infinie: in he ependiure funcion (2), an infinie reservaion price raised o he negaive power ( ) simply vanishes. To solve for finie reservaion prices in he ranslog case, i is essenial o simplify he ranslog by imposing he addiional symmery requiremens: N~ 0, and 0 for i j, N ~ ii ij N ~ wih i, j =,, N ~. (43) I is readily confirmed ha he resricions in (43) saisfy he homogeneiy condiions (4), and also guaranee ha he reservaion prices are finie. Because N ~ is a fied number, (43) simply says ha he mari has a negaive consan on he diagonal, and a posiive consan on he offdiagonal, chosen so ha he rows and columns sum o zero. The resricions in (43) are no familiar from he ranslog lieraure, bu are essenial o solve for reservaion prices for goods no available. Noe ha we have no resriced he i > 0 parameers, hough hey mus sum o uniy as in (4), so here are N ~ free i parameers. 0 In addiion, we have he free parameer 0 in (40) as well as > 0 in (43), so here are a oal of N ~ free parameers in his symmeric ranslog funcion. Tha is he same number of free 0 Feensra (2003) adds an addiional symmery resricion on he i parameers, bu Bergin and Feensra (2009) show ha Theorem 5 below can be obained wihou ha resricion.

27 26 parameers in our non-symmeric CES funcion (), where we allowed for N ~ parameers a i > 0 (possibly changing over ime) along wih he elasiciy >. So in describing he ranslog case as symmeric we are comparing i o he empirical version ha does no use (43); while in describing he CES funcion as non-symmeric we are comparing i o he heoreical version in monopolisic compeiion models ha assumes a i, i =, N ~. In fac, boh he CES funcion in () and he ranslog in (40) have he same number of free parameers, or degree of symmery, which we have chosen o be racable in a monopolisic compeiion framework. The usefulness of he symmeric resricions in (43) is shown by he following resul: Theorem 5 (Feensra, 2003; Bergin and Feensra, 2009) Using he symmery resricions (43), suppose ha only he goods i=,,n are available, so he reservaion prices ~ ~ p j for j=n+,, N are used. Then he uni-ependiure funcion equals: N N N 0 ai ln pi b 2 ij ln pi ln p j i i j ln e a, (44) where: b ii (N ) 0, and bij 0 for i j wih i, j =,,N, (45a) N N i i N N i a, for i =,,N, (45b) i a N ~ in 2 i N N ~ i N 2 i. (45c) Noice ha he ependiure funcion in (44) looks like a convenional ranslog funcion defined over he goods i=,,n, while he symmery resricions coninue o hold in (45a), bu are defined now using he number of available goods N, which can change over ime. As N grows, for eample, we will find ha he price elasiciy of demand also grows (as shown

28 27 below), because goods are closer subsiues. To inerpre (45b), i implies ha each of he coefficien i is increased by he same amoun o ensure ha he coefficiens a i sum o uniy over i=,,n. The final erm a 0, appearing in (45c), incorporaes he coefficiens i of he unavailable producs. If he number of available producs N rise, hen a 0 falls, indicaing a welfare gain from increasing he number of available producs. Theorem 5 is a promising sar owards using he ranslog funcion in monopolisic compeiion models, and shows how he funcional form changes as N grows. For heoreical work, his resul is all ha is needed. Bu for empirical work, he gains from new varieies suggesed by (45c) does no allow for he direc measuremen of welfare gain, because i depends on he unknown parameers i. We now repor resuls from Feensra and Weinsein (2009), who develop an alernaive formula for he welfare gain ha depends on he observable ependiure shares on goods, and can herefore be implemened. Le us disinguish wo periods - and, and coninue o assume ha he goods i=n+,, N ~ are no available in eiher period. The goods {,,N} are divided ino wo (overlapping) ses: he se i I is available in period = -,, and has N goods; wih union I I {,..., N} and inersecion I I. We shall le I I I denoe any non-empy subse of heir inersecion, wih N 0 goods wihin he se I. We need o solve for he reservaion prices p~ i for new goods available only in period, and i p~ for disappearing goods available only in period -. These reservaion prices are again defined by he respecive shares equaling zero, where he share equaions are obained by differeniaing (44) wih respec o ln pi, while making use of (45): s = a ln p ln p i i i, I i, (46)

29 28 N i i ) where a ( is a ime-effec which ensures ha ( a ), and N i I i I i ln p ln p is he average log-price of all available goods in period. I i In general, he Törnqvis price inde is eac for he ranslog ependiure funcion (Diewer, 974), which means ha he raio of he uni-ependiure funcions is measured by: N ln e (si si )(ln pi ln pi ). e 2 (47) i Using his formula, and subsiuing for he reservaion prices for new and disappearing goods, we obain he following resul: Theorem 6 (Feensra and Weinsein, 2009) Le I I I denoe any non-empy subse of goods available in boh periods, and using he symmery resricion (43) and share equaions (46) o solve for he reservaion prices for new and disappearing goods. Then he raio of uni-ependiure funcions equals: ln e (si si )(ln pi ln pi ) V, e 2 (48a) ii i i N (48b) i I i I where, V s s s s, ii 2 i 2 i and he shares si and s i are defined as: s i si s N i, for i I, and = -,. (49) ii To inerpre his resul, noice ha he consruced shares s i apply o he N goods wihin he se I, i.e. a subse of hose available in boh periods. The consruced shares simply ake he

