Globalization and Synchronization of Innovation Cycles*

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1 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Globalizaio ad Syhroizaio of Iovaio Cyles By Kimiori Masuyama, Norhweser Uiversiy, USA Irya Sushko, Isiue of Mahemais, Naioal Aademy of Siee, Ukraie Laura Gardii, Uiversiy of Urbio, Ialy Updaed o Absra: We propose ad aalyze a wo-oury model of edogeous iovaio yles. I auarky, iovaio fluuaios i he wo ouries are deoupled. As he rade oss fall ad ira-idusry rade rises, hey beome syhroized. This is beause globalizaio leads o he aligme of iovaio ieives aross firms based i differe ouries, as hey operae i he ireasigly global (hee ommo) marke evirome. Furhermore, syhroizaio ours faser (i.e., wih a smaller reduio i rade oss) whe he oury sizes are more uequal, ad i is he larger oury ha diaes he empo of global iovaio yles wih he smaller oury adusig is rhyhm o he rhyhm of he larger oury. These resuls sugges ha addig edogeous soures of produiviy fluuaios migh help improve our udersadig of why ouries ha rade more wih eah oher have more syhroized busiess yles. Keywords: Edogeous iovaio yles ad produiviy o-movemes; Globalizaio, Home marke effe; Syhroized vs. Asyhroized yles; Syhroizaio of oupled osillaors; Basis of araio; Two-dimesioal, pieewise smooh, oiverible maps JEL Classifiaio Numbers: C6 (Dyami Aalysis), E3 (Busiess Fluuaios, Cyles), F (Model of Trade wih Imperfe Compeiio ad Sale Eoomies), F44 (Ieraioal Busiess Cyles), F6 (Eoomi Impas of Globalizaio), O3 (Iovaio ad Iveio) K. Masuyama haks he semiar ad oferee pariipas a (i hroologial order) NU Maro Bag luh, Chiago Fed, SNF Siergia-CEPR Coferee i Asoa, Zurih, IMT-Lua, EUI, Bologa, he 8 h bieial workshop of MDEF (Modelli Diamii i Eoomia e Fiaza) i Urbio, TNIT (Toulouse Nework for Iformaio Tehology) Aual Meeig i Cambridge (MA), Prieo Trade, NU Maro, NYU Maro, Chiago Moey ad Bakig, Hiosubashi Coferee o Ieraioal Trade ad FDI for heir feedbak. He akowledges he suppor of TNIT. The work of I. Sushko ad L. Gardii has bee suppored by he COST Aio IS04. Page of 58

2 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio. Iroduio How does globalizaio affe maroeoomi o-movemes aross ouries? A vas maoriy of researh approahes his quesio by assumig ha produiviy movemes i eah oury are drive by some exogeous proesses. As already demosraed by iovaio-based models of edogeous growh, however, globalizaio a hage he growh raes of produiviy. I his paper, we demosrae ha globalizaio a also hage syhroiiy of produiviy fluuaios aross ouries i a wo-oury model of edogeous fluuaios of iovaio aiviies. The iuiio we wa o apure a be simply saed. Imagie ha here are wo sruurally ideial ouries. I auarky, eah of hese ouries experiees edogeous fluuaios of iovaio, due o sraegi omplemeariies i he imig of iovaio amog firms ompeig i heir domesi marke, whih auses emporal luserig of iovaio aiviies ad hee aggregae fluuaios. Wihou rade, edogeous fluuaios i he wo ouries are obviously disoeed. As rade oss fall ad firms based i he wo ouries sar ompeig agais eah oher, he iovaors from boh ouries sar respodig o a ireasigly global (hee ommo) marke evirome. This leads o a aligme of iovaio ieives, hereby syhroizig iovaio aiviies, ad hee produiviy movemes, aross ouries. To apure his iuiio i a raspare maer, we osider a model ha osiss of he followig wo buildig bloks. Our firs buildig blok is a model of edogeous fluuaios of iovaios, origially proposed by Judd (985). I his lassi arile, Judd developed hree dyami exesios of he Dixi-Sigliz moopolisi ompeiive model, i whih iovaors ould pay a oe-ime fixed os o irodue a ew (horizoally differeiaed) variey. Firs, he showed ha he equilibrium raeory overges moooially o a uique seady sae uder he assumpio Empirially, Frakel ad Rose (998) ad may subseque sudies have esablished ha ouries ha rade more wih eah oher have more syhroized busiess yles. The evidee is pariularly srog amog developed ouries as well as amog developig ouries, while i is less so bewee developed ad developig ouries. Sadard ieraioal RBC models have diffiuly explaiig his, ad i is easy o see why. Wih exogeous produiviy shoks drivig busiess yles i hese models, more rade leads o more speializaio, whih meas less syhroizaio, o he exe ha he shoks have seor-speifi ompoes. Some aemps o resolve suh rade-omoveme puzzle by appealig o verial speializaio aross ouries have me limied suess, ad some auhors suggesed ha i would help o improve heir performaes if globalizaio would also syhroize produiviy movemes ha drive busiess yles aross ouries: see, e.g., Kose ad Yi (006). We hope ha our model offers oe suh heoreial igredie. Page of 58

3 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio ha iovaors hold moopoly over heir iovaios idefiiely. The, he ured o he ases where he iovaors hold moopoly oly for a limied ime, so ha eah variey is sold iiially a he moopoly prie ad laer a he ompeiive prie. The assumpio of emporary moopoly drasially hages he aure of dyamis ad geeraes edogeous fluuaios. This is beause, wih free ery o iovaio, eah iovaor eeds o reover his os of iovaio by earig eough reveue durig his moopoly. Ceraily, i is disouragig for him o see ohers eerig he marke a he same ime, beause he has o ompee wih heir iovaios. (This meas o sraegi omplemeariies bewee oemporaeous iovaios.) Neverheless, he impa of suh oemporaeous iovaios is relaively mued, beause hey are also sold a he moopoly pries. Wha is eve more disouragig is for him o see he iovaios irodued i he ree pas sar beig sold ompeiively, as heir iovaors lose heir moopoly. Thus, a iovaor would raher eer he marke whe ohers do, so ha he eoys his moopoly while hey sill hold heir moopoly, isead of waiig ad eerig he marke afer hey lose heir moopoly. Or o pu i differely, he full impa of iovaios ours wih a delay, whih reaes sraegi omplemeariies i he imig of iovaio (despie ha here is o sraegi omplemeariies i iovaios). This leads o a emporal luserig of iovaio, geeraig aggregae fluuaios of produiviy. Judd developed wo models ha formalize his idea, of whih we use he oe, skehed by Judd (985; Se.4) ad examied i greaer deail by Deekere ad Judd (99; DJ for shor) for is aalyial raabiliy. Wha makes i aalyially raable is he assumpio ha ime is disree ad ha he iovaors hold heir moopoly for us oe period, he same period i whih hey irodue heir varieies. Wih his assumpio, he sae of he eoomy i eah period (how sauraed he marke is from pas iovaios) is summarized by oe variable (how may varieies of ompeiive goods he eoomy has iheried). Ad he ery game played by iovaors i eah period beomes effeively sai beause hey do o expe o ear ay profi i he fuure (alhough he ouome of his game will affe he ouome of he games i he fuure). 3 Sie he profi from iovaig i ay period is dereasig i he aggregae iovaios i he same period, he free ery odiio pis dow he ouome of his sai This versio of he Judd model has bee exeded o a wo-oury, wo-faor model by Grossma ad Helpma (988). I also provided he foudaio for he edogeous growh lieraure developed by Romer (990) ad ohers. 3 Furhermore, i obviaes he eed for priig he owership share of he iovaig firms, beause heir profis are us eough o over he iovaio os, so ha here is o divided o pay ou. Page 3 of 58

4 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio ery game uiquely. As a resul, he equilibrium raeory a be obaied uiquely by ieraig a oe-dimesioal (D) map from ay iiial odiio. This map urs ou o be isomorphi o he skew e map. Tha is, i is oiverible ad pieewise liear (PWL) wih wo brahes. I depeds o wo parameers; σ (he elasiiy of subsiuio bewee goods) ad δ (he survival rae of he exisig goods). 4 A higher σ ireases he exe o whih a pas iovaio, whih is ompeiively sold, disourages iovaors more ha a oemporaeous iovaio, whih is moopolisially sold. A higher δ meas more of he pas iovaios survive ad arry over o disourage urre iovaios. For a suffiiely high σ ad/or a suffiiely high δ, sraegi omplemeariies i he imig of iovaio are srog eough o ause emporal luserig of iovaio ha makes he uique seady sae usable ad he equilibrium raeory fluuae forever, sarig from almos all iiial odiios. For a moderaely high σ ad/or δ, he equilibrium raeory asympoially overges o a uique period- yle, alog whih he eoomy aleraes bewee he period of aive iovaio ad he period of o iovaio. For a muh higher σ ad/or δ, eve he period- yle is usable, ad he raeory overges o a haoi araor. Sie he equilibrium raeory is uique, fluuaios are drive eiher by mulipliiy or by self-fulfillig expeaios. This feaure of he model makes i useful as a buildig blok o examie he effes of globalizaio o he aure of fluuaios aross wo ouries. 5 Our seod buildig blok is Helpma ad Krugma (985; Ch.0; HK for shor), a model of ieraioal rade i horizoally differeiaed (Dixi-Sigliz) varieies wih ieberg rade oss bewee wo sruurally ideial ouries, whih may differ oly i size. This model has wo key parameers; he disribuio of oury sizes ad he degree of globalizaio, whih is iversely relaed o he rade os. I his model, he equilibrium umber of firms based i eah oury is proporioal o is size i auarky (wih ifiiely large rade oss). As rade oss fall, horizoally differeiaed goods produed i he wo ouries muually peerae 4 I a model of horizoal iovaio (or expadig variey), ew goods are added o old goods wihou replaig hem, so ha he marke ould eveually beome so sauraed ha iovaios would sop permaely. Oe way o avoid his is o le he eoomy grow i size, exogeously as i Judd (985) or edogeously as i Masuyama (999, 00). Here, we assume isead, by followig DJ (99), ha he exisig goods are sube o idiosyrai obsolesee shoks, so ha oly a osa fraio of hem, δ, arries over o he ex period. 5 I is worh poiig ou ha he disree ime speifiaio is o resposible for ausig fluuaios. Ideed, Judd (985; Se.3) developed a oiuous ime model i whih eah iovaor holds moopoly for a fixed duraio of ime, T > 0 (i.e., a oe-hoss shay speifiaio), ad showed ha he eoomy aleraes bewee he periods of aive iovaio ad he periods of o iovaio alog ay equilibrium raeory for almos all iiial odiios whe T is suffiiely large (bu fiie). Page 4 of 58

