Vertical Specialization and International Business Cycle Synchronization *

Transcription

1 Federal Reserve Bank of Dallas Globalzaton and Monetary Polcy Insttute Workng Paper No. 2 Vertcal Specalzaton and Internatonal Busness Cycle Synchronzaton * Costas Arkolaks Yale Unversty and NBER Ananth Ramanarayanan Federal Reserve Bank of Dallas October 2008 Revsed: September 2009 Abstract We explore the mpact of vertcal specalzaton trade n goods across multple stages of producton on the relatonshp between trade and busness cycle synchronzaton across countres. We develop an nternatonal busness cycle model n whch the degree of vertcal specalzaton vares wth trade barrers. Wth perfect competton, we show analytcally that fluctuatons n measured total factor productvty are not lnked across countres through trade. In numercal smulatons, we fnd lttle dependence of busness cycle synchronzaton on trade ntensty. An extenson of the model to allow for mperfect competton has the potental to resolve these shortcomngs. JEL codes: F4, E32 * Costas Arkolaks, Department of Economcs, Yale Unversty, P.O. Box , New Haven, CT costas.arkolaks@yale.edu. Ananth Ramanarayanan, Research Department, Federal, Reserve Bank of Dallas, 2200 N. Pearl St, Dallas, TX Ananth.ramanaryanan@dal.frb.org. We are grateful to Crstna Arellano and Tmothy Kehoe for valuable advce. For helpful comments, we also thank two anonymous referees, as well as Smona Cocuba, Jonathan Eaton, Patrck Kehoe, Steve Reddng, Olga Tmoshenko and Ke-Mu Y. A prevous verson of ths paper crculated under the ttle Endogenous Specalzaton, Intermedate Goods, and Internatonal Busness Cycles. The vews n ths paper are those of the authors and do not necessarly reflect the vews of the Federal Reserve Bank of Dallas or the Federal Reserve System.

2 I. Introducton In recent emprcal work, several authors have documented a lnk between nternatonal trade and cross-country busness cycle synchronzaton. Frankel & Rose (998) establshed that country pars that trade more exhbt on average hgher correlatons n ther busness cycles, as measured by uctuatons n GDP. However, Kose & Y (200) and (2006) have llustrated what they call a trade-comovement puzzle: standard nternatonal real busness cycle models along the lnes of Backus, Kehoe & Kydland (994) cannot quanttatvely account for the relatonshp between trade and busness cycle comovement. In ths paper, we develop an nternatonal busness cycle model augmented wth vertcal specalzaton -.e., the producton of goods n multple stages spread across countres - and quanttatvely assess ts ablty to generate stronger busness cycle synchronzaton between countres that trade more. In addressng the emprcal facts behnd the trade-comovement puzzle, several authors have suggested that t s not only the volume of trade, but partcular features of specalzaton patterns and ndustral structure assocated wth ncreased trade that lead to busness cycle synchronzaton. Frankel & Rose (998) conectured that ntra-ndustry trade tends to make countres more correlated, whle Kose & Y (200) have suggested that vertcal specalzaton may be the key lnkage that synchronzes busness cycles of countres wth close trade relatonshps. The ntuton s that f closer trade relatonshps are characterzed by tghter lnks n the chan of producton, uctuatons n one economy should be transmtted more to the other. 2 Recent papers that have con rmed ths ndng are Clark & Van Wncoop (200) and Baxter & Koupartsas (2005). 2 Indeed, Ng (2007) nds that drect measures of blateral vertcal specalzaton are related to ncreased busness cycle correlaton, and that ntra-ndustry trade plays no sgn cant role once vertcal specalzaton s taken nto account. In addton, D Govann & Levchenko (2008), usng cross-country ndustry-level data, 2

3 The man nnovaton of our work s to ntroduce producer heterogenety wth two stages of producton nto an nternatonal busness cycle model along the lnes of Backus et al. (994). In ths respect, our model s smlar to Y (2003) and (2009), extended to an envronment wth aggregate uncertanty. In each stage, the degree of specalzaton by each country s endogenously determned. As a result, each country requres nputs from the other to produce nal output. Snce ths lnk s stronger when countres trade a wder range of goods, ths vertcal specalzaton provdes a potental mechansm for the model to generate ncreased busness cycle correlaton wth hgher trade. We rst consder a model wth perfect competton, as n the Rcardan model of Eaton & Kortum (2002). Qualtatvely, ths model generates an ncrease n GDP correlaton wth hgher trade ntensty, but falls short quanttatvely compared to the data. There are two ndngs that explan ths result: one analytcal, and one numercal. The analytcal ndng s that, when standard natonal ncome accountng methods are used to construct real GDP, uctuatons n measured Total Factor Productvty (TFP) n each country depend only on domestc shocks, to a rst-order approxmaton. Hence, trade lnks do not transmt technology shocks across countres drectly nto measured productvty. Changes n real GDP, though, are accounted for by changes n TFP as well as changes n factor nputs; the numercal ndng s that the correlaton across countres of changes n nputs s not su cent to generate substantal correlaton n real GDP, nor a sgn cant dependence of busness cycle correlaton on trade ntensty. These results allow us to dentfy the features of the model that lead standard accountng methods to understate the mpact of trade on nternatonal busness cycle transmsson. Most nd that the correlaton of output n ndustres wth vertcal producton lnkages s more senstve to trade. 3