30 29 observed shares s i (which sum o uniy over he se I ) and addiively increase each of hem by he amoun needed so ha hey sum o uniy across he producs ii 2 ( i i i i i I. This ransformaion of shares means ha he erm s s )(ln p ln p ), appearing in (48a), is he Törnqvis price inde defined over producs available in boh periods. The erm V defined in (48b) is herefore he era impac on he eac price inde from having he new and disappearing goods. In he CES case repored in Theorem 2, he welfare impac from a changing se of goods depends on he share of new producs as compared o disappearing producs. Now from (48b) we see ha he welfare impac depends on he sum of squared shares for new and disappearing goods (appearing firs on he righ), and also he sum of shares squared for new and disappearing goods (appearing second on he righ, divided by N ). As new goods become available, wih shares eceeding ha for disappearing goods, hen here will be a welfare gain. Since we epec he squared shares o change less han shares hemselves in response o he addiion of new goods, i follows ha we migh epec he welfare gains o be smaller in he ranslog han in he CES case. This is consisen wih he idea ha he welfare gain under he demand curve, area A in Figure, should be smaller when he reservaion price is finie. However his inuiion is no enirely correc because we canno idenify V as he oal welfare effec of new goods. Raher, new goods can also conribue o lower prices for eising goods by reducing heir markups. In oher words, he ranslog ependiure funcion allows for a pro-compeiive effec of impors ha is enirely absen in he CES case. So impors from new supplying counries can raise welfare from boh he parial welfare effec V and he procompeiive effec: he oal welfare impac would have o sum hese wo effecs.

31 30 Pro-compeiive Effec To illusrae he pro-compeiive effec, we follow Feensra and Weinsein (2009) in wriing he opimal prices for firms as he familiar markup over marginal coss: i pi ci, i where c i denoes he ime-dependan marginal coss, and i is he elasiciy of demand. Since demand equals q E s / p, for given ependiure E, he elasiciy of demand is i i i compued from he share equaion (46) as: ln si (N ) i ln p. (5) i sin Noice ha he elasiciy of demand is inversely relaed o he share of he firm s i. If all firms have he same share hen s i = /N, and he elasiciy becomes (N ), which i increases as he number of available producs rises. This confirms ha goods are sronger subsiues as more of hem become available, as we assered above. Using he elasiciy (5) in he pricing equaion, and aking logs, we obain: ln p i ln c i si N ln. (52) (N ) The pricing equaion in (52) will be he key o idenifying he pro-compeiive effec. Noice ha a reducion in he share s i of good i lowers he markup and price of ha good. Feensra and Weinsein (2009) show ha he U.S. marke share falls for many goods, leading o a fall in markups. In order o cumulae all hese various pro-compeiive effecs we subsiue (52) ino he Törnqvis price inde in (48) o obain:

32 3 s s )(ln p ln p ) (s s )(ln c ln c ) P, (53) ii 2 ( i i i i 2 i i i i ii where, s N / (N ) P i (s s ) ln 2 i i sin / (N ). (54) ii The firs erm on he righ of (53) is a Törnqvis inde defined over he marginal coss of goods available in boh periods, while he second erm P indicaes he change in he average markups on hese goods. We epec hese markups o fall as new producs becomes available, so P < 0, which is he pro-compeiive effec. To summarize, he change in he eac price inde for he ranslog case can be decomposed ino hree erms: ln e (si si )(ln ci ln ci ) V P. e 2 i I (55) The firs erm reflecs he drop in marginal coss for producs supplied in boh periods; he second erm V in (48b) reflecs he parial welfare effec of new and disappearing goods; and he hird erm P in (54) reflecs he change in markups for goods available boh periods. Feensra and Weinsein (2009) implemen he decomposiion in (55) for he Unied Saes, using boh domesic sales and impor sales for each good. Tha is, hey allow consumers o subsiue beween domesic and impor varieies of each good, unlike Broda and Weinsein (2006), wih finie reservaion prices for each variey. Increased shares of impors and reduced U.S. shares can lead o reduced U.S. markups, unlike he CES case. For boh reasons, he ranslog case offers a promising heoreical and empirical framework ha goes beyond he limiaions of he CES case. 5. Conclusions [TO BE COMPLETED]

33 32 Appendi Using L M f Mf M f and he full employmen condiion, we have ha: e e L M Evaluaing hese inegrals: * [q( ) / ] ( )d M [q ( ) / ] ( ) d *, * [q( ) / ] ( )d * q( *) * q( *) * * * q( *) * ( ) * ( ) f, ( ) ( )d ( *) d * where he firs line uses q( ) ( / *) q( *) and he las line uses q( *) / * ( ) f. Likewise, [ q ( ) / ] ( )d f * ( ). ( ) Subsiuing hese in o he full employmen condiion above we obain: L Mf M f, ( ) from which i follows ha M e L( ) / f. e

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