5 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio eah oher s home marke (Two-way flows of goods), ad he equilibrium disribuio of firms beome ireasigly skewed oward he larger oury (Home Marke Effe ad is Magifiaio). By ombiig he DJ mehaism of edogeous fluuaios of iovaios wih he HK model of ieraioal rade, we show: The sae spae of our wo-oury model of he world eoomy is wo-dimesioal (i.e., how may ompeiive varieies eah oury has iheried, whih deermies how sauraed he wo markes are from pas iovaios) ad represes he global marke odiio for he urre iovaors i he wo ouries. For eah iiial odiio, he equilibrium raeory is uique ad obaied by ieraig a wo-dimesioal (D), pieewise smooh (PWS), oiverible map, whih has four parameers (he wo omig from DJ ad he wo omig from HK). I auarky, wih ifiie rade oss, he dyamis of wo ouries are deoupled i he sese ha he D-sysem a be deomposed io wo idepede D-sysems, whih are isomorphi o he origial DJ model. Uder he same parameer odiio ha esures he isabiliy of he seady sae i he DJ model, he dyamis of he wo ouries may overge o eiher syhroized or asyhroized fluuaios, depedig o he iiial odiios; As rade oss fall, ad he goods produed i wo ouries muually peerae eah oher s home marke, he dyamis beome syhroized i he sese ha he basi of araio 6 for he syhroized yle expads ad eveually overs a full measure of he sae spae, ad he basi of araio for he asyhroized yle shriks ad eveually disappears. 7 To pu i differely, as rade oss fall, he iovaio dyamis beomes more likely o overge o he syhroized -yle, ad for a suffiiely small rade os, i overges o he syhroized -yle for almos all iiial odiios. 6 I he ermiology of he dyamial sysem heory, he se of iiial odiios ha overge o a araor (e.g., a araig seady sae, a araig period- yle, a haoi araor, e.) is alled is basi of araio. 7 For hese resuls, we impose he parameer odiios ha esure he exisee of a uique, sable period- yle i he DJ model. As poied ou above, he equilibrium raeory i he DJ model overges o a haoi araor uder he parameer odiios ha esure he isabiliy of he (uique) period- yle. Alhough we have obaied some ieresig resuls for hese ases, we have hose o o disuss hem here parly beause he sable -yle ase is suffiie for oveyig he eoomi iuiio behid he syhroizaio mehaism ad parly beause we wa o avoid makig his paper more ehially demadig i order o keep i aessible o a wider audiee. Page 5 of 58

6 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Syhroizaio ours faser (i.e., wih a smaller reduio of rade oss) whe he wo ouries are more uequal i size. Furhermore, eve a small oury size differee speeds up syhroizaio sigifialy. Ad he larger oury ses he empo of global iovaio yles, wih he smaller oury adusig is rhyhm o he rhyhm of he larger oury. The iuiio behid hese resuls should be easy o grasp. Wih globalizaio, he markes beome more iegraed. As a resul, a wave of iovaios ha ook plae i oe oury i he pas disourages iovaios oday o oly i ha oury, bu also i he oher oury, ausig syhroizaio of iovaio aiviies aross he wo ouries. Furhermore, as iovaio aiviies beome syhroized, he marke odiios i he wo ouries beome more similar, whih furher auses syhroizaio. The larger oury plays a more impora role i seig he rhyhm of global iovaio yles, beause he iovaors based i he smaller oury rely more heavily o he reveue eared i he expor marke o reover he os of iovaio ha hose based i he larger oury. Relaed Lieraure: To he bes of our kowledge, his is he firs aemp o explai how globalizaio may syhroize produiviy fluuaios aross ouries. Neverheless, i is relaed o several srads of lieraure. Firs, i is relaed o sai models of ieraioal rade, pariularly hose of ira-idusry rade ad home marke effes. This is oe of he ore maerials of ieraioal rade. We have hose HK as oe of our buildig bloks, beause i is perhaps he mos sadard exbook reame. Seod, here are ow a large body of lieraure ha sudy he effes of globalizaio i iovaio-drive models of edogeous growh: see Grossma ad Helpma (99), Rivera-Baiz ad Romer (99), Aemoglu ad Ziliboi (00), Veura (005), Aemoglu (008; Ch.9), Aemoglu, Gaia ad Ziliboi (04) ad may ohers. All of hese examie he effes of globalizaio o produiviy growh raes alog he balaed growh pah. Third, here are may losed eoomy models of edogeous fluuaios of iovaio, whih ilude Shleifer (986), Gale (996), Jovaovi ad Rob (990), Evas, Hokapoa ad Romer (998), Masuyama (999, 00), Wälde (00, 005), Fraois ad Lloyd-Ellis (003, 008, 009), Jovaovi (006), Bramoullé ad Sai-Paul (00), ad Behabib (04), i addiio o Judd (985) ad Deekere ad Judd (99). We have hose DJ as oe of our buildig bloks beause of is raabiliy ad he uiqueess of he equilibrium Page 6 of 58

7 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio raeory. 8 We oeure ha he basi iuiio--globalizaio syhroizes iovaio aiviies aross ouries, as he iovaors everywhere respod o he ireasigly global (ad hee ommo) marke evirome--, should go hrough i a muh wider lass of models of edogeous iovaio yles. Amog hese sudies, Masuyama (999, 00) embed he DJ mehaism io a losed eoomy edogeous growh model wih apial aumulaio similar o Rivera-Baiz ad Romer (99) ad showed ha he wo egies of growh, iovaio ad apial aumulaio, move asyhroously. This is beause here is oly oe soure of edogeous fluuaios; apial aumulaio merely respods o he fluuaios of iovaio. 9 I oras, our model has wo soures of edogeous fluuaios. To pu our oribuio i a broader oex, we offer a ew model of syhroizaio of oupled osillaors. The sube of oupled osillaors is oered wih he effes of ombiig wo or more sysems ha geerae self-susaied osillaios, i pariular, how hey muually affe heir rhyhms. I is a maor opi i aural siee, ragig from physis o hemisry o biology o egieerig, wih housads of appliaios. 0 We are o aware of ay previous example from eoomis. To he bes of our kowledge, his is he firs wo-oury, 8 Perhaps i migh be isruive o ompare he DJ model wih he Shleifer model. I he Shleifer model, here are o osly iovaio aiviies. Isead, every period, a osa fraio of he ages reeives a idea exogeously, whih hey ould impleme o ear profi. Oe implemeed, i will be quikly imiaed so ha he age wih a idea a ear profi for oly oe period. Furhermore, he profi depeds o he size of he marke. If he ages aiipae ha a boom is immie, hey are willig o pospoe he implemeaio of he idea. Bu a boom ours i he period whe may ages impleme heir ideas ad ear heir profis, whih hey have o sped durig he same period. I oher words, he profi from iovaio i a give period ireases wih he aggregae iovaios i ha same period. This geeraes sraegi omplemeariies bewee oemporaeous iovaios i he Shleifer model. Aiipaios of a immie boom ould be self-fulfillig, whih ould geerae a ylial equilibrium. However, his is oe of muliple equilibria. The ylial equilibrium o-exiss wih a saioary equilibrium, i whih every age implemes his or her idea immediaely. I oras, i he DJ model, differe iovaios ompee wih eah oher, so ha he profi from iovaio dereases wih he umber of iovaios. Thus, here is o sraegi omplemeariy bewee oemporaeous iovaios, whih esures he uiqueess of he equilibrium pah. Wha reaes sraegi omplemeariies i he imig of iovaio i he DJ model is a delay effe of iovaios. Pas iovaios are more disouragig ha oemporaeous iovaios so ha iovaors would o wa o iovae afer ohers iovaed. To quoe Shleifer (986, foooe ), Judd s mehaism is almos he opposie of mie; iovaios i his model repel raher ha ara oher iovaios. 9 See also Gardii, Sushko, ad Naimzada (008) for a omplee haraerizaio of he Masuyama (999) model. 0 Jus o ame a few, osider he Moo, wih is roaio aroud is ow axis ad is revoluio aroud he Earh. These wo osillaios are perfely syhroized i he same frequey, whih is he reaso why we observe oly oe side of he Moo from he Earh. Or osider he Lodo Milleium Bridge. I is opeig days, hudreds of pedesrias ried o adus heir fooseps o laeral movemes of he bridge. I doig so, hey iadverely syhroized heir fooseps amog hemselves, whih aused he bridge o swig widely, forig a losure of he bridge. See Srogaz (003) for a popular, o-ehial iroduio o his huge sube. Of ourse, here may have bee aemps o borrow a exisig model of oupled osillaors from siee ad give a eoomi ierpreaio o is variables. The problem of his approah is ha i would be hard o give ay Page 7 of 58

8 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio dyami geeral equilibrium model of edogeous fluuaios. Ideed, his is oe of he oly wo dyami geeral equilibrium models, whose equilibrium raeory a be haraerized by a dyamial sysem, whih a be viewed as a ouplig of wo dyamial sysems ha geerae self-susaied equilibrium fluuaios. The oher oe is our ompaio piee, Masuyama, Sushko, ad Gardii (forhomig), whih develops a wo-seor, losed eoomy model, where eah seor produes a Dixi-Sigliz omposie of differeiaed goods, as i DJ. Whe he osumers have Cobb-Douglas preferees over he wo omposies, iovaio dyamis i he wo seors are deoupled. For he ases of CES preferees, i is show ha, as he elasiiy of subsiuio bewee he wo omposies ireases (dereases) from oe, fluuaios i he wo seors beome syhroized (asyhroized), whih amplifies (dampes) he aggregae fluuaios. The above wo are amog he few eoomi examples of D dyami sysems, defied by PWS, oiverible maps. 3 Masuyama, Kiyoaki, ad Masui (993) is also relaed i spiri i ha hey oo osider globalizaio as a ouplig of wo games of sraegi omplemeariies. They developed a wooury model of urrey irulaio. The ages are radomly mahed ogeher, ad urrey irulaio is modeled as a game of sraegi omplemeariies, where a age aeps a erai obe as a meas of payme if he expes hose he would ru io i he fuure o do he same. I auarky, ages are mahed oly wihi he same oury, so ha wo ouries play wo separae games of sraegi omplemeariies, hee differe urreies may be irulaed i he wo ouries. The, globalizaio ireases he frequey i whih ages from differe ouries are mahed ogeher. Ieresigly, he ages from he smaller oury, o hose from he larger oury, are he firs o adus heir sraegies ad o sar aepig a foreig sruural ierpreaio o he parameers of he sysem. Imporaly, we derive a sysem of oupled osillaors from a fully speified eoomi model, ad we eed o aalyze his sysem, beause i is ew ad differe from ay sysem ha has bee sudied before. Furhermore, he oury size differee has orivial effes i our model, ad plays a impora role i our aalysis. We are o aware of ay previous sudies, whih odu a sysemai aalysis of he role of size differee bewee oupled osillaors. Some may fid his resul surprisig, beause he presee of omplemeary (subsiues) seors is ommoly viewed as a amplifyig (moderaig) faor. However, his resul is o iosise wih suh a ommo view, whih is oered abou he propagaio of exogeous produiviy shoks from oe seor o ohers. This resul is oered abou how produiviy i various seors respods edogeously o a hage i he marke odiio. Seors produig subsiues (omplemes) respod i he same (opposie) direio, hereby amplifyig (moderaig) he aggregae fluuaio. 3 See Mira, Gardii, Barugola ad Cahala (996) for a iroduio o D oiverible maps i geeral, ad see Sushko ad Gardii (00) for PWS examples. Page 8 of 58