4 notable of these features s the fact that perfect competton prevents e cency d erences across producers from translatng nto d erences n measured productvty. To address ths ssue, we ntroduce mperfect competton followng Bernard, Eaton, Jensen & Kortum (2003), allowng for markups to vary across producers wth d erent e cency. In ths model, e cency d erences do translate nto measured productvty d erences, and we nd that ths new channel gves the model the potental to generate cross-country correlatons n measured TFP, and thus real GDP, that ncrease wth trade ntensty. Ths paper s most closely related to recent papers that attempt to account for the tradecomovement relatonshp. Bursten, Kurz & Tesar (2008) show that allowng for producton sharng among countres can delver tghter busness cycle synchronzaton. 3 Our results suggest that for lookng at real GDP, relaxng key assumptons such as constant elastcty of substtuton n preferences and perfect competton could be crucal for changng the mplcatons of trade ntensty for busness cycle behavor. Drozd & Nosal (2008) devate from the standard neoclasscal framework and address the lnk between trade and comovement n a model featurng a low short-run prce elastcty of trade coexstng wth a hgh long-run prce elastcty. 4 The rest of the paper s organzed as follows: Secton II. lays out our model and secton III. dscusses the structure of vertcal specalzaton. Secton IV. dscusses the dynamcs of real GDP and measured productvty. Secton V. revews some data on vertcal specalzaton and lnks varous measures to our model. Secton VI. descrbes our numercal experments wth 3 Followng a d erent approach, Huang & Lu (2007) argue that multple stages of producton ncrease busness cycle comovement n the presence of nomnal rgdtes. 4 In related work, Bergn, Feenstra & Hanson (2007) combne a model of outsourcng of producton wth non-ces preferences to account for the varance of output n outsourcng ndustres n Mexco compared to the US. 4

5 perfect competton, and Secton VII. dscusses the extenson wth mperfect competton and varable markups. Secton VIII. concludes. II. Setup of the Model Goods are produced n two stages wth the second stage of producton (producton of nal goods ) usng goods produced n the rst stage ( ntermedate goods ). The presence of Rcardan comparatve advantage, whch we model as n Eaton & Kortum (2002), leads countres to endogenously specalze across a contnuum of goods n each stage. The tme horzon s n nte and dscrete, and perods are ndexed by t = 0; ; : : :. In each country, there are two sectors of producton: a tradeable sector and a nontradeable sector. There s a contnuum of measure one of goods n both stages of tradeable producton. To economze on notaton, except where noted below, we ndex both ntermedate and nal goods n the tradeable sector by!, although an ntermedate good labelled! and a nal good labelled! are dstnct commodtes. We use subscrpts to refer to stages of producton, s = ; 2, and tme perods, and we use superscrpts to refer to countres, ; = ; 2. When a varable has a double superscrpt, the rst ndex refers to the source country and the second refers to the destnaton. Producton of Tradeable Goods Each rst-stage ntermedate nput! can be produced n country usng a constant returns to scale technology combnng physcal captal and labor wth e cency denoted by A tz (!). A t s a country-spec c tme-varyng productvty shock common to all ntermedate goods producers n country, and z (!) s a good-spec c e cency that s constant over tme. 5

6 Output of each ntermedate good! produced by country s gven by: y t (!) = A tz (!) k t (!) `t (!), () where k t (!) and `t (!) denote captal and labor, respectvely, used n the producton of good!, and 2 (0; ). The mnmum unt cost of producng ntermedate good! n country s: q t (!) = q t z (!), (2) where q t s the cost of the nput bundle scaled by aggregate productvty n the rst stage: q t = (r t) (w t) A t ( ). (3) Here, wt denotes the wage and rt s the rental rate of captal n country. Purchasers of ntermedate goods n each country buy each good from the source country that o ers the lowest prce after accountng for trade costs. We adopt the standard ceberg cost formulaton, so that delverng one unt of any stage-s good from country to country requres shppng s unts, wth s = and s. Therefore, the prce at whch country purchases ntermedate good! s gven by: p t (!) = mn q t (!) : = ; 2. (4) 6

7 The technology for producng output of nal good! s: 5 y 2t (!) = A 2tz 2 (!) k 2t (!) `2t (!) ( ) m t (!;! 0 ) d! 0, (5) where k2t (!), and `2t (!) denote captal and labor used n the producton of nal good!, A 2t and z2 (!) are tme-varyng and good-spec c e cency for nal goods, respectvely, and m t (!;! 0 ) s the use of ntermedate good! 0 n the producton of nal good!. The parameter > 0 s the elastcty of substtuton between d erent ntermedate nputs. We de ne the aggregate composte of ntermedate goods used n producton of nal tradeable goods as: MT t = m t (!;! 0 ) d! 0 d!, (6) where the subscrpt T dstngushes varables for the tradeable sector from correspondng varables for the nontradeable sector, dscussed below. The unt cost of producton for nal good! s gven by q 2t (!) = q 2t z 2 (!), (7) where q 2t = A 2t (r t) (w t) ( )! P t. (8) 5 Unless otherwse noted, ntegraton s over the entre set of goods n the relevant stage of producton. 7

8 Here, the term P t denotes the prce for the ntermedate goods bundle: Pt = p t (!) d! =( ). (9) Smlar to rst-stage goods, nal goods are also purchased n each country from the source o erng the lowest prce adusted for trade costs. Thus, the prce of nal good! n country s: p 2t (!) = mn q 2t (!) 2 : = ; 2. Fnal goods are purchased by households to form composte consumpton and nvestment of tradeable goods, whch we denote X T t = x T t (!) d!, (0) C T t = c T t (!) d!. () Therefore, the prce ndex for nal tradeable goods has the same form as (9), and s gven by: P2t = p 2t (!) d! =( ). (2) In the next secton, we mpose further structure on the dstrbuton of good-spec c e cences, and derve how these assumptons shape the pattern of trade and the prces pad for goods n each country. 8

