Inernionl Trnsmission of Invesmen-Speci c Technology Shocks wih Incomplee sse Mrkes Drf Copy - No for Circulion or Ciion Enrique Mrínez-Grcí y ederl Reserve Bnk of Dlls irs Drf: Sepember, Curren Drf: ebrury, bsrc I build symmeric wo-counry model h incorpores nominl rigidiies, locl-currency pricing nd monopolisic compeiion disoring he goods mrkes. The model is similr o he frmework developed in Mrínez-Grcí nd Søndergrd (8, 8b, bu i lso inroduces fricions in he sses mrkes by resricing he nncil sses vilble o wo unconingen nominl bonds in zero-ne supply nd by dding qudric coss on inernionl borrowing (see, e.g., Benigno nd Thoenissen (8 nd Benigno (9. The echnicl pr of he pper conins hree bsic clculions. irs, I derive he equilibrium condiions of he open economy model under locl-currency pricing nd incomplee sse mrkes. Second, I compue he zero-in ion (deerminisic sedy se nd discuss wh hppens wih non-zero ne foreign sse posiion. Third, I derive he log-linerizion of he equilibrium condiions round he deerminisic sedy se. The quniive pr of he pper ims o give brod overview of he role h incomplee inernionl sse mrkes cn ply in ccouning for he persisence nd voliliy of he rel echnge re. I nd h he simulion of he incomplee nd complee sse mrkes models is lmos indisinguishble whenever he business cycle is driven primrily by eiher non-persisen monery or persisen produciviy (bu no permnen shocks. In urn, sse mrke incompleeness hs more sizeble welh e ecs whenever he cycle is driven by persisen (bu no permnen invesmen-speci c echnology shocks, resuling in signi cnly lower rel echnge re voliliy. JEL Clssi cion:, 7, I would like o hnk Jne Koech, Mrí Teres Mrínez-Grcí, Jens Søndergrd, Jin Wng, Mrk Wynne, nd mny ohers for helpful discussions. I lso grefully cknowledge he suppor of he ederl Reserve Bnk of Dlls. ll remining errors re mine lone. The views epressed in his pper do no necessrily re ec hose of he ederl Reserve Bnk of Dlls, or he ederl Reserve Sysem. y Enrique Mrinez-Grci, ederl Reserve Bnk of Dlls. Correspondence: N. erl Sree, Dlls, T 7. hone: + ( 9-. : + ( 9-9. E-mil: enrique.mrinez-grci@dl.frb.org. Webpge: hp://dllsfed.org/reserch/bios/mrinez-grci.hml.
KEY WORDS: Rel echnge res, Locl-currency pricing, Tylor rules, Incomplee sse mrkes, Inernionl borrowing coss.
Inroducion inding mcroeconomic model h cn simulneously replice he voliliy nd persisence of he rel echnge re (RER hs been chllenge. One srnd of he lierure in priculr hs focused on imperfecions in he goods mrkes (nominl rigidiies s possible source of RER ucuions. No surprisingly, mny so-clled New Open Economy Mcro (NOEM models hve been wrien for he purpose of invesiging he sylized fcs of he RER by looking more closely he pricing decisions of rms. Mos of hose models, however, isole he fricions in he goods mrkes nd ke for grned h inernionl sse mrkes ough o be complee. The funcioning of inernionl sse mrkes deermines he een o which households cn e cienly insure mongs hemselves o smooh heir consumpion in he presence of counry-speci c shocks. sse mrkes re viewed s crucil for he propgion nd rnsmission of business cycle ucuions cross counries. ence, nurl quesion becomes how sensiive he resuls in he NOEM lierure re o he ssumpion of complee inernionl sse mrkes. In his pper, I eend he full- edge NOEM model wih cpil ccumulion of Mrínez-Grcí nd Søndergrd (8, 8b by bndoning he ssumpion of complee sse mrkes in order o provide rcble frmework h would esily lend iself o quniive evluion of he role of he complee mrkes ssumpion. I dop sndrd incomplee inernionl sse mrkes srucure h resrics he nncil sses vilble o jus wo unconingen nominl bonds in zero-ne supply (issued in wo counries dding qudric cos on inernionl borrowing ied o he rel ne foreign sse posiion of he home counry (see, e.g., Benigno