9 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio urrey, ad as a resul, ha he larger oury s urrey emerges as a vehile urrey of he world rade. The res of he paper is orgaized as follows. Seio develops our wo-oury model of edogeous fluuaios of iovaio, ad derives he D-PWS, oiverible map ha govers he equilibrium raeory. Seio 3 osiders he ase of auarky, where he D sysem a be deomposed io wo idepede D-PWL, oiverible maps, whih are isomorphi o he origial sysem obaied by DJ. I Seio 3., we offer a deailed aalysis of his D map, hereby revisiig he DJ model. We also irodue he oio of syhroized ad asyhroized yles as well as heir basis of araio i seio 3.. Seio 4 he reurs o he D sysem i order o sudy he effes of globalizaio, or a ouplig, o iovaio dyamis i he wo ouries. Firs, i Seio 4., we show ha he effes of globalizaio o he ross-oury disribuio of his model a is uique seady sae ad alog syhroized fluuaios are ideial wih hose of he HK model. For he res of he paper, we assume he parameer odiio ha esures he exisee of a sable period- yle i he DJ model. I Seio 4., we osider he symmeri ase where he wo ouries are of equal size. The, i Seio 4.3 we ur o he asymmeri ases o sudy he role of oury size differees o he syhroizaio effes of globalizaio. I Seio 4.4, we offer some ime series plos of equilibrium raeories, whih eapsulae he key prediios of he model. They also help o illusrae rasie behaviors. We olude i Seio 5.. Model Time is disree ad idexed by {0,,,...}. The world eoomy osiss of wo ouries, idexed by or k = or. The represeaive household of oury ielaially supplies he sigle oradable faor, labor, by rae, L (measured i is effiiey ui) a he wage w. The wo ouries are sruurally ideial, ad may differ oly i labor supply, so we le L L wihou loss of geeraliy. The household osumes he sigle oradeable fial good, whih is ompeiively produed by assemblig he wo ypes of radeable iermediae ipus, wih he followig Cobb-Douglas ehology: () Y k C k o X k X k, (0 < α < ), Page 9 of 58

10 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio where o X k is he homogeeous ipu, produed wih he liear ehology ha overs oe ui of labor io oe ui of oupu. This ipu is ompeiively supplied ad radeable a zero os so ha he law of oe prie holds for his ipu. By hoosig his ipu as he umeraire, we have w, ad ype of he ipus, () X x ) k w = holds wheever oury produes he homogeeous ipu. The seod X k, is a omposie of differeiaed ipus, aggregaed as k ( d, (σ > ), where x ( ) is he quaiy of a differeiaed ipu variey v used i he fial goods k produio i oury k i period ; σ > is he dire parial elasiiy of subsiuio bewee a pair of varieies, ad is he se of differeiaed ipu varieies available i period, whih hages over ime due o iovaio as well as obsolesee. These differeiaed varieies a be lassified depedig o where hey are produed ad wheher hey are supplied ompeiively or moopolisially. Thus, = = m ( ), where is he se of all m differeiaed ipus produed i i period : m is he se of ew ipu varieies irodued ad produed i ad sold exlusively (ad hee moopolisially) by heir iovaors for us oe period. Ad is he se of ompeiively produed ipu varieies i i period, whih were irodued i he pas. Hee, predeermied i period. Demads for Differeiaed Ipus: m is edogeously deermied i i period, while Assumig he balaed rade, he demad urves for hese ipus by he fial goods seor i k are derived from () as: is (3) x ( ) = k pk ( v) pk( v) X k P k P k Y Pk Y P k k = pk ( v) P k wk L P k k, where p ( ) k give by is he ui prie of variey v i k; P k is he prie idex for differeiaed ipus i k, (4) P p ) k k ( d, Page 0 of 58

11 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Y ad P P is he prie of he fial good. The ui prie of variey v depeds o k, beause k k of he (ieberg) rade oss. Tha is, o supply oe ui of a variey i k, uis eed o be shipped from. The he effeive ui prie i k is p ( ) = p ( ) k p ( ) for. Iserig his expressio o (3), he oal demad for eah variey a be obaied as: (5) D ( ) = k k x k ( v) = A p ( )), for where ( k k k wk Lk (6) A k ( P ) k, wih ( ) k k, may be ierpreed as he demad shif parameer for a variey produed i, wih k ( k ) beig he weigh aahed o he aggregae spedig i oury k. We follow HK ad assume ; so ha ; ( ). Thus, [0, ) measures how muh he fial goods produers sped o a impored variey, relaive o wha hey would sped i he absee of he rade os; ad i is iversely relaed o, wih 0 for ad for. This is our measure of globalizaio. Differeiaed Ipus Priig: Produig oe ui of eah variey of differeiaed ipus requires uis of labor, so ha he margial os is equal o w for. Sie all ompeiive ipus produed i he same oury are pried a he same margial os, ad hey all eer symmerially i produio, we ould wrie, from (5), as: (7) p ( ) w p ; D (v) A ( p ) y for, ( = or ), where p ad y are he (ommo) ui prie ad oupu of eah ompeiive variey produed i oury ad period. Eq. (5) shows ha all moopoliss fae he same osa prie elasiiy of demad,. Thus, hey all use he same marked-up rae. Hee all moopolisi varieies produed i he same oury are pried equally, ad produed by he same amou beause hey all eer symmerially i produio. Thus, (8) p w m ( ) p ; D (v) / m A ( p ) m y for, m Page of 58

12 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio where m p ad m y are he (ommo) ui prie ad oupu of eah moopolisi variey produed i oury ad period. From (7) ad (8), p y (9) ; m m p y p y ; m m p y (, e). Thus, a ompeiive variey is heaper, ad hee produed ad sold more ha a moopolisi variey. Furhermore, he fial goods produer speds more o a ompeiive variey ha o a moopolisi variey by he faor, >. Usig (7)-(9), he prie idies i (4) a be wrie as: where m m P N p N p k N ( N m a be furher wrie as: k k N N m p p m k w ) deoe he measure of ( ). Usig ad, his (0) P / w M w where k m () M N N /, k k M, m is he effeive oal ipu varieies produed i available o he fial goods produers, i.e., whih apures he degree of ompeiio ha iovaors would have o fae upo eerig. Noe ha he measure of moopolisi varieies is disoued by o over i o he ompeiive variey equivale i eq.(). Thus, a ui measure of ompeiive varieies has he same effe wih measure of moopolisi varieies. Wih >, a ompeiive variey is more disouragig o iovaors ha a moopolisi variey. Noe ha is moooe ireasig i σ, wih as ad e , as, ad ye, i varies lile wih over a empirially releva rage, wih.37 a 4 ad.6 a 4. For his reaso, we se. 5 for all of our umerial demosraios. 4 Noe also ha he measure of foreig varieies is muliplied by < o over i o he domesi variey equivale i eq.(0). Iroduio of New Varieies:, 4 I urs ou ha we eed (i.e., ) for geeraig edogeous fluuaios. Page of 58

13 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio I eah period, ew varieies of differeiaed ipu varieies may be irodued by usig f uis of labor per variey i eah oury. Followig DJ, we assume ha iovaors hold moopoly over heir iovaios for oly oe period, he same period i whih heir varieies are irodued. Wih free ery o iovaio aiviies, he e beefi of iovaio mus be equal o zero, wheever some iovaios ake plae, ad i mus be egaive wheever o iovaio akes plae. Thus, he followig omplemeariy slakess odiio holds: m m m N 0 ; ( p w ) y w f 0, m m m where oe of he wo iequaliies holds wih he equaliy: 0. I oher words, eiher he N zero profi odiio or he o-egaiviy osrai o iovaio mus be bidig i eah oury. Noe ha he gross beefi of iovaio is equal o he moopoly profi eared i he same period i whih a ew variey is irodued, beause iovaors lose is moopoly afer oe period. By usig (7)-(9) ad (), hese odiios a be furher rewrie as m () N ( M N ) 0; A ( w ) = y f y m. For he remaider of his paper, we follow HK ad osider he ase of ospeializaio, where boh ouries always produe he homogeeous ipu, whih esures w, for all. (See Appedix A for a suffiie odiio for he o-speializaio.) By seig w w i eq. (6) ad (0), eq. (), beomes m L L k (3) N ( M N ) 0; f, ( k). ( M M k ) ( M M k / ) Thus, iovaio is aive i oury, if ad oly if he reveue for a ew variey irodued i oury, give i he square brake, is us eough o over he os of iovaio. 5 The firs erm i he brake is he reveue from is domesi marke,, equal o is aggregae spedig o differeiaed ipus,, divided by he effeive ompeiio i faes a home, M M ) L ( k m m = N N ( N N ). Noie ha he measure of ompeiive varieies is muliplied by k k, relaive o he moopolisi varieies, ad ha he measure of he foreig varieies are muliplied by, relaive o he home varieies, due o he disadvaage he foreig varieies 5 Noe ha, from eq. (8), he gross profi per ui of he reveue is ( p w ) / p /. Page 3 of 58