9 Technology structure for tradeable goods We follow the probablstc representaton of Eaton & Kortum (2002) for good-spec c e - cences. For each country, stage s, and good!, z s (!) s drawn from a Fréchet dstrbuton wth cumulatve dstrbuton functon: F s (z) = e T s z, (3) where Ts > 0 and >. E cency draws are ndependent across goods, stages, and countres. The probablty that a partcular stage-s good! can be produced n country wth e cency less than or equal to z s gven by Fs (z). Snce draws are ndependent across the contnuum of goods, Fs (z) also denotes the fracton of stage-s goods that country s able to produce wth e cency at most z. As n Eaton & Kortum (2002), the cross country d erences n Ts re ect absolute advantage n the producton of goods n each stage: a country wth a hgher Ts draws e cences for all goods n a gven stage from a better dstrbuton. The parameter determnes the dsperson of e cency draws, and hence governs heterogenety across goods and leads to comparatve advantage wthn each stage of producton. In addton, snce the terms Ts may d er for s = ; 2, our technologcal structure allows for comparatve advantage across stages, determned by the rato T=T 2 across countres. A country wth a hgher T=T 2, for example, s relatvely more productve n ntermedate nputs. As Eaton & Kortum (2002) show, the dstrbuton of prces of stage- goods that country o ers to country s equal to G st (p) = e T s(q st) p, where q st = qst s. The probablty that country s able to purchase a certan good at prce below p s the probablty that ether 9

10 source country o ers country a prce below p, that s, the probablty that mn q st; q st p. Therefore, the overall dstrbuton of prces of stage-s goods avalable n country s G st (p) = e st p, (4) where st T s q st + T s q st. (5) Snce the only dmenson of heterogenety across goods n a gven stage s e cency, we can aggregate across goods by aggregatng across e cency levels or across prces. Wth ths transformaton, the prce ndex for stage-s goods can be wrtten as (P st) = R 0 p dg st (p), whch, usng the dstrbutons (4) s: =( ) + P st = = st, (6) where s the Gamma functon, (a) = R 0 t a e t dt. The probablty that country buys a certan good from country, or alternatvely the fracton of goods that country buys from country, s gven by the probablty that q st mn qst; q st. As Eaton & Kortum (2002) show, ths s equal to: st = T s q st st. (7) In addton, because the dstrbuton of stage-s goods actually purchased by country from country s equal to the overall prce dstrbuton G st, the fracton st of goods purchased 0

11 from country s also equal to the fracton of country s total expendtures on stage-s goods that t spends on goods from country. Producton of nontradeable goods Each country also produces a nontradeable good accordng to the followng technology: Y Nt = A Nt KNt L Nt ( ) m Nt(!) d!, (8) where m Nt (!) s the quantty of ntermedate good! purchased for use n the producton of the nontradeable good. The composte ntermedate nputs used n nontradeable producton s MNt = m Nt(!) d!. (9) The unt cost of producng nontradeable goods s P Nt = A Nt (r t) (w t) ( )! P t. (20) Households Each country s populated by an n ntely-lved representatve household whch values sequences of consumpton of tradeable nal goods, consumpton of nontradeable goods, and lesure, accordng to the followng preferences: E X t t=0 CT t C Nt L " t ( "),

12 where L denotes the fracton of tme devoted to labor suppled n domestc producton, C T denotes the tradeable composte consumpton de ned n () and C N denotes consumpton of nontradeable goods. E denotes the expectaton over the entre tme horzon, and 2 (0; ) s the household s dscount factor. The parameter 2 (0; ) determnes the fracton of aggregate expendture on tradeable goods and " > 0 determnes the household s elastcty of substtuton across dates and states of the world. The household also purchases tradeable and nontradeable nvestment goods, XT t and XNt, that are bundled to augment the aggregate captal stock K t: K t+ = X T t X Nt + ( )K t, (2) where s the deprecaton rate of captal and the fracton of nvestment expendtures are spent on tradeable goods. We assume that the household spends a constant share of ncome on nontradeable consumpton and nontradeable nvestment to avod needng excessvely hgh transportaton costs to generate realstc levels of trade. The household receves ncome from sellng labor servces and rentng captal to rms. We assume that countres do not trade nancal assets - that s, trade n goods s balanced n each perod - so the total expendture of each country s constraned by ts ncome: 6 P 2t C T t + X T t + P Nt C Nt + X Nt w t L t + r tk t. (22) 6 The ndngs n Heathcote & Perr (2002) suggest that models wth nancal autarky generate nternatonal comovement that s closer to the data than models wth complete nancal markets. Models wth complete markets have a strong rsk-sharng channel from whch we wsh to abstract to solate the mechansms n our model. 2

13 Market Clearng and Equlbrum Let st denote the set of stage-s goods whch country produces. We de ne total captal stocks and labor n each country = ; 2 and n each tradeable stage s = ; 2 as K st = L st = kst(!)d!, (23) st `st(!)d!. st At each date t, the total supply of captal and labor by households equals the demand from producers, K t = K t + K 2t + K Nt, (24) L t = L t + L 2t + L Nt. We use the expendture shares de ned n (7) to wrte aggregated market clearng condtons for goods n each stage and sector. Country s expendture on goods from country equal a fracton st for stage-s goods. Snce producers are perfectly compettve, the ncome pad to factors n the rst stage equals the value of rst-stage producton n country, and the ncome pad to factors n producton of nal goods equals a fracton of the value of nal goods producton. Therefore, for tradeable goods, w tl t + r tk t = tp t MT t + MNt + tp t M T t + M Nt, (25) w tl 2t + r tk 2t = 2tP 2t CT t + XT t + 2tP 2t C T t + X T t, 3

14 and for nontradeable goods, w tl Nt + r tk Nt = P Nt C Nt + X Nt, = ; 2. (26) An equlbrum conssts of stochastc processes for prces and quanttes such that households utlty s maxmzed subect to (2) and (22), producers costs are mnmzed, and the market clearng condtons for goods and factor nputs are sats ed each perod. We solve for an equlbrum n terms of consumpton, nvestment, and ntermedate nput expendtures, captal and labor supples, factor prces, and composte prce ndces, wth quanttes aggregated across the contnuum of goods n the case of the two tradeable stages. We use a standard lnear approxmaton method to solve for recursve decson rules n the neghborhood of the model s determnstc steady state. 7 III. Vertcal Specalzaton Structure We now bre y compare the vertcal specalzaton structure n our model to related ways of specfyng ths feature. We wll focus our comparson on three recent contrbutons, Eaton & Kortum (2002), Bursten et al. (2008), and Y (2009). Eaton & Kortum (2002) assume that each good s used as both an ntermedate and a nal good. 8 Intermedate nputs are combned to produce output n the same way as n our second stage technology, summarzed n (5). Ther modelng of ntermedate nputs s smple n that t does not ncorporate any noton of specalzaton nto d erent stages of 7 Smlar approaches are used by Ghron & Meltz (2005) and Nakno (2008), n models wth a contnuum of goods. 8 A smlar ntermedate nputs structure has been explored by Krugman & Venables (995). 4