nd Thoenissen (8 nd Benigno (9. This se-up represens cler deprure from he complee inernionl sse mrke ssumpion, bu i lso ensures h he soluion of he model would be sionry up o rs-order pproimion (see lso he discussion on closing smll open economy models in Schmi-Grohé nd Uribe (. more in-deph eplorion of he comple role of sse mrkes my revel imporn insighs h cnno be esily mpped ino his reduce form chrcerizion of inernionl sse mrkes, bu h goes beyond he scope of his pper nd is lef for fuure reserch. The emphsis of he pper is clerly on he echnicl side nd he drf is predominnly focused on he chrcerizion of he rs principles of his open-economy seing, nd on he mny issues surrounding he deerminion of sedy se nd rs-order pproimion of he equilibrium condiions specilly whenever he rde blnce migh be di eren hn zero in sedy se. Noneheless, I eensively documen he simuled momens of he model under complee nd incomplee sse mrkes nd provide some bsic insighs bou he implicions of hese eperimens h complemen (nd re ne he work in Mrínez- Grcí nd Søndergrd (8, 8b. The inuiion in Mrínez-Grcí nd Søndergrd (8, 8b is rher srighforwrd. dding cpil gives households in boh counries mrgin of ineremporl djusmen, hereby mking he consumpion nd RER phs smooher. Cpil ccumulion conribues o signi cnly lower he consumpion nd RER voliliy in he NOEM model irrespecive of he shocks driving he cycle. djusmen coss slow he response of invesmen o shocks, mking i coslier for households o djus ineremporlly nd pushing he voliliy of consumpion nd he RER up. Wihou severely consrining he use of cpil ccumulion, only in response o monery shocks we cn epec su cienly volile RERs. owever, he persisence of he RER flls shor if monery shocks re he primry driver of he business cycle (see lso Chri e l. (. igh RER persisence ends o occur in response o persisen produciv-
iy shocks if he Tylor (99 monery policy rule lso hs very ineril componen. Wih non-persisen monery shocks, he persisence is ofen less bu i is lso ied o he speci cion of he Tylor (99 monery policy nd he djusmen cos funcion. Sill, deviions from he lw of one price re lrger nd more imporn o ccoun for RER ucuions when he model is primrily driven by monery shocks (see, e.g., Bes nd Devereu (. When I depr from he ssumpion of complee inernionl sse mrkes, which imposes perfec inernionl risk-shring nd igh link beween he RER nd relive consumpion, I nd h bond economy subjec o inernionl borrowing coss nd he workhorse NOEM model wih complee sse mrkes genere very similr inernionl business cycle perns in response o produciviy nd monery shocks (see lso Ber nd Crucini (99, ehcoe nd erri ( nd Chri e l. (. sse mrke incompleeness, however, ends o resul in signi cnly lower RER voliliy whenever he business cycles re primrily driven by (persisen bu no permnen invesmen-speci c echnology (IST shocks. Invesmen-speci c echnology (IST shocks, however, lso induce ecessive invesmen voliliy nd counercyclicl consumpion h re inconsisen wih he d. Ineresingly, he opiml decision o pospone consumpion o inves more in response o posiive IST shock leds he RER o pprecie on impc while domesic oupu increses, bu he opposie occurs wih eiher produciviy or monery shocks. The reminder of he pper is srucured s follows: secion describes my wo-counry NOEM model wih cpil ccumulion nd incomplee sse mrkes, while secion oulines he opimliy condiions, secion chrcerizes he zero-in ion sedy se, nd secion derives in deil he log-linerizion of he equilibrium condiions of he model. Secion covers furher re nemen of he model o incorpore vrible cpil uilizion, nd secion 7 summrizes he prmeerizion sregy used for he simulions. Secion 8 highlighs he quniive ndings, nd secion 9 concludes. There is lso n ppendi h describes he se of log-liner equilibrium condiions