14 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio suffer i heir expor marke,. The seod erm i he brake is he reveue from is expor marke, k, equal o is aggregae spedig o differeiaed ipus, Lk, divided by he effeive m m ompeiio i faes abroad, ( M M / ) = N N ( N N )/. Noie ha he k measure of he foreig varieies are muliplied by /, relaive o he home varieies, due o he advaage he foreig varieies eoy i heir domesi marke, k. Obsolesee of Old Varieies: All ew varieies, irodued ad supplied moopolisially by heir iovaors i period, are added o he exisig old varieies of differeiaed ipus whih are ompeiively supplied. Eah of hese varieies is sube o a idiosyrai obsolesee shok wih probabiliy, (0,). Thus, a fraio (0,) of hem survives ad arries over o he ex period ad beome ompeiively supplied, old varieies. 6 m (4) N N N N ( M N ) This a be expressed as:. (0,). ( = or ) Dyamial Sysem: To proeed furher, le us irodue ormalized measures of varieies as: fn ( L ) ; i L The, eqs.(3) a be rewrie as: m fn ( L ) ad m L (5) i ( m ) 0 ; m h m ), ( k where h ( ) 0 is impliily defied by m k fm ( L ) L k k i h ( m k s ) m k h ( m k sk ) m k, / wih s L /( L L ), he share of oury. Eq.(4) a be wrie as: (6) i ( m ) m ( ) 6 I addiio, we ould assume ha labor supply i eah oury may grow a a ommo, osa faor, G ; L L0( G). The, he measures of varieies per labor would follow he same dyamis by replaig wih / G. I urs ou ha we eed G / e for geeraig edogeous fluuaios. To see wha his implies, le ( d) T ad G ( g) T, where T is he period legh i years, d he obsolesee probabiliy per year ad g he aual growh rae of he exogeous ompoe of TFP. The, log( G / ) T log[( g)/( d)] ( g d) T log( e ) Page 4 of 58

15 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Noie ha eq.(5) may be ierpreed as he equilibrium odiios of he sai iovaio games simulaeously played i he wo ouries. Codiioal o he urre global marke odiio, (, ) R, whih shows how sauraed he wo markes are from pas iovaios, he iovaors i eah oury deide wheher o irodue ew varieies. For ay [0,), he Nash ouomes of hese games i period are uique. Ideed, eq.(5) a be solved for is uique soluio, m ( m, m ) R as a fuio of (, ) R. 7 Iserig his soluio io eq.(6) geeraes he marke odiio i he ex period, (, ) R. Thus, we obai he D-dyamial sysem ha govers he equilibrium law of moio for (, ) R, whih we sae formally as follows. R Theorem: For eah iiial odiio,, 0 0 0, 0 =, he equilibrium raeory, 0, is obaied by ieraig he D-dyamial sysem, F( ) give by: (7) where s ( ) ( ) for D s ( ) ( ) for D for h ( ) ( ) h ( ) ( ) for, ; LL, R s ( ) HH R h( k D HL F, : R R, R s ( ); h ( ) D LH, R h ( ); s( ) s s s ( ) s ( ) mi,, wih 0.5 s s ad h ( k ) 0 defied ) 7 Oe may woder wha happes if ρ =. The, he wo markes beome fully iegraed, ad here will o home marke advaage; he loaio of iovaio o loger maers. As a resul, eq.(5) o loger has a uique soluio; ad m ( m, m ) R, ad hee i ( i, i ) R beome ideermiae. However, m m ad hee i i is uiquely deermied by, ad hee he dyamis of he world aggregaes follows he same D-dyamis obaied by DJ. Effeively, he world eoomy beomes a sigle losed eoomy. Page 5 of 58

16 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio s sk impliily by. h ( ) h ( ) / k k k k See Appedix B for he derivaio of eq.(7). Oe we obai he equilibrium raeory for (, ) R by ieraig his D-sysem, i is sraighforward o obai he equilibrium raeory for may oher variables of ieres. For example, from eq.(5) ad eq.(7), he dyamis of iovaios, i heir ormalized form, i m ) )/ a be ( ( derived as: i ( s( ) ) ; i ( s( ) ) for D LL, i 0 i 0 for D HH, (8) i 0 i ( h ( ) ) for D HL, i ( h ( ) ) ; i 0 for D LH. Likewise, i a be show ha Toal faor produiviies (TFPs), Z Y / L w / P follow log( Zk) 0 log( zk) wih: z ( ) s z ( ) s for D LL, k k k k k (9) z z for D HH, z ) z h ( h ( ) z Some Prelimiary Observaios: z h ) for D HL, ( for D LH. h ( ) Sarig from he ex seio, we will odu a sep-by-sep aalysis of he D-sysem, eq. (7). However, i is worh offerig some prelimiary observaios abou his sysem. Firs, i is haraerized by he four parameers: (, e) ; (0, ) ; (0, ) ; ad s [0.5, ). (The firs wo ome from DJ, ad he seod wo from HK.) Seod, i is a oiuous, piee-wise smooh sysem, osisig of four smooh maps defied over four domais, depedig o whih of he wo iequaliies i eq.(5) hold wih he equaliies i eah oury. Third, is dereasig i dereasig i i D LH ad D LL ad ireasig i i D LL ad D HL ad ireasig i i D HH ad D HL. Similarly, is i D LH ad D HH. This suggess, amog Page 6 of 58

17 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio ohers, ha he map is oiverible. Fourh, if / s( )/ s( ), / s ( )/ s( ) Thus, he ray,, ) R / s ( ) / s ( ), is forward-ivaria. Oe he raeory (. reahes here, i says here forever. However, i is o bakward-ivaria, beause he map is oiverible. 8 Figure illusraes he four domais ad heir boudaries for s / s. For 0 D HH, boh markes are so sauraed ha here is o iovaio, i 0 ad i 0 obsolesee shoks, origi i his domai. For. Due o he ad, so ha he map is oraig oward he D LL, eiher marke is sauraed ha iovaio is aive ad he zero profi odiio holds i boh markes. Due o he obsolesee shoks, he uique seady sae of his sysem is loaed i his domai,, D LL. For D HL, he oegaiviy osrai is bidig i oury ad he zero-profi odiio is bidig i oury. Iovaio is hus aive oly i oury, give by i ( h ( ) ). Beause ρ > 0, whih implies h ( ) 0, iovaio i oury is disouraged by he ompeiive varieies based i ' oury (a higher higher ), bu o as muh as by he ompeiive varieies based i oury (a ), beause ρ <, whih implies h ( ). Hee, he iso-iovaio urves for oury i his domai, ' i h ( ) / for i > 0 (o draw i Figure ), are dowwardslopig wih heir slopes less ha oe i absolue value. Furhermore, i beomes seeper as varies from zero o oe. So is he border bewee D HL ad D LL, h ). Likewise, i D LH, he iso-iovaio urves for oury, ( h ( ) i / for i > 0 (o draw i Figure ), are dowward-slopig wih heir slopes greaer ha oe i absolue value. Furhermore, i beomes less seep as varies from zero o oe. So is he border bewee D LH ad D LL, h ). ( Before proeedig, we offer some words of auio o he reader ausomed o see he D-phase diagram for a ordiary differeial equaio i wo variables. Our model is i disree ime, so ha a raeory geeraed by ieraig eq.(7) a be represeed as a sequee of pois, whih hop aroud i he sae spae. I ao be represeed as a oiuous flow. This 8 A se, S R, is forward-ivaria, if F( S) S, ad is bakward-ivaria, if F ( S) S. Page 7 of 58

18 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio is why we did o draw ay isolie urves or ay arrows idiaig he direio of movemes. They are o pariularly useful for udersadig he dyamis; ideed, hey ould be misleadig. 3. Auarky ad Deoupled Iovaio Dyamis h We begi our aalysis of eq.(7) wih he ase of auarky, 0. The, s () s ad ( m ) s. Hee, eq. (7) beomes: k s ( ) for s ( ) for for s ( ) s ( ) for, D LL R s ;, s D HH R s ;, s D HL R s ;, s D LH R s ;, s as illusraed i Figure. No surprisigly, he dyamis of he wo ouries are urelaed i auarky, ad hee he D sysem a be deoupled o wo idepede D sysems: f L ( ) ( s ( ) ) for s ; (0) f ( ( 0 ; e ) ) f H ( ) for s From (8) ad (9), iovaio ad TFP move as: i max s,0 ; z maxs,.. 3. D-Aalysis of The Skew Te Map: Revisiig Deekere-Judd (99) Page 8 of 58

19 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 3 illusraes he D-sysem ha govers he dyamis of eah oury, eq. (0), whih is isomorphi o he origial DJ sysem. (We drop he oury idies i his subseio.) I is a PWL, oiverible map wih he followig wo brahes: 9 The H-brah, defied over s, is upward-slopig, ad loaed below he 45º lie. Wih oo may ompeiive varieies, he marke is oo sauraed for iovaio. Hee, he oegaiviy osrai is bidig, i 0. Wih o iovaio ad, he map is oraig over his rage. The L-brah, defied over s, is dowward-slopig. Wihou oo may ompeiive varieies, here is aive iovaio, so ha he zero-profi odiio is bidig. Noie ha i is dowward slopig beause. Beause old, ompeiive varieies are more disouragig ha ew moopolisi varieies, ui measure of addiioal ompeiive varieies his period would rowd ou measure of ew varieies so ha he eoomy will be lef wih fewer ompeiive varieies i he ex period. This effe is sroger whe differeiaed varieies are more subsiuable (a higher σ ad hee, a higher θ). Sie, he uique seady sae, s s, ( ) is loaed i L-brah, where he slope of he map is equal o. Hee, he uique seady sae is sable ad ideed globally araig for. For, i is usable. For his ase, here exiss a absorbig ierval, J [ s, f L ( s)], idiaed by he red box i Figure 4. Iside he red box, here exiss a uique period -yle, s s ( ) ( ) L H, ha aleraes bewee he L- ad he H-brahes. This is also illusraed i Figure 4. The graph of he d ierae of he map, f f ( ) f ( ), show i blue, rosses he 45 lie hree imes. The red do idiaes he usable seady sae, f '( ) f '( ) ( ), where he slope of he d ierae is. The wo blue dos, oe i he L-brah ad he oher i he 9 The map of his form is alled he skew e map, whih has bee fully haraerized i he applied mah lieraure: see, e.g., Sushko ad Gardii (00, Seio 3.) ad he referees herei. Page 9 of 58