15 producton. Specalzaton n d erent stages does exst n Bursten et al. (2008) who employ an Armngton-aggregator structure n whch ntermedate nputs from d erent countres are combned wth a xed elastcty of substtuton nto a nal good. However, the pattern of specalzaton s xed n that countres do not specalze n the producton of the goods n whch they are most e cent. An alternatve model wth endogenous specalzaton s the one used n Y (2003) and (2009), n whch there are two stages, but each second-stage good uses exactly one rst-stage good as an nput, not a bundle of all the goods. Our ntermedate nput structure can be consdered as a combnaton of Eaton & Kortum (2002) and the two-stage structure of Y (2009), and our assumpton that all nal producers use the same bundle of nputs allows us to solve the model usng the results n Eaton & Kortum (2002). Our model ncorporates n a parsmonous way two of the mportant features of Y s setup - endogenous specalzaton and trade n ntermedate nputs - that could create addtonal lnkages n the producton processes of countres through ncreased trade. There are two d erences between our setup and that n Y (2009). Frst, n our model, the degree of vertcal specalzaton depends on trade barrers, but s the same for all goods. In Y (2009), only a fracton of goods are produced wth vertcal specalzaton, and ths fracton depends on trade ntensty. Second, n Y (2009), for the goods that are vertcally specalzed, a decrease n the prce of mports would make both the nput and output prce n the second stage drop by a proportonal amount. Ths mechansm generates large changes n trade shares for goods that are vertcally specalzed, or goods that become vertcally specalzed as a result. In our model, snce each good uses the same composte of ntermedate nputs, a drop n mport prces causes the nput prce ndex and the output prce of any nal good to 5

16 drop by only a fracton of the mport prce drop, and the elastcty of trade shares to mport prces s the same for all goods. IV. Real GDP and Productvty n the Model Compared to standard nternatonal busness cycle models, our model contans addtonal potental channels of nternatonal transmsson of uctuatons through trade. The presence of vertcal specalzaton lnks countres producton processes, and the degree of specalzaton ncreases wth trade ntensty, so that countres that trade more have producton processes that are more closely lnked. In standard nternatonal busness cycle models n whch all value-added s created n a sngle stage of producton, technology shocks are transmtted across countres through the mperfect substtutablty of goods: a country that receves a favorable shock demands more mports as well as domestc goods for consumpton and nvestment, so ts tradng partner produces more. In our model, there s an addtonal e ect from havng two stages of producton: a country wth a favorable shock o ers ts tradng partner lower prces for ntermedate nputs, whch makes producton of nal goods more e cent, as the same amount of output can be produced wth lower nput expendtures. If the degree of vertcal specalzaton ncreases wth trade ntensty, then there s more potental for a country to bene t from foregn technology mprovements n ths way. Addtonally, due to endogenous specalzaton, countres produce the goods n whch they are most e cent, so that foregn productvty mprovements cause ncreased specalzaton nto more e cently produced goods at home. To assess the potental mpact of these mechansms on busness cycle comovement, we need to construct a measure of real value added, or GDP. In order to compare the model s 6

17 predctons to data, we construct a measure of real GDP from the model s output comparable wth standard natonal accountng methods. In our model, snce there s a contnuum of goods and changng trade patterns, we have some choce regardng how to compute aggregate quanttes. Snce our results depend heavly on the method of GDP measurement, we dwell on ths pont a bt here: n the rst subsecton below, we explan the de ntons of aggregate statstcs we use, and n the second subsecton, we derve some of the mplcatons of our choces for uctuatons n aggregate productvty. Natonal Accounts Statstcs n the Model We construct an analogue of real GDP, that s, GDP measured n base perod prces, as reported n actual data by natonal statstcal agences. In order to do ths n a way that s as close as possble to the methods used by these agences, we follow the recommendatons n the UN s System of Natonal Accounts 993 (SNA93). We de ne nomnal GDP at current prces as aggregate value-added, or the d erence between the total value of gross output less total expendtures on ntermedate nputs. Gross output at current producer prces s gven by: t = qt (!) yt (!) d! + q2t (!) y2t (!) d! + PNtY Nt, (27) t 2t and expendture on ntermedate nputs, valued at purchaser prces, s gven by: It = 2t p t (! 0 ) m t (!;! 0 ) d! 0 d! + p t (!) m Nt (!) d!. (28) 7

18 GDP at current prces s then Y t = t I t. To construct real GDP as measured n the data, we reconstruct the above formulas usng constant, base-perod prces for each good. Real gross output s: t = ~q 0 (!) yt (!) d! + ~q 20 (!) y2t (!) d! + PN0Y Nt, (29) t 2t where ~q s0 (!) s a base perod producer prce, de ned below. Real expendtures on ntermedate nputs are: It = 2t p 0 (! 0 ) m t (!;! 0 ) d! 0 d! + p 0 (!) m Nt (!) d!. (30) where p 0 (!) s the base perod purchaser s prce. Real GDP s then de ned as Yt = t It. E ectvely, we are usng what natonal statstcal agences refer to as a double-de aton method, de atng the current prce values of gross output and ntermedate consumpton each by ther own de ators. A practcal problem that ths method rases s that country-spec c perod-0 producer prces, ~q s0 (!), are not de ned for all goods, due to the fact that specalzaton patterns change n the model. For example, t may be the case that good! s produced n country n perod t, but was not produced by country n perod 0; country would have mported the good n perod 0. In ths case, t s not obvous at what prce we should value country s output of good! n perod t when calculatng real gross output. On the one hand, the SNA 93 recommends (paragraph 6.53) usng average prce changes of smlar products as a proxy for the change n prce of a new good between the base perod and the current perod; 8