nd includes ll he Tbles nd igures in he pper. The Monery Open Economy wih Incomplee Mrkes. The Ineremporl Consumpion nd Svings roblem I specify sochsic, wo-counry generl equilibrium model. Ech counry is populed by coninuum of in niely lived (nd idenicl households in he inervl [; ]. In ech period, he domesic households uiliy funcion is ddiively seprble in consumpion, C, nd lbor, L. Domesic households mimize, + E (C + + ' (L + +' ; ( where < < is he subjecive ineremporl discoun fcor. The elsiciy of ineremporl subsiuion sis es h > ( while he inverse of he risch elsiciy of lbor supply sis es h ' >. I ssume h domesic households re ble o rde wo nominl risk-less (unconigen bonds denomined in domesic nd foreign currency, respecively. ence, he domesic household mimizes is lifeime uiliy in ( subjec o he sequence of budge consrins described by, (C + + T + B + + I I S B+ + I S B + B + S B + W L + Z K + r ; (
nd he lw of moion for cpil given by, K + ( K + V ( ; ; K ; ( where W is he domesic nominl wge, is he domesic consumpion price inde (CI, r re he nominl pro s genered by he domesic rms, nd nd T is lump-sum nominl levied on he domesic households. Moreover, is domesic rel invesmen, K snds for domesic rel cpil, Z de nes he nominl renl re of cpil, nd V is n eogenous, invesmen-speci c echnology (IST shock. B + is he pyo in period + of he (unconingen risk-free bond denomined in unis of he domesic currency, cquired by he domesic household he end of period. The implici nominl gross ineres re on his bond is I. B + is he pyo in period + of he (unconingen risk-free bond denomined in foreign currency nd cquired by he domesic household he end of period. The implici nominl gross ineres re on his bond is I, while S denoes he nominl echnge re. s in Benigno (9, I lso ssume h here is qudric cos funcion (e.g. n inernionl borrowing cos h penlizes chnges in he rel foreign sse posiion of he domesic economy, SB +, whenever i devies from consn rel reference level of. The qudric borrowing cos is hen re-scled by I muliplied by he prmeer > o is size. (for nlyicl convenience nd The foreign households mimize heir lifeime uiliy (nlogous o ( subjec o lw of moion for cpil similr o he one described in (. I follow Benigno nd Thoenissen (8 in ssuming h foreign households re ble o rde only on (unconigen risk-free bonds denomined in foreign currency. ence, he foreign households budge consrin is described by, (C + + T + I B+ B + W L + Z K + r + T r : ( B + is he pyo in period + of he (unconingen risk-free bond denomined in unis of he foreign currency, cquired by he foreign household he end of period. s before, he implici nominl gross ineres re on his bond is I. I ssume h he foreign households receive he pro s from he foreign rms in equl proporion, r. oreign households lso coun s revenue he inernionl borrowing coss pid by he domesic households in rding foreign bonds, i.e., T r S I S B + ; ( where T r epresses he inernionl borrowing coss pid by he domesic households in unis of he foreign currency. This symmery in he nncil mrke srucure beween domesic nd foreign households is mde for simpliciy. or n eension of his se-up in which domesic nd foreign households cn rde in bonds denomined in boh currencies, see Benigno (9. I cn re-inerpre he model presened here s polr cse of Benigno (9 in which he coss of inernionl borrowing re prohibiively high for he foreign households, bu no for he domesic households. These modelling ssumpions inroduce rher sndrd iscl policy is no fully incorpored ino he model. owever, I include blnced budge rule (no governmen borrowing nd lump-sum es o nnce subsidy o rms men o neurlize he mrk-up disorion ssocied wih he ssumpion of monopolisic compeiion. In cse no subsidy ws inroduced, hen he lump-sum would simply be equl o zero, i.e. T.