20 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio H-brah, idiae he wo pois o he period- yle, L fh( H ) fh fl( L ) ad f ( ) f f ( ). The slope of he d ierae a hese pois is f ' f ' H L L L H. Hee, for H iiial odiios (i.e., uless he iiial odiio is equal o, he period -yle is sable ad araig from almos all Thus, he araig -yle exiss if ad oly if or oe of is pre-images). 0 L H. I words, i exiss if ad oly if he survival rae of he exisig varieies is high eough ha iovaio his period is disouraged by high iovaio oe period ago, bu o high eough ha i is o disouraged by high iovaio wo periods ago. For, he uique period -yle is usable. For his rage, DJ oed ha he d ierae of he map is expasive over he absorbig ierval, i.e., f '( ) for all differeiable pois i J, from whih hey observed i heir Theorem ha he sysem has ergodi haos by appealig o Lasoa ad Yorke (973; Theorem 3). I fa, we a say more. From he exisig resuls o he skew e map, i a be show ha his sysem has a robus haoi araor ha osiss of oe ierval, wo iervals, four iervals, or more geerally, m - iervals, (m = 0,,, ). Figure 5 summarizes he asympoi behavior of he equilibrium raeory govered by eq. (0) i he (δ, σ)-plae. Noie ha edogeous fluuaios our wih a higher σ (hee a higher θ), whih makes ompeiive varieies eve more disouragig o iovaors ha moopolisi varieies, whih makes he delayed impa of iovaio aused by 0 The pre-images of a poi,, are all he pois ha map io afer a fiie umber of ieraios. Noe ha he usable seady sae,, has ouably may pre-images beause our map is oiverible. Oe of hem, f H ( ), is show i Figure 4. I oras, may exisig examples of haos i eoomis are o araig, pariularly hose relyig o he Li- Yorke heorem of period-3 implies haos. This heorem saes ha, o he sysem defied by a oiuous map o he ierval, he exisee of a period-3 yle implies he exisee of a period- yle for ay, as well as he exisee of a aperiodi (haoi) fluuaio for some iiial odiios. The se of suh iiial odiios may be of measure zero. For suh a haoi fluuaio o be observable, i has o be araig, so ha a leas a posiive measure of iiial odiios mus overge o i. Furhermore, mos examples of haoi araors i eoomis are o robus (i.e., hey do o exis for a ope regio of he parameer spae), beause he se of parameer values for whih a sable yle exiss is dese, ad he se of parameer values for whih a haoi araor exiss is oally disoeed (alhough i may have a posiive measure). Moreover, a rasiio from period- yle o haos ofe requires a ifiie asade of bifuraios, as hese are geeral feaures of a sysem geeraed by everywhere smooh maps, whih mos appliaios assume. Our sysem a geerae a haoi araor, whih is robus ad a rasiio for he sable -yle o haos is immediae, beause our sysem is pieewise liear. Sushko ad Gardii (00) disuss more o hese issues. Page 0 of 58

21 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio he loss of moopoly by he iovaor more sigifia, ad wih a higher δ, whih makes more ompeiive varieies survive o disourage urre iovaors. 3. A D-View of Auarky: Syhroized vs. Asyhroized -Cyles Alhough he iovaio dyamis of he wo ouries i auarky a be idepedely aalyzed, i is useful o view hem oily as a D-sysem o provide a behmark agais whih o observe he effes of globalizaio sudied i he ex seio. We fous o he ase where, so ha he D sysem of eah s oury has a usable seady sae, ad a sable period -yle, ( ) s s L H ( ) ( ), whih aleraes bewee he L- ad H-brahes (i.e., i aleraes bewee he period of aive iovaio ad he period of o iovaio). As a Dsysem, he wo-oury world eoomy has: A usable seady sae,, A pair of sable period -yles: o Syhroized -yle: ; D LL, L L D LL, H H D HH, alog whih iovaio i he wo ouries are aive ad iaive a he same ime. Furhermore,, i, ad i he same direio aross he wo ouries. For his reaso, we shall all i he syhroized -yle. o Asyhroized -yle: D L, H LH aive oly i oe oury. Furhermore,, i, ad H, L D HL Z, move, alog whih iovaio is Z, move i he opposie direio aross he wo ouries. For his reaso, we shall all i he asyhroized -yle. 3 A pair of saddle -yles: L, D LL H, D ad HL, H D., L LL D LH Noie also ha he oly sable yle is a period- yle i he DJ model. This is due o he resriio o he relaive slope of he wo brahes, f '/ f ' e L H. I geeral, he skewed e map a geerae a sable yle of ay posiive ieger, if he slopes of he ireasig ad dereasig brahes are uresried. 3 Laer we will all ay -yle ha aleraes bewee D HH ad D LL syhroized ad ay -yle ha aleraes bewee D HL ad D LH asyhroized also i asymmeri ases. Page of 58

22 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio I Figure 6, he ligh gree do idiaes he usable seady sae, he dark gree dos he wo saddle -yles, ad he blak dos he wo sable -yles. The red area illusraes he basi of araio for he syhroized -yle ad he whie area he basi for he asyhroized - yle. Noie ha eiher basi of araio is oeed, whih is oe of he feaures of a oiverible map. 4 The boudaries of hese basis are formed by he losure of he sable ses of he wo saddle -yles Globalizaio ad Ierdepede Iovaio Dyamis: D Aalysis We ow ur o he ase > 0 o sudy he effes of globalizaio. 4. A Brief Look a he Uique Seady Sae: Reierpreig Helpma-Krugma (985) Firs, we look a he uique seady sae of eq.(7), s ( ), s ( ), ( ), whih is sable ad globally araig if. A his seady sae, iovaios ad he effeive measures of he varieies produed i eah oury are give by: ( ) i i s ( ), s ( ) ; m m s ( ), s ( ), ( ), Figure 7a shows how he share of oury i hese variables depeds o is size a he seady sae. I he ierior, i is equal o: i m ( ) s s s( ). i i m m Noie ha he slope is (+ρ)/( ρ) >. Thus, a disproporioaely larger share of ipu varieies is produed ad a disproporioaely large share of iovaio is doe i he oury ha has he 4 To see why he wo basis of araio show he hess board paers i Figure 6, osider he dyamial sysem defied by he d ierae of he map, eq.(0), whose graph is show i blue i Figure 4. I has wo sable fixed pois, L ad H, whose basis of araio are give by aleraig iervals, whih are separaed by is usable fixed poi,, is immediae pre-image, f H ( ), ad all of is pre-images. If boh ouries sar from he basi of araio for L ( H ), hey overge o he syhroized -yle i whih hey boh iovae i every eve (odd) period. O he oher had, if oe oury sars from he basi of araio for L ad he oher sars from he basi of araio for H, hey overge o he asyhroized -yle i whih oe oury iovaes i every eve period ad he oher iovaes i every odd period. 5 The sable se of a ivaria se (say, a fixed poi, a yle, e.) is he se of all iiial odiios ha overge o i. I is eessary o ake he losure i order o ilude he usable seady sae ad all of is pre-images. Page of 58

23 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio larger domesi marke ad hee he larger oury beomes he e exporer of he differeiaed ipus varieies (Home Marke Effe), wih he smaller oury beomig he e exporer of he homogeeous ipu. Furhermore, his effe beomes magified if he rade os beome smaller (i.e. wih a larger ρ), as show i Figure 7b. 6 Thus, he seady sae of our model shares he same properies wih he equilibrium of he sai HK model. Oe migh hik ha he omparaive seady sae aalysis of his kid would make sese oly if he seady sae is sable, i.e.,. I fa, he above omparaive aalysis is also iformaive eve whe he seady sae is usable, beause globalizaio auses syhroized yles ad he share of oury asympoially overges o he same seady sae value, s, as will be show i Seio 4.3. For he remaider of his paper, we assume ha he uique seady sae is usable,. Ideed, we will fous o he ases where he dyamis of eah oury overges o he sable period- yle i auarky,. 4. Syhroizaio Effes of Globalizaio: Symmeri Cases I his seio, we assume ha he wo ouries are of equal size ( s / D-sysem defied by eq.(7), beomes symmeri as follows. () / ( ) for D / ( ) for D for h( ) ( ) h( ) ( ) for, LL R, / HH R h( k ) ), so ha he D HL D LH where h ( ) 0 is defied impliily by. h( ) h( ) /, R / ; h( ), R h( ); / 6 Noe ha he graph i Figure 7b is a orrespodee a ρ = (he lak of lower hemi-oiuiy), beause he equilibrium alloaio is ideermiae if ρ =, as poied ou earlier. Page 3 of 58

24 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 8 shows he symmeri D sysem, wih he blue arrows illusraig how he four domais hage wih. Firs, he diagoal, (, R, is forward-ivaria, ) ad he dyamis o is idepede of. I fa, i is he skew e map, give by eq. (0) wih s /. Seod, has o effe o D LL. Third, i D LH, a higher redues iovaio i, give by i h( ) ), as he ompeiive varieies produed i,, disourages ( iovaors i. This also auses D LH o shrik ad D HH o expad, wih he boudary, h( ), iiially verial (as / ) a = 0, ils ouer-lokwise as ireases, ad approahig o as. A higher also ils he iso-iovaio urves i D LH, h( ) / (o draw; horizoally paralleled o he boudary bewee D LH ad D HH ), i i he same way. Likewise, a higher redues iovaio i i D HL. This auses D HL o shrik ad D HH o expad, wih he boudary, h( ), iiially horizoal (as / ) a = 0, ilig lokwise as ireases, ad approahig o as. I has he same ilig effe o he iso-iovaio urves i D HL, h( ) / (o draw; verially paralleled o i he boudary bewee D HL ad D HH ). Take ogeher, his implies ha a higher auses he aligme of iovaio ieives aross he wo ouries, i he sese ha boh a higher ad a higher have similar disouragig effes o he iovaors i boh ouries. For, eah oury would have a usable seady sae, / ( ) ad a sable -yle, / / L L ( ) H H i ( ) auarky, = 0. Thus, as already poied ou i Seio 3., he world eoomy osisig of he wo ouries i auarky has he wo sable -yles. Oe of hem is he syhroized -yle, L L DLL H, H DHH L H DLH H, L DHL,. The oher is he symmeri asyhroized yle,,. Now, le rise. Sie he diagoal is ivaria, ad has o effe o he dyamis i, D, D, exiss for all (0, ) D LL ad D HH, he syhroized -yle, L L LL H H HH. Ideed, i is idepede of ad is loal sabiliy is o affeed. Page 4 of 58

25 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio I addiio, here exiss a uique symmeri asyhroized -yle, a a a a L H DLH H, L DHL,, for all (0, ). To see his, if i exiss, a a a a saisfy, from eq. (), = h ( H ) ( ) h( H ) ( ) a H wrie more ompaly as: a a h( H ) H, where (,). = By iserig his expressio io he defiiio of h, we obai a a () L H. / Noe ha implies implies a L L H a L ad a H mus, whih a be a H ad ha / / a a = h ). This proves he exisee ad he uiqueess of he H H ( a H a a a a, D, D. symmeri asyhroized -yle, L H LH H L HL For = 0, his -yle is equal o L, H H, L as varies, ad is o equal o, L, H H L usable for a suffiiely large. More formally, Proposiio: Le s 0. 5, (, e). However, i moves oiuously, for > 0. Furhermore, i beomes, ad. For all (0, ), here exiss a a a a a, D, D, give by uique symmeri asyhroized -yle, L H LH H L HL where a a a a a H ; L H H h( H ) (,) ad h ( ) 0 solves. Furhermore, h( ) h( ) / i) For 0 < ( ) /, i is a sable fous; ii) For / ( ), i is a sable ode; iii) For ( ), i is a saddle, / / where ( ) is a oiuous, ireasig fuio wih ( 0) 0 ad / Page 5 of 58