19 adaptng ths nterpretaton n our model would mean usng country s prce of sellng the good to tself n perod 0, qs0 (!). On the other hand, f we use the prce at whch the good was mported n perod 0, then the product account (gross output less ntermedate nputs) s consstent wth the expendture account (consumpton plus nvestment less net exports), because the base perod consumer prce s unambguously equal to the mport prce. We proceed wth the latter assumpton, but ths choce has essentally no e ect on our results. Measured TFP correlaton In Secton VI., we perform several numercal experments to evaluate the extent to whch ncreased trade ntensty a ects busness cycle synchronzaton n the presence of vertcal specalzaton. In ths secton, however, we show that, n one mportant respect, trade n our model does not make countres more correlated: changes n measured TFP, constructed n our model usng real GDP as de ned above, are not lnked across countres through nternatonal trade. That s, although measured TFP s an endogenous obect n our model, there s no endogenous lnk between TFP across countres, and hence no dependence on trade ntensty of the correlaton n TFP. When measurng aggregate accountng statstcs as n the data, the dynamcs of TFP are pnned down by exogenous factors alone. In addton, the ntuton behnd the result suggests that t holds for a wde class of dynamc models wth endogenous specalzaton and trade n ntermedate goods. Wthout comovement n TFP, our model can produce endogenous comovement n GDP, but to a small degree, as we show n the next secton. 9 9 Ths s consstent wth the ndng n Kose & Y (2006), that the standard model can account for the trade-comovement puzzle f t s exogenously mposed that the correlaton of TFP ncreases wth trade ntensty. 9

20 Our result can be stated as follows: Proposton Let varables wth ^ denote log devatons from the steady state. Measured TFP, de ned by: A t = follows, to a rst order approxmaton: Y t (K t) (L t), (3) ^A t = ( ) ^A t + ^A 2t + ( ) ^A Nt. (32) A sketch of the proof of ths and the next proposton s provded n the appendx. The proposton s related to Hulten (978), who showed that, n a mult-sector neoclasscal closed economy wth ntermedate nputs, aggregate TFP evolves as a weghted average of sectoral productvty growth, wth the weghts gven by sectoral gross output shares of GDP. Equaton (32) shows that aggregate TFP n our model behaves exactly as n a three-sector verson of the closed economy models that Hulten consders, wth only one of the sectors supplyng ntermedate nputs. The ntuton for ths stark result s straghtforward, gven the assumptons used n aggregatng value-added across goods to construct real GDP. Frst, because we use good-spec c prces n valung the output of each producer, e cency d erences across goods do not show up n the value of output: n the presence of perfect competton, a producer wth hgher e - cency smply charges a proportonally lower prce, so the measured value of output per unt of nput does not vary across producers. 0 Bernard et al. (2003) and Gbson (2006) rase the 0 Ths s only true for goods that are produced n the same country n a perod t and the base perod. However, to a rst-order approxmaton, the goods that swtch country of producton contrbute zero to 20

21 same pont regardng the e ect of reallocaton that occurs n response to trade lberalzaton n models wth heterogeneous producers and monopolstc competton. Second, by usng base perod prces, the uctuatons n the value of ncome from movements n factor costs and prces do not show up n the computaton of real GDP, a pont made by Kehoe & Ruhl (2008). Even wth ntermedate goods, the gans n e cency that result from purchasng mported nputs at cheaper prces are removed by valung nputs at base perod prces. To show that the accountng method s the reason that foregn productvty shocks do not a ect domestc TFP, n the next proposton we show that a welfare-based measure of productvty does capture the e ects of foregn productvty shocks through trade. Proposton 2 De ne welfare-based real ncome and welfare-based productvty as: ~Y t = ~A t = (wtl t + rtk t) (P2t) (PNt ) ( ), (33) ~Y t (Kt) (L t). Then, for some constant, ~A t = ( )= t 2t = At A 2t A Nt. (34) A welfare-based measure of productvty ncreases when the share of goods that a country buys from tself, st, decreases n ether stage. Foregn productvty mprovements generate changes n real GDP. Interestngly, the dstrbutons of e cences n each stage do not play any role n dervng ths result. As long as we look at rst-order e ects, where the goods that swtch contrbute zero to changes n real GDP, the proof goes through under any spec caton of the e cency dstrbutons. Equaton (32) shows that, f producton technologes are not Cobb-Douglas, TFP correlatons could rse wth trade ntensty f an ncrease n trade shfts countres output shares toward sectors that are more correlated. 2

22 such a decrease n the share of goods purchased domestcally. In ths sense, the welfare-based productvty measure re ects transmsson of productvty shocks across countres. V. Vertcal Specalzaton and Trade Before proceedng to our numercal experments, we revew some measures of vertcal specalzaton n the data, and show how to construct correspondng measures n the model. A commonly cted measure of vertcal specalzaton s the one n Hummels, Ish & Y (200). The authors de ne an ndex of vertcal specalzaton for a gven sector as the rato of mported nputs to gross output. To construct an economywde measure of vertcal specalzaton, they aggregate ths ndex across sectors, weghted by each sector s share of exports, to capture the degree to whch mported nputs are mportant n exportng sectors. Ther measure for country s then gven by: VS = P II Exports P Exports, (35) where denotes a sector, II denotes mported ntermedates n sector, and denotes gross output n sector. When the exports of a sector are zero, that sector does not a ect the ndex. When mported ntermedate nputs n a sector are zero, that sector does not add to the numerator of the ndex. Countres are more vertcally specalzed f they export more n sectors that use mported ntermedates ntensvely. Usng Input-Output tables Hummels et al. (200) report ncreasng vertcal specalzaton n nternatonal trade evaluated usng ths ndex; some numbers taken from ther paper are llustrated n the columns labelled VS ndex n Table. 22