incomplee sse mrke srucure (see, e.g., Benigno nd Thoenissen (8, nd Benigno (9 h serve o close he model down while inducing sionriy. The domesic- nd foreign-currency denomined bonds re issued respecively by he domesic nd foreign governmens in zero-ne supply. The bond mrke clering condiions cn be epressed s, B ; ( B + B : (7 I ssume h here is no rde in eiher domesic or foreign shres of rms. Sole ownership of he locl rms ress in he hnds of he locl households. Money is purely uni of ccoun, bu monery policy hs poenil economic impc by reguling shor-erm nominl ineres res o ec he in ion re in he presence of nominl rigidiies. Embedded in his model is lso he ssumpion h boh fcor mrkes (for lbor nd cpil re homogenous nd perfecly compeiive wihin counry, bu segreged cross counries. In oher words, fcors cn be used for producion purposes in ny rm wihin he sme counry, bu hey re immobile cross borders. The cpil ccumulion in ( my be subjec o djusmen coss, (. I consider hree specil cses: he cpil djusmen cos (CC cse, he invesmen djusmen cos (IC cse, nd he no djusmen coss (NC cse. The NC funcion is simply, ( ; ; K : (8 The NC funcion for he foreign lw of moion for cpil is he obvious counerpr. This implies h in sedy se ; ; K, ; ; K, nd ; ; K. The CC nd IC djusmen cos funcions, however, require more deiled descripion. Cpil djusmen Cos (CC uncion. The cpil djusmen cos (CC funcion (see, e.g., Chri, e l., implies h he funcion ( in ( kes he following form, K K ; (9 K where K is he corresponding invesmen-o-cpil rio, is he depreciion re ppering in he lw of moion for cpil, nd mesures he curvure of he cos funcion. mong he properies of his or more deils nd oher lernives o close down he model, see Schmi-Grohé nd Uribe (.
djusmen cos funcion h re relevn for my model, I noe h, K K K K K K K K K K 7 ; K K 7 : The sme djusmen cos formul s in (9 pplies o he foreign households problem. I ssume h in sedy se he IST shocks re heir uncondiionl men of V V. ence, in sedy se he djusmen coss dissipe nd he invesmen-o-cpil rio is equl o he depreciion re (i.e., K nd K s in he NC cse. (. Invesmen djusmen Cos (IC uncion. This implies h (, (, nd I lso eplore he invesmen djusmen cos funcion (IC used mong ohers by Smes nd Wouers ( nd Chrisino, e l. (. The IC speci cion implies h he funcion ( in ( kes he following form, ; ( where is he corresponding gross re of invesmen, nd mesures he curvure of he cos funcion. mong he properies of he IC funcion h re relevn for my model, I noe h, 7 ; The sme djusmen cos formul s in ( pplies o he foreign households problem. In sedy se, he djusmen coss dissipe gin becuse he ne growh of invesmen is zero. Under his djusmen cos speci cion i is cosly o chnge he level of invesmen nd he cos increses wih he size of he chnge, bu here re no djusmen coss in sedy se. ence, he sedy se epression of he lw of moion for cpil is he sme s in he sndrd NC cse. This lso implies h (, (, nd (. ggregion Rules nd he rice Indees. 7 : I ssume h invesmen, like consumpion, is composie inde of domesic nd impored foreign vrieies. The home nd foreign consumpion bundles of he domesic household, C nd C, s well s he invesmen bundles, nd, re ggreged by mens of CES
inde s, C Z Z C (h (h Z dh ; C C (f df ; ( Z dh ; (f df ; ( while ggrege consumpion nd invesmen, C nd, re de ned wih noher CES inde s, C C + + C ; ( : ( The elsiciy of subsiuion cross vrieies produced wihin counry is >, nd he elsiciy of inremporl subsiuion beween he home nd foreign bundles of vrieies is >. The shre of he home goods in he domesic ggregors is, while he shre of foreign goods is. I ssume he shres re homogeneous, i.e. +. Similrly, I cn de ne he ggregors for he foreign household. The only di erence being h he shre of he home goods in he foreign ggregors is, while he shre of foreign goods in he foreign