26 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio ( ). See Appedix C for he proof. This proposiio says ha he uique symmeri asyhroized -yle exiss for all (0, ), bu i is sable for 0, ) ad usable for (,), where ( (0,) is give by ( ). Thus, for a suffiiely large (or a suffiiely small rade os), he sable asyhroized -yle disappears. Furhermore, eve before is disappearae, a higher expads he basi of araio for he syhroized -yle ad redues ha for he asyhroized -yle for 0, ). ( Figures 9a- show his umerially wih hree differe values of = 0.7, = 0.75, ad = I all hree ases, a irease i ause he red area (he basi of araio for he syhroized -yle) o expad ad he whie area (he basi of araio for he symmeri asyhroized -yle) o shrik. These figures show ha he red area fills mos of he sae spae a = 0.8. However, he symmeri asyhroized -yle is sill sable a = 0.8, so ha he whie area sill oupies a posiive (hough very small) measure of he sae spae. Oly a a higher value of, he symmeri asyhroized -yle loses is sabiliy. For, he red area overs a full measure of he sae spae (i.e., he syhroized -yle beomes he uique araor ad he equilibrium raeory overges o he syhroized -yle for almos all iiial odiios). 8 7 Reall ha he sable -yle exiss i auarky for, whih implies ( , ) for =.5. If we raslae his i erms of T (he period legh i years), d (he obsolesee probabiliy per year) ad g (he aual growh rae of he exogeous ompoe of TFP), ( / )log( ) ( g d ) T < log( ) a a a a Tehiially speakig, he symmeri asyhroized -yle, L, H D LH H, L D HL, udergoes a subriial pihfork bifuraio a. Reall ha he losure of he sable ses of he symmeri pair of saddle -yles form he boudaries of he red ad whie areas. A = 0, his symmeri pair of saddle -yles are give by L, DLL H, DHL ad H DLH, L DLL ross he boudary of D LL a ' '' " ' H, L D L H ' " '' ' L, H D LH HL ad D LH H, L D HL,. As rises, hey move ad simulaeously, afer whih hey beome a symmeri pair of saddles of he form,,. Thus, for (, ), here exis hree asyhroized -yles; a symmeri pair of asymmeri asyhroized -yles, whih are saddles, ad he symmeri asyhroized -yle, whih is sable. The, as, he symmeri pair of he saddle -yles merge wih he symmeri asyhroized -yle ad disappear, afer whih he laer beomes a saddle. However, he ierval, (, ), seems very arrow. Aordig o our alulaio, < ρ < < ρ < for δ = 0.7; < ρ < < ρ < for δ = 0.75; ad < ρ < ; < ρ < for δ = 0.8. Page 6 of 58

27 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio 4.3 Syhroizaio Effes of Globalizaio: Asymmeri Cases We ow ur o he ases where he wo ouries differ i size; s 0.5 s s. We oiue o assume so ha, i auarky, eah oury has a usable seady sae, ad a sable period -yle. Thus, viewed as a D-sysem, he world eoomy has a usable seady sae, a pair of sable -yles, oe syhroized ad oe asyhroized, whose basis of araio are already show i Figure 6 as Red ad Whie, ad he boudaries of he wo basis are give by he losure of he sable ses of a pair of saddle -yles, as already poied ou i Seio 3.. Now, le rise. The blue arrows i Figure 0a illusrae he effes of a higher, whih are abse i he symmeri ase. Tha is, hese effes are i addiio o hose illusraed by he blue arrows i Figure 8 for he symmeri ase. Wih uequal oury sizes, s 0. 5 s, a higher ireases s ) s ( ), whih is ohig bu he magifiaio of he home ( marke effe i he HK model. This auses he ray, / s ( )/ s ( ), o roae lokwise, ad he border poi of he four domais, s ), s ( ), o move souheas. This oiues uil ( s /s, whe D ad LL D, vaish. For s HL /s show i Figure 0b., here is o iovaio i oury, as As log as s / s, iovaio will ever sop i eiher oury. For his 0 s s s s D,, D, rage, here always exiss he sable syhroized -yle, L L LL H H HH where s ( ) s s s ( ) L ; H ( ). ( ) Alog his syhroized -yle, he world eoomy aleraes bewee D ad LL D HH o he ray, / s ( )/ s ( ), ad hee he share of oury is equal o s ( ). a a a a D,, ad says, D, for There also exiss a sable asyhroized -yle, L H LH H L HL a small eough. For, i disappears. 9 Furhermore, eve before is disappearae, 9 A, he sable asyhroized -yle ollides wih oe of he (o loger symmeri) pair of saddle - yles o-exisig for, ad hey boh disappear via a fold (border ollisio) bifuraio. Page 7 of 58

28 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio a higher auses he basi of araio for he syhroized -yle o expad ad he basi of araio for he asyhroized -yle o shrik. Furhermore, his ours more rapidly wih a higher s. Figures a-d illusrae hese umerially, for four differe values of s = 0.55, = 0.6, = 0.7, ad = 0.8, for = Noie ha he red beomes domia faser for a higher s = 0.6. These figures also show a sudde appearae of ifiiely may red islads iside he whie area us before he disappearae of he asyhroized -yle. 30 for = 0.7 ad = 0.8. The resuls are very similar We have also esimaed, he riial value a whih he sable asyhroized -yle disappears, leavig he syhroized -yle as he uique araor. This is repored i his Table (. 5 for all). s TABLE s / s = 0.7 = 0.75 = Noie ha delies very rapidly as s ireases from 0.5, bu i hardly hages wih δ. Noie also ha i is sigifialy less ha s /s. Tha is, as we redue he rade oss, he asyhroized -yle disappears muh earlier ha he smaller oury sops iovaig. Figure show he graph of he riial value as a fuio of s for δ = 0.7, = 0.75, ad = shows ha he riial value delies sharply, as s ireases from 0.5. Thus, eve a small differee i oury sizes would ause syhroizaio o our very rapidly. Eah 30 This is due o a oa bifuraio, where a riial urve rosses he basi boudary, afer whih a ew se of ouably ifiie pre-images are reaed, aoher ommo ourree i sysems wih oiverible maps. 3 The hree graphs vary lile wih δ. We would o be able o ell hem apar, if we were o superimpose hem. Page 8 of 58

29 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio A ieresig quesio is his. Suppose ha he wo ouries are iiially ou of sy i auarky. Ad whe globalizaio auses hem o syhroize, whih oury ses he empo of global iovaio yles. Or o pu i differely, whih oury aduss is rhyhm o syhroize? Is i he smaller oury or he larger oury? 3 To aswer his quesio, we look a he d ierae of he map, F F( ) F ( ), ad is four sable seady saes, whih are he four pois o he wo sable -yles. I Figure 3, we use he followig four olors o idiae he four basis of araio for he four sable seady saes of he d ierae. Red: Basi of araio for he sable seady sae i D LL. This orrespods o he se of iiial odiios ha overges o he syhroized -yle alog whih he raeory visis D LL i eve periods ad D HH i odd periods. Azure: Basi of araio for he sable seady sae i D HH. This orrespods o he se of iiial odios ha overges o he syhroized -yle alog whih he raeory visis D HH i eve periods ad D LL i odd periods. Whie: Basi of araio for he sable seady sae i D LH. This orrespods o he se of iiial odiios ha overges o he asyhroized -yle alog whih he raeory visis D LH i eve periods ad D HL i odd periods. Gray: Basi of araio for he sable seady sae i D HL. This orrespods o he se of iiial odiios ha overge o he asyhroized -yle alog whih he raeory visis D HL i eve periods ad D LH i odd periods. Syhroizaio meas ha Red ad Azure expad, while Whie ad Gray shrik. Figure 3 shows ha, as ρ goes up, ad syhroizaio ours by Red ivadig Whie ad Azure ivadig Gray, isead of Red ivadig Gray ad Azure ivadig Whie, ad we observe he emergee of verial slips of Red ad Azure. We have experimeed wih may differe values of parameers, bu his paer has bee always observed. This meas ha he empo of syhroized fluuaios is diaed by he rhyhm of oury, whih is he larger oury ad ha oury, he smaller oury, aduss is rhyhm o he rhyhm of he larger oury. 4.4 Three Effes of Globalizaio: Some Traeories 3 We hak Gadi Barlevy for posig his quesio o us. Page 9 of 58

30 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Fially, we plo some raeories ha eapsulae he key prediios of he model i Figure 4. They also help o illusrae some rasie behaviors. 33 We fix s = 0.7, =.5, ad = 0.75, as i Figures ad 3. Wih hese parameer values, here exiss he sable asyhroized -yle for < = i addiio o he sable syhroized -yle. The laer beomes he uique araor for > = We geerae he plos uder he assumpio ha he wo ouries are iiially i auarky ( = 0), wih he iiial odiio very lose o is -periodi poi i D LH, so ha, osillaes alog he asyhroized -yle a = 0, = (0.5339, ) DLH = (0.79, 0.88) DHL for, L H he firs 0 periods, wih, H L / osillaig bewee (i eve periods) ad 0.34 (i odd periods). The, we le = 0. or = 0.3 afer he h period o. 34 Sie = 0.854, he sable asyhroized -yle disappears ad he wo ouries would almos surely overge o he syhroized -yle, L, L LL H H HH s s D s s, D, give by (0.60, 0.55 ) DLL (0.835, ) DHH for = 0.; (0.6646, ) DLL (0.886, ) DHH for = 0.3. The upper paels of Figure 4 show he plos of (red), (gree), ad / (blak). As umps from = 0 o = 0. (o he lef pael) or o = 0.3 (o he righ pael), shifs up ad shifs dow, ad so does /, demosraig he Home Marke Effe. Furhermore, / quikly sabilizes ad overges (o 0.5 for = 0. o he lef; o for = 0.3 o he righ). Noie ha oiues he paers of up ad dow, wihou ierrupio as hages. Thus, he bigger oury oiues o iovae i every eve period. I oras, slides dow wo oseuive periods (for = 0.) ad four oseuive periods (for = 0.3) immediaely afer he hage. As a resul, he smaller oury, whih iovaed i every odd period uder auarky, ow sars iovaig every eve period o syhroize o he rhyhm of he bigger oury. 33 We hak Bob Luas for his suggesio o ilude plos like hese. 34 Wih θ =.5, σ Thus, τ.35 for ρ = 0. ad τ.5 for ρ = 0.3. Page 30 of 58