23 TABLE ABOUT HERE. In our model we do not explctly map the contnuum of goods nto a sectoral categorzaton. For the purposes of constructng a measure of vertcal specalzaton comparable to the above ndex, we can thnk of all the goods as beng assgned to one sector n the data. 2 The share of ntermedates n gross output for all the second stage goods s whle the share of the value of mported ntermedates out of the total value of ntermedates s t. Therefore, n our model, the analogue of the ndex n Hummels et al. (200) s gven by: 2 E VS = t E + E2, (36) where s, s = ; 2 denote gross output n each stage and E s denotes country s exports from n each stage. Addtonally, we can use nput-output tables to drectly construct measures n the data analogous to the shares of ntermedate and nal expendtures spend by a country on domestc goods, the and 2. Table also shows these statstcs. As trade has ncreased sgn cantly over tme n these countres (a decrease n the fracton spent on domestc goods), both the shares of mported ntermedate and nal goods have ncreased. However, the mported share of ntermedate nputs ncreased more than the mported share of nal goods. In the next secton, we evaluate the ablty of versons of our model wth and wthout vertcal specalzaton to generate hgher synchronzaton of busness cycles between countres that have hgher trade ntensty. 2 Alternatvely, a random assgnment of subsets of goods along the contnuum from each stage nto sectors would gve the same result. 23

24 VI. Numercal Experments We set several parameters to standard values n the nternatonal busness cycle lterature. We nterpret one model perod as one quarter, and set the dscount factor = 0:99 so that the steady state real nterest rate s 4% per year. We set to 0:3, so 30% of value added s pad to captal. The deprecaton rate s set to 2:5% per perod. The utlty parameter s set to 0:34, so that about =3 of the total tme endowment s suppled as labor n the steady state. We set the share of tradeable output n GDP to match the share of nontradeable expendtures n the model to the share of nal expendtures on servces n US nput-output tables, gvng = :3. We assume that the technology shocks are equal across sectors wthn a country, A t = A 2t = A Nt A t, and that each country s aggregate technology follows an AR() process n logs, log A t+ = log A t +" t+, for = ; 2, where " t s a mean-zero normally dstrbuted..d. nnovaton wth standard devaton ". We set = 0:9 and " = 0:0. Both values are close to those used, for example, by Backus et al. (994), and generate average autocorrelaton and standard devaton of HP- ltered TFP n our benchmark experments of 0:59 and 0:07, respectvely. Notably, we do not buld any correlaton nto the shocks to technology across countres or across sectors, so that we solate the degree to whch our model endogenously generates cross-country correlaton n measured real GDP. 3 The model contans two parameters related to the elastcty of substtuton between domestc and foregn goods. As n Eaton & Kortum (2002), the role of the parameter n determnng the elastcty of trade s concealed by the role of. Whle governs subst- 3 Addng postve correlaton n the exogenous shocks would ncrease the levels of cross-country correlatons, but does not a ect ther dependence on trade ntensty. 24

25 tutablty n the ntensve margn - wthn goods that are contnuously traded - governs the heterogenety across goods, and hence determnes the extent to whch the extensve margn of trade n new goods responds to varatons n technology or trade costs. At the aggregate level, the elastcty of substtuton between mported and domestc goods n our model s. As Ruhl (2008) notes, measures of ths aggregate elastcty n the data d er dependng on the source of prce varaton: measures from tme seres data gve small elastctes of the magntude typcally used n nternatonal busness cycle models. On the other hand, estmates from cross-secton data relatng trade patterns to tar and non-tar barrers nd elastctes that are much hgher. Snce the parameter governs the elastcty n response to both types of prce varaton n our model, we balance between the two measures by choosng a value of = 3:6 because t s the lowest of the three estmates from Eaton & Kortum (2002). For the elastcty of substtuton that determnes the substtuton between d erent goods, we use the benchmark value of Backus et al. (994) of = :5, whch s consstent wth estmates of the elastcty of substtuton between foregn and domestc goods by SITC commodty groups. 4 The parameter determnes the share of gross output n nal goods pad to ntermedate nputs. We consder two versons of our model, a one-stage benchmark wth =, and a verson wth vertcal specalzaton, wth = 0:5, whch s approxmately the rato of ntermedate nputs to gross output n US nput-output tables. In the experments below, we choose the technology parameters T s and the trade costs s to generate d erent steady state specalzaton and nternatonal trade patterns. We subse- 4 Recent estmates by Broda & Wensten (2006) place the medan value of ths elastctty across all sectors to be 2.5 and the average value much hgher than that. The model constrans us to set > but rasng the elastcty does not substantally alter our results. 25

26 quently look at the degree of busness cycle comovement across these d erent patterns. 5 Results We look at varous cross-country correlatons of H-P ltered varables n d erent versons of the model. In addton to real GDP and TFP, we also look at the correlatons between countres of welfare-based real ncome, ~ Y ; aggregate labor supply; real consumpton expendtures at base perod prces, Ct = R p 20 (!) c T t (!) d! + P N0 C Nt ; real nvestment expendtures at base perod prces, X t = R p 20 (!) x T t (!) d! + P N0 X Nt ; and real value added n d erent stages of producton: Y t = Y 2t = ~q 0 (!) yt (!) d!, (37) t ~q 20 (!) y2t (!) d! p 0 (! 0 ) m t (!;! 0 ) d! 0 d!. 2t 2t In lght of Proposton 2, we show the alternatve ncome measure, ~ Y, to get a sense of how large the e ects of ncreases n trade and vertcal specalzaton are on comovement, even though t does not appear n tradtonal natonal accountng statstcs. TABLE 2 ABOUT HERE. We rst look at a verson of the model wth no vertcal specalzaton whch we denote as the benchmark. The goal s to subsequently ntroduce d erent experments where vertcal specalzaton arses for d erent reasons and compare the propertes of these versons of the model wth the benchmark. The left half of Table 2 shows statstcs for the benchmark 5 In an appendx avalable onlne, we perform senstvty analyss wth respect to the elastcty parameters and, and the correlaton of shocks " t+, and show that the nature of our man results do not change. 26