31 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio The effes o produiviy a be see i he middle paels of Figure 4, whih plo (red) ad z (gree), ad i he boom paels, whih plo z / z (red) ad z / z (gree). The middle paels show how boh ouries beefi immediaely from he produiviy gais from rade. 35 This also shows up i he boom paels, as he huge spikes i he produiviy growh upo he hage. Noie ha produiviy i he wo ouries fluuae asyhroously before he hage; he, afer he spikes aused by he hage, hey sar syhroizig. z 5. Coludig Remarks This paper is he firs aemp o demosrae how globalizaio a syhroize produiviy fluuaios aross ouries. To his ed, we proposed ad aalyzed a wo-oury model of edogeous iovaio yles, buil o he work of Deekere ad Judd (99) ad Helpma ad Krugma (985). I auarky, iovaio dyamis i he wo ouries are deoupled. As rade oss fall ad ira-idusry rade rise, hey beome more syhroized. This is beause globalizaio leads o he aligme of iovaio ieives aross iovaors based i differe ouries, as hey operae i he ireasigly global (hee ommo) marke evirome. Syhroizaio ours faser (i.e., wih a smaller reduio i he rade os) whe he wo ouries are more uequal i size. Furhermore, eve a small oury size differee speeds up he syhroizaio sigifialy. Ad i is he larger oury ha diaes he empo of global iovaio yles, wih he smaller oury adusig is rhyhm o he rhyhm of he larger oury. This is beause he iovaors based i he smaller oury rely more heavily o he profi eared i is larger expor marke o reover he os of iovaio ha hose based i he larger oury. Our resuls sugges ha addig edogeous soures of fluuaios would help improve our udersadig of why ouries ha rade more wih eah oher have more syhroized busiess yles. We hose he Deekere-Judd model of edogeous iovaio yles as oe of our buildig bloks due o is raabiliy ad he uiqueess of he equilibrium raeory. We believe ha he basi iuiio should go hrough wih a muh wider lass of models of 35 Noie ha he produiviy i he smaller oury overshoos is log ru level. This is due o he legay of he small oury iovaig i auarky a a level ha ao be susaiable afer he globalizaio. Here, his legay effe is relaively small beause globalizaio ours i he period i whih oury would iovae if i remaied i auarky. Isead, if globalizaio ours i he period immediaely afer oury iovaed i auarky, a overshooig would be more prooued. Page 3 of 58

32 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio edogeous iovaio yles. 36 As log as globalizaio auses iovaors based i differe ouries o ompee agais eah oher i a ommo marke evirome, i should syhroize heir iovaio aiviies, regardless of he speifi mehaism hrough whih ieives o iovae are affeed. I he Deekere-Judd model, more ompeiive marke evirome disourages iovaios. The, as he marke evirome beomes more ompeiive i oe oury, all iovaors aroud he world who hope o make some profi by sellig o ha oury would be disouraged i a globalized world, bu oly loal iovaors would be disouraged i a less globalized world. I some oher models of iovaio, more ompeiive marke evirome migh eourage iovaios. The, as he marke evirome beomes more ompeiive i oe oury, all iovaors aroud he world who hope o make some profi by sellig o ha oury would be eouraged i a globalized world, bu oly loal iovaors would be eouraged i a less globalized world. Thus, regardless of wheher more ompeiio eourages or disourages iovaios, iovaors based i differe ouries would respod o a hage i he marke odiio i oe oury i he same direio i a more globalized world, bu o i a less globalized world. Thus, globalizaio should ause syhroizaio of iovaio aiviies aross ouries. Wha seems more ruial i our aalysis is he assumpio ha he ouries are sruurally similar. Wha if he ouries are sruurally dissimilar? For example, wha if globalizaio auses verial speializaio hrough some ypes of verial supply hais? Imagie ha here are wo idusries, oe Upsream ad oe Dowseam, eah produig he Dixi-Sigliz omposie as i he Deekere-Judd model. Ad suppose ha oe oury has omparaive advaage i U ad he oher i D. Our oeure is ha i would lead o asyhroizaio of iovaio yles. This is beause, ulike he wo ouries i he HK model, whih produe ad rade highly subsiuable, horizoally differeiaed goods, verial hais make he produio sruure of he wo ouries omplemeary. The, as he goods iovaed i he pas i oe oury lose heir moopoly, hey beome heaper, whih disourage he iovaors i ha oury, bu eourages he iovaors i he oher oury, 36 Of ourse, he prediio would have o be eessarily weaker if we used a model of iovaio yles, i whih a ylial equilibrium pah o-exiss wih a saioary equilibrium pah. Neverheless, oe should be able o sae he prediio i erms of he disappearae of he asyhroized yle uder globalizaio, alhough boh he syhroized yle ad he saioary equilibrium survive uder globalizaio. Page 3 of 58

33 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio whih produes heir omplemeary goods. 37 If his oeure is ofirmed, i is eraily empirially o iosise, beause he evidee for he syhroizig effe of rade is srog amog developed ouries, bu less so bewee developed ad developig ouries. Fially, we would like o sress ha iovaio migh be us oe hael hrough whih globalizaio a ause a syhroizaio of produiviy fluuaios aross ouries. We hope o explore oher possible haels as well i our fuure researh. For example, may ree sudies o maroeoomis of fiaial friios have demosraed he possibiliy of produiviy fluuaios due o redi yles i losed eoomy models. I a wo oury versio of suh a model, globalizaio migh lead o ross-oury spillovers of peuiary exeraliies, whih auses a syhroizaio of redi yles, ad hee produiviy omovemes, aross ouries. 37 This may ome as a surprise o hose familiar wih he exisig sudies ha ry o explai syhroizaio of busiess yles wih verial speializaio. However, i is o oradiory, beause hese sudies look a he propagaio effes of a oury speifi produiviy shok from oe oury o aoher. Here, we are osiderig how produiviy of differe ouries respods edogeously o a hage i he global marke odiio. I his paper, we showed ha produiviy movemes syhroize whe he wo ouries produe highly subsiuable goods. We oeure ha produiviy movemes would be asyhroized whe he wo ouries produe omplemes. (Our oeure is based o he resuls i our ompaio piee, Masuyama, Sushko, ad Gardii (forhomig), i whih we have ivesigaed a losed eoomy, wo-seor exesio of he Deekere-Judd model ad foud ha iovaio yles i he wo seors are asyhroized i he omposies produed i he wo seors beome more ompleme.) Page 33 of 58

34 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Referees: Aemoglu, Daro, Iroduio o Moder Eoomi Growh, Prieo Uiversiy Press, Prieo, 008. Aemoglu, Daro, Gio Gaia ad Fabrizio Ziliboi, Offshorig ad Direed Tehologial Chage, 03. Aemoglu, Daro, ad Fabrizio Ziliboi, Produiviy Differees, Quarerly Joural of Eoomis, 6, 00, Behabib, Jess, Muliple Equilibria i he Aghio-Howi Model, Researh i Eoomis 68, 04, -6. Bramoullé, Ya, ad Gilles Sai-Paul, Researh Cyles, Joural of Eoomi Theory, 45 (00): Deekere, Raymod, ad Keeh Judd, Cylial ad Chaoi Behavior i a Dyami Equilibrium Model, Chaper 4 i Jess Behabib, ed., Cyles ad Chaos i Eoomi Equilibrium Prieo Uiversiy Press, Prieo, 99. Evas, George, Seppo Hokapoa ad Paul M. Romer, Growh Cyles, Ameria Eoomi Review, 88, Jue 998, Fraois, Parik, ad Huw Lloyd-Ellis, Aimal Spiris hrough Creaive Desruio, Ameria Eoomi Review, 93, 003, Fraois, Parik, ad Huw Lloyd-Ellis, Implemeaio Cyles, Ivesme, ad Growh, Ieraioal Eoomi Review, 49, (Augus 008), Fraois, Parik, ad Huw Lloyd-Ellis, Shumpearia Cyles wih Pro-ylial R&D, Review of Eoomi Dyamis, (009): Frakel, Jeffrey A., ad Adrew K. Rose, The Edogeeiy of he Opimal Currey Area Crieria, Eoomi Joural, 08 (July 998): Gale, Douglas, Delay ad Cyles, Review of Eoomi Sudies, 63 (996), Gardii, Laura, Irya Sushko, ad Ahmad Naimzada Growig Through Chaoi Iervals, Joural of Eoomi Theory, 43 (008), Grossma, Gee, ad Elhaa Helpma, Produ Developme ad Ieraioal Trade, Joural of Poliial Eoomy, 97 (Deember 989), Grossma, Gee, ad Elhaa Helpma, Iovaio ad Growh i he Global Eoomy, MIT Press, Cambridge, 99 Helpma, Elhaa, ad Paul Krugma, Marke Sruure ad Ieraioal Trade, MIT Press, Cambridge, 985. Jovaovi, Boya, Asymmeri Cyles, Review of Eoomi Sudies, 73 (006), Jovaovi, Boya, ad Rafael Rob, Log Waves ad Shor Waves: Growh Through Iesive ad Exesive Searh, Eoomeria 58 (990), Judd, Keeh L., O he Performae of Paes, Eoomeria, 53 (May 985), Kose, M. Ayha, ad Kei-Mu Yi, Ca he Sadard Ieraioal Busiess Cyle Model Explai he Relaio Bewee Trade ad Comoveme, Joural of Ieraioal Eoomis, 68 (Marh 006): Lasoa, A. ad J. Yorke, O he Exisee of Ivaria Measures for Pieewise Moooi Trasformaois, Trasaios of he Ameria Mahemaial Soiey, 86, (973), Masuyama, Kimiori, "Growig Through Cyles," Eoomeria, 67 (Marh 999): Masuyama, Kimiori, "Growig Through Cyles i a Ifiiely Lived Age Eoomy, Joural of Eoomi Theory, 00 (Oober 00): Page 34 of 58

35 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Masuyama, Kimiori, Nobuhiro Kiyoaki, ad Akihiko Masui, "Toward a Theory of Ieraioal Currey," The Review of Eoomi Sudies 60 (April 993): Masuyama, Kimiori, Irya Sushko, ad Laura Gardii, Ierdepede Iovaio Cyles, forhomig. Mira, Chrisia, Laura Gardii, A. Barugola ad JC Cahala, Chaoi Dyamis i Two- Dimesioal Noiverible Maps, World Sieifi (996) Rivera-Baiz, L.A., ad P. M. Romer, Eoomi iegraio ad eoomi growh, Quarerly Joural of Eoomis, 06, 99, Romer, Paul M., Edogeous Tehologial Chage, Joural of Poliial Eoomy, 98, 990, S7-S0 Shleifer, Adrei, Implemeaio Cyles, Joural of Poliial Eoomy, 94, Deember 986, Srogaz, Seve H., Sy: How Order Emerges from Chaos i he Uiverse, Naure, ad Daily Life, (003) Sushko, Irya, ad Laura Gardii, Degeerae Bifuraios ad Border Collisios i Pieewise Smooh D ad D Maps, Ieraioal Joural of Bifuraio ad Chaos, 0 (00), Veura, Jaume, A Global View of Eoomi Growh, Chaper i Phillippe Aghio ad Seve N. Durlauf, eds., Hadbook of Eoomi Growh, Volume B, 005. Wälde, Klaus, The Eoomi Deermias of Tehology Shoks i a Real Busiess Cyle Model, Joural of Eoomi Dyamis ad Corol, 7 (00), -8. Wälde, Klaus, Edogeous Growh Cyles, Ieraioal Eoomi Review 46 (Augus 005), Page 35 of 58

36 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Appedies: Appedix A: The suffiie odiio for he o-speializaio Coury produes he homogeeous ipu if ad oly if he oal labor demad by is m m differeiaed ipus seor falls shor of is labor supply. Tha is, L N ( y ) N ( y f ) m m m N ( p y ) N ( p y ) m m p x p x m N N M w y, or L / M y. From p x w eq.(), his iequaliy is guaraeed if fm / L m / s. Thus, boh ouries always s s produe he homogeous ipu if 0 mi, alog he sequee, saisfyig eqs. (5) m m ad (6), whih is bouded so ha he upper boud is srily posiive. Q.E.D. Appedix B: Derivaio of eq.(7) from eqs.(5) ad eq. (6) We disuss oly he ase of 0 s / s, whih implies 0.5 s( ) = s ( ) <. The ase of s / s, whih implies s ( ) s ( ), is similar (ad simpler). s sk Firs, oe ha h ( m k ) 0, defied by, has he h ( m ) m h ( m ) m / followig properies, as see i Figure 5. They are hyperbole, moooe dereasig wih h ( ) as m 0ad h ( 0) ad h ( m k ) 0 as mk s / sk. m h( m ) ad m h ( m ) ierse a (, ) s ( ), ( ) s i he posiive quadra. m h ( h ( m )) implies m ( s ) ad m h ( h ( m )) implies m s ( ). We ow osider eah of he four ases i eq.(5). i) Suppose m for boh = ad. The, from (5), m h( m ) ad m = h ( m ), hee m s (). Iserig hese expressios i eq. (6) yields he map for he ierior of D LL. ii) Suppose m h( m ) ad m h( m ). The, from (5), m for boh = ad, hee h ( ) ad h ( ). Iserig hese expressios i (6) yields he map for he ierior of D HH. iii) Suppose m h( m ) ad m. The, from (5), m ad m = h ( m ), hee h ( h ( )), whih implies ( s ) ad h ( ). Iserig hese expressios i (6) yields he map for he ierior of D HL. iv) Supposig m ad m h ( m ) similarly yields he map for he ierior of D LH. Fially, i is sraighforward o show ha he map is oiuous a he boudaries of hese four domais. Q.E.D. k m k k k k k Page 36 of 58

37 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Appedix C: Proof of Proposiio Sie he uique exisee of he symmeri asyhroized -yle has bee show i he a a a a ex, we oly eed o ivesigae is loal sabiliy properies. From (, ) F(, ) ad ( a a, L H) = F ( a, a ) H L, he Jaobia marix a he asyhroized -yle a be alulaed as: 0 J = 0 ( ) a where h' ( ) 0. Is eigevalues are he roos of is haraerisi fuio, H 4 F( ) ra( J) de( J) {( ) } ( ) H L They are omplex ougaed if [ ra( J)] 4de( J) { ( ) } 4 ( ) 0 <. Is modulus is de( J ) ( ), hee he -yle is a sable fous i his rage. For <, [ ra( J)] 4de( J), so ha F ( ) 0 has wo real roos. A, hey are boh equal o ( ). For a higher, he wo real roos are 4 disi, ad saisfy ( ), if F () {( ) } ( ) 0 0 [ ( )] /. Tha is, for ( ) <, he -yle is a sable ode. For >, F ( ) 0 ad 0 saddle. To obai, differeiae he defiiio of h,, h( ) h( ) / wih respe o o have h'( ) h'( ) / 0. ( h( ) ) ( h( ) / ) L H, so ha he -yle is a a a a By evaluaig his expressio a H, ad usig h'( H ) ad ( a H h H ), / 0 ( ) ( / ) from whih, / / a / h ( ) ( ). Q.E.D. ' H Page 37 of 58

38 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure : The Sae Spae ad The Four Domais of he D Sysem (for s / s ). 0 Iovaio Aive i D HH No Iovaio D LH Iovaio Aive i Boh O D LL D HL Iovaio Aive i Page 38 of 58

39 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure : The Sae Spae ad The Four Domais of he S-Sysem i Auarky ( 0 ). Iovaio Aive oly i No Iovaio Iovaio Aive i Boh Iovaio Aive oly i O Page 39 of 58

40 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 3: D-Sysem: The Skew Te Map 45º O Aive Iovaio No Iovaio Page 40 of 58

41 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 4: The Usable Seady Sae, The Absorbig Ierval, ad he Sable -Cyle for 45º The hik blak lies show he graph of he skew e map, f, eq.(0). The hi blak lies show how he graph of he d ierae of he map, f, show i he hik blue lies, a be osrued from he graph of f. The red do is he seady sae,, whih is usable for. The red box idiaes he absorbig ierval, whih exiss for. The blue box idiaes he period- yle (wih he blue dos idiaig he wo pois o he period- yle, ad L ), whih is sable for. Noie ha H ad L are he wo sable H seady saes uder f. Noe ha has wo immediae pre-images uder f, give by < ( fh ). Likewise, < ) f H. The wo iervals, ( ) ( has four immediae pre-images uder f L, ( ) ad (, ) f ( H f, give by ) f L < ( ), belog o he basi of araio for uder L f. The ierval, (, ), as well as a ierval immediaely below ( f L ) ad a ierval immediaely above ) uder f ( f H, belog o he basi of araio for H. This way, we a see why he wo basis are o oeed, give by aleraig iervals, ad heir boudaries are formed by he pre-images of he usable seady sae, <. Page 4 of 58

42 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 5: Bifuraio diagram i he (δ, σ)-plae ad Is Magifiaio ~ Q m (m = 0,,, ) idiae he parameer regios for he exisee of a haoi araor ha osiss of figure. m iervals. The boom figure is a magifiaio of he red box area i he op Page 4 of 58

43 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 6 : Syhrozied vs. Asyhroized -Cyles: A D-view of he World Eoomy wih he wo-ouries i auarky Page 43 of 58

44 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 7: Seady Sae Aalysis wih s s 0. 5 Figure 7a: Home Marke Effe s / O / s Figure 7b: Globalizaio ad Magifiaio of he Home Marke Effe s s / O s /s ρ Page 44 of 58

45 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 8: Symmeri ( s / ) D Sysem (ρ = 0) (ρ = ) Iovaio Aive i D LH D HH No Iovaio (ρ = 0) Iovaio Aive i Boh D LL D HL O Iovaio Aive i (ρ = ) Page 45 of 58

46 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 9a: Syhroized versus Asyhroized -Cyles: s 0. 5,. 5, 0. 7 Red (he basi for he syhroized -yle) beomes domia. The symmeri asyhroized -yle beomes a sable ode a ρ =.870; ad a saddle a ρ = Page 46 of 58

47 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 9b: Syhroized versus Asyhroized -Cyles: s 0. 5,. 5, Red (he basi for he syhroized -yle) beomes domia. The symmeri asyhroized -yle beomes a sable ode a ρ =.87867, ad a saddle a ρ = Page 47 of 58

48 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 9: Syhroized versus Asyhroized -Cyles: s 0. 5,. 5, 0. 8 Red (he basi for he syhroized -yle) beomes domia. The symmeri asyhroized -yle beomes a sable ode a ρ =.884; a saddle a ρ = Page 48 of 58

49 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 0a: Asymmeri ( s / ) D Sysem: 0 s / s A higher has addiioal effes of shifig iovaio owards (ad away from ), show by blue arrows. Iovaio Aive i D HH No Iovaio D LH Iovaio Aive i Boh O D LL D HL Iovaio Aive i Page 49 of 58

50 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 0b: Asymmeri ( s / ) D Sysem: for s / s. No iovaio i ; ad 0. Iovaio i : max{ h ( ), } ( ) max{, } ( ). Asympoially, he dyamis is give by a D-skew e map o he horizoal axis. D HH O D LH Iovaio Aive i No Iovaio Page 50 of 58

51 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure a:asymmeri Syhroized & Asyhroized -Cyles: s 0. 55,. 5, By ρ =.36, ifiiely may Red islads appear iside Whie. By ρ =.39, he sable asyhroized -yle ollides wih he basi boudary ad disappears, leavig he Syhroized -yle as he uique araor. Page 5 of 58

52 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure b: Asymmeri Syhroized & Asyhroized -Cyles : s 0. 6,. 5, By ρ =.7, ifiiely may Red islads appear iside Whie regio. By ρ =.30, he sable asyhro. -yle ollides wih is basi boudary ad disappears, leavig he Syhroized -yle as he uique araor. Page 5 of 58

53 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure :Asymmeri Syhroized & Asyhroized -Cyles: s 0. 7,. 5, By ρ =.65, ifiiely may Red islads appear iside Whie. By ρ =.9, he sable asyhroized -yle ollides wih is basi boudary ad disappears, leavig he Syhroized -yle as he uique araor. Page 53 of 58

54 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure d: Asymmeri Syhroized & Asyhroized -Cyles: s 0. 8,. 5 ; By ρ =.0, ifiiely may Red islads appear iside Whie. By ρ =., he sable asyhro. -yle ollides wih is basi boudary ad disappears, leavig he Syh. -yle as he uique araor. Page 54 of 58

55 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure : Criial Value of ρ a whih he Sable Asyhroized -yle disappears (as a fuio of s ) Page 55 of 58

56 Kimiori Masuyama, Globalizaio ad Syhroizaio of Iovaio Figure 3: Four Basis of Araio: s 0. 7,. 5, As ρ rises, Red ivades Whie, ad Azure ivades Gray, ad verial slips of Red ad Azure emerge. Page 56 of 58