27 model, wth =, and wth symmetrc countres, so that T s = T 2 s. When we vary trade costs to generate steady state trade to GDP ratos between 3% and 5%, we see that the correlaton of real GDP across countres ncreases very slghtly, from.0 to.04. One way to compare ths numbers to measures n the data s by lookng at the mpled slope of the GDP correlaton - trade ntensty relatonshp: the d erence n GDP correlaton over the d erence n the log of the trade to GDP rato. Kose & Y (2006) estmate ths slope at 0.09 n cross-country data. The slope for our benchmark model s 0.09, about one fth of Kose and Y s estmate. Our experments are conducted for trade ntenstes exceedng those of the typcal country par n Kose and Y s data, so ths slope should be consdered an upper bound on what our model can produce. Kose and Y also show that a three-country extenson of Backus et al. (994) calbrated to match blateral trade shares for select countres can account for at most about % of ths slope (n ther baselne calbraton). Con rmng Proposton, the correlaton of TFP across countres s close to zero, and essentally does not rse wth trade ntensty. The correlatons of labor, nvestment, and consumpton rse across trade ntenstes, from.02 to.0 for labor and nvestment, and from.0 to. for consumpton. In addton, consstent wth Proposton 2, the correlatons of the alternatve measure of real ncome Y ~, and the correspondng productvty measure A, ~ ncrease wth trade ntensty much more than that of real GDP. The mpled slope of these correlatons wth trade ntensty s 0.056, nearly trple the real GDP slope. The rght half of Table 2 shows our model wth vertcal specalzaton, n three cases n whch the trade costs s and technology parameters T s are the same across countres and across stages of producton. As trade ntensty ncreases n ths model from 3% to 5%, the 27

28 steady state vertcal specalzaton measure ncreases as well, from 0.26% to.32%. 6 The patterns of cross country correlatons are broadly smlar to the one-stage model, except that the ncreases are smaller as trade ntensty rses. Even for the correlatons of ~ Y and ~A, real ncome and ts assocated productvty measure, the ncreases across trade ntensty are smaller than n the one stage benchmark. Ths result makes sense, gven proposton 2, and the fact that technology shocks mght change specalzaton to a d erent degree n each stage of producton. Wth two trade shares, t and 2t, there s another source of varaton n ~ A compared to the one-stage benchmark. In fact, wth symmetrc countres, the presence of two stages of producton actually reduces, rather than enhances the dependence of comovement on trade. Another reason that correlatons rse wth trade ntensty to a smaller degree n the model wth vertcal specalzaton s that our measure of trade ntensty s a measure of the gross output of goods traded, not value added. As Kose & Y (200) pont out, for a gven trade ntensty, less value-added s traded n the model wth vertcal specalzaton than n the model wthout. If we nstead vary transport costs so that the value-added of mports relatve to GDP s 3%, 9%, and 5%, then the correlatons of real GDP are.0,.02, and.03, respectvely. TABLE 3 ABOUT HERE. In Table 3 we show statstcs for two varatons of our model wth 5% trade ntensty, n whch trade costs d er across stages of producton. The column labelled Low VS 6 Whle the fractons n our model results come close to the data for, for example, the US (compare to n Table ), the V S measures our model predcts are much smaller than n the data. Ths suggests that sectoral d erences that are emphaszed by the V S measure of Hummels et al. (200) may be mportant. 28

29 has relatvely hgh trade costs for rst stage goods, so that trade ntensty n these goods s dampened, whle the column labelled Hgh VS has relatvely low trade costs for rst stage goods. The results ndcate that a hgher degree of vertcal specalzaton does not sgn cantly a ect the busness cycle correlatons we consder f countres are symmetrc. TABLE 4 ABOUT HERE. Fnally, n Table 4 we consder a case n whch countres are asymmetrc: we choose the Ts terms so that country has a comparatve advantage n the producton of stage goods, whle country 2 has a comparatve advantage n stage 2 goods, and both countres have the same steady state GDP. In the left column, labeled low specalzaton, the degree of comparatve advantage s smaller than n the rght column, labeled hgh specalzaton, so that the vertcal specalzaton measure for country 2 s larger, and for country s smaller, n the rght column. We see agan that most of the cross-country correlatons change lttle across these cases. The excepton s the correlaton of real value added across countres n d erent stages: whle the correlaton of real value added n the sector n whch each country specalzes s only.02 n the low specalzaton case, ths correlaton ncreases to.3 n the hgh specalzaton case. The extent to whch countres are negatvely correlated n the sectors n whch they do not have comparatve advantage ncreases as well. The negatve change slghtly outweghs the postve change here, so that even though t apples to sectors that are small n each country, the overall e ect s that the correlaton of aggregate real GDP s smlar under low specalzaton or hgh specalzaton. In these results, we have shown that several versons of our model can generate moderate ncreases n busness cycle comovement wth ncreases n trade ntensty. However, for the 29

30 cases we have consdered here, these ncreases are small, and the addton of vertcal specalzaton does not contrbute sgn cantly to magnfyng them. Whle asymmetrc countres n our model clearly do dsplay the tght lnks across sectors that vertcal specalzaton mples, addtonal mechansms would be needed to translate these lnks to aggregate real GDP and measured TFP. In the next secton, we examne one such mechansm. VII. An Extenson wth Imperfect Competton Havng establshed that our model wth perfect competton does not provde a lnk between trade and TFP correlatons, we follow Bernard et al. (2003) by ntroducng n our model a sort of mperfect competton that breaks the lnk between prces and producers e cences, and show that t has the potental to lnk trade ntensty to correlatons n measured TFP, as well as real GDP. We modfy the structure of our model as follows: n each country, there are a large number of potental producers of each good, and country s k th most e cent producer of good! n stage s has e cency zks (!). The lowest cost among producers n both countres of delverng good! to country s then p st (!) = mn fq st s =z s (!) : = ; 2g. Under Bertrand competton, each good s sold n a country by the producer wth the lowest-cost of sellng there, but ths producer charges a prce equal to ether the second best producer s margnal cost or a xed markup over ts own margnal cost, whchever s smaller. The second lowest cost s p 2st (!) = mn 2 fqst s =zks (!) : = ; 2; k = ; 2g, where mn 2 refers to the second smallest. Therefore, the prce charged for good! n country s p st (!) = mn p 2st (!) ; mp st (!), (38) 30

31 where m = =( ) s the monopolstc markup. In ths model, we assume that >, so that the monopolstc markup s postve. The only relevant e cences for determnng prces and producton patterns are zs (!) and z2s (!). We assume that these e cences are ontly dstrbuted accordng to the analogue of the Fréchet dstrbuton n Bernard et al. (2003): the ont probablty that zs (!) z and z 2s (!) z 2, for z 2 z, s gven by F s (z ; z 2 ) = + T s z2 z e Ts z 2. The margnal dstrbuton of z s stll gven by (3), so the parameters T s and play the same role as n the prevous setup. Followng the dervatons n Bernard et al. (2003), the CES prce ndces for each stage are: P st = + ( ) m =( ) st =, (39) where st and the trade shares st are de ned n the same way as n (5) and (7). Fnally, the model wth Bertrand competton d ers n that producers earn pro ts, so consumers budget constrants are mod ed to nclude pro ts and the market clearng condtons are mod ed. As Bernard et al. (2003) show, aggregate pro ts n each stage are a fracton = ( + ) of total revenues, so that the market clearng condtons for each stage of tradable goods can be wrtten: wtl t + rtk t = + tpt wtl 2t + rtk 2t = + 2tP 2t MT t + MNt + tp t M T t + M Nt, (40) CT t + XT t + 2tP 2t C T t + X T t. In the case of Bertrand competton, e cency does not map one-to-one to prces, as 3

32 evdent n (38). In lght of the dscusson n Secton IV., ths dsconnecton gves the model more potental to generate cross-country correlaton n measured TFP and real GDP. 7 Table 5 dsplays the analogues of the results n Table 2 for ths extenson of the model. As we go from 3% to 5% trade ntensty, the correlaton of real GDP ncreases from.0 to.06 n the one stage benchmark, and from.04 to.09 n the vertcal specalzaton model. The slope of the GDP correlaton n the vertcal specalzaton model s 0.03, about one thrd of the slope estmated n the data n Kose & Y (2006). Ths ncrease s also seen n the correlaton of measured TFP, whch ncreases from.02 to to.06 n the one stage model, and from.04 to.0 n the vertcal model. The latter result was qualtatvely mssng n the model wth perfect competton. In the vertcal specalzaton model wth mperfect competton, the mpled slope of the TFP correlaton and trade relatonshp s 0.037, about 40% of the slope that Kose & Y (2006) estmate n the data, whch s Thus, the model wth mperfect competton, through ts e ects on the measurement of real GDP, qualtatvely changes the way that measured TFP depends on trade n our model, and has the potental to quanttatvely change the relatonshp between trade ntensty and GDP correlatons. An extenson of ths setup to a model wth more than two countres, calbrated to match blateral trade shares, would be needed to provde a better quanttatve evaluaton of ths mechansm s ablty, combned wth endogenous specalzaton, to solve the trade-comovement puzzle. TABLE 5 ABOUT HERE. 7 Wth mperfect competton, the expressons we derved above for real GDP are no longer vald, and analogous expressons are much more ntractable. To compute aggregate statstcs, we draw e cences for one hundred thousand goods, compute the relevant varables for producers of each good every perod, and calculate aggregates at base-perod prces as sample averages. 32

33 VIII. Concluson We have developed a model vertcal specalzaton n nternatonal trade n a settng wth aggregate uctuatons. We asked whether ths framework can account for the trade comovement puzzle dent ed by Kose & Y (200) and (2006). Whle the framework we develop does not resolve the puzzle, our work helps to take mportant steps n understandng the reasons behnd ts persstence under d erent modelng frameworks: we prove that wth perfect competton, measured TFP does not depend on trade or vertcal specalzaton ntensty. Whle vertcal specalzaton as we have spec ed t provdes an ntutve reason for countres that trade more to be more correlated, addtonal mechansms are needed to account for the extent to whch ths channel a ects the behavor of measured busness cycle statstcs. Our prelmnary results from a model wth varable markups ndcate that ths feature, combned wth vertcal specalzaton, s a promsng avenue for further exploraton to lnk trade and TFP uctuatons. 33

34 Year V S ndex 2 Year. V S ndex 2 Canada 97 :20 :77 :7 990 :27 :70 :57 France 972 :8 :8 : :24 :70 :72 Germany 978 :8 :79 :8 990 :20 :75 :73 Japan 970 :8 :90 : : :89 :9 Unted Kngdom 979 :25 :74 : :26 :64 :62 Unted States 972 :06 :94 : : :88 :8 Table : Vertcal Specalzaton Measures from Hummels et al. (200) and domestc expendture shares for ntermedate and nal goods from OECD Input-Output Tables Benchmark ( = ) Vert. Spec. ( = :5) varable trade/gdp trade/gdp 3% 9% 5% 3% 9% 5% corr (Y ; Y 2 ) corr (A ; A 2 ) corr (L ; L 2 ) corr (X ; X 2 ) corr (C ; C 2 ) corr( Y ~ ; Y ~ 2 ) corr( A ~ ; A ~ 2 ) ; 2 ( ; :0) ( ; :30) ( ; :50) (:04; :04) (:; :) (:9; :9) V S ndex (%) Table 2: Model busness cycle correlatons (see text for varable de ntons). 34