ggregor is. The model inroduces home-produc bis in consumpion (Wrnock, s well s in he composiion of invesmen. By ssumpion, invesmen goods cn only be used for locl producion fer ggregion. This is lso he cse becuse of composiionl di erences cross counries. owever, ll locl nd foreign vrieies cn be rded inernionlly for eiher consumpion or invesmen purposes. Moreover, he symmery of he ggregors implies h he corresponding price indees re idenicl for invesmen nd consumpion bundles. ence, he relive price beween consumpion nd invesmen is one s re eced in he budge consrin (h is, in equion (. Under sndrd resuls on funcionl seprbiliy, he CI indees which correspond o my speci cion of he domesic ggregors in ( ( nd heir foreign counerprs re, h h + + i i ; ( h + i ; (
nd he price sub-indees re, Z Z Z Z ( (h dh ; (7 ( (f df ; (8 ( (h dh ; (9 ( (f df ; ( where nd re he price sub-indees for he home- nd foreign-produced bundle of goods in unis of he home currency. Similrly for where S denoes he nominl echnge re. nd. I de ne he rel echnge re s, RS S ; (. The rice-seing roblem Ech rm supplies he home nd foreign mrke, nd ses prices in he locl currency (henceforh, loclcurrency pricing or LC pricing. irms engge in hird-degree price discriminion cross mrkes (re-selling is infesible nd, furhermore, enjoy monopolisic power in heir own vriey. ricions in he goods mrke re modelled wih nominl price sickiness à l Clvo (98. ime ny rm (wheher domesic or foreign is forced o minin is previous period prices in he domesic nd foreign mrkes wih probbiliy < <. Insed, wih probbiliy (, he rm receives signl o opimlly rese ech price. I ssume h producion employs (homogeneous of degree one Cobb-Dougls echnology, i.e. Y (h (K (h (L (h ; 8h [; ] ; ( Y (f (K (f (L (f ; 8f [; ] ; ( where is he (ggrege domesic produciviy shock nd is he (ggrege foreign produciviy shock. The lbor shre in he producion funcion is represened by. By consisency nd mrke clering i follows h he ggrege cpil ccumuled by households in he domesic nd foreign counry is K R K (h dh nd K R K (f df respecively, while ggrege lbor is L R L (h dh nd L R L (f df respecively. Solving he cos-minimizion problem of ech individul rm yields n e ciency condiion linking he cpil-o-lbor rios o fcor price rios s follows, K K (h L L (h W ; 8h [; ] ; Z ( K L K (f L (f W Z ; 8f [; ] ; ( These epressions reduce o he sndrd cse of liner-in-lbor echnologies if. 7
s well s chrcerizion for he (pre-subsidy nominl mrginl coss, MC ( (W (Z ; ( MC ( (W (Z : (7 The lbor force is homogenous wihin counry nd immobile cross borders, nd he nionl lbor mrkes re perfecly compeiive. Wges equlize in ech counry (bu no necessrily cross counries, i.e. W (h W for ll h [; ] nd W (f W for ll f [; ], nd so does he renl re on cpil, i.e. Z (h Z for ll h [; ] nd Z (f Z for ll f [; ]. Then, since he producion funcion is homogeneous of degree one (consn reurns-o-scle, his implies h ll locl rms choose he sme cpil-o-lbor rio even hough hey end up producing di eren mouns. Moreover, he fcors of producion re compensed ccording o heir mrginl produc cross ll rms. I inroduce governmen subsidy in ech counry h is proporionl o he locl rms producion coss, i.e. MC Y (h nd MC Y (f respecively in he domesic nd foreign counries. The pre-subsidy producion coss of rm re simply frcion of he (pre-subsidy nominl mrginl coss, MC nd MC, in ( (7 imes he oupu of h rm. Governmens only subsidize he producion of rms loced in heir own counry, independenly of wheher he goods re sold loclly or epored. Governmens se no impor ri s or subsidize he locl demnd over he epor demnd, hence no disoring he inernionl relive prices (e.g., he erms of rde nd he rel echnge re. Using he pre-subsidy mrginl coss described in ( rms in boh counries s follows, (7, I cn wrie he pos-subsidy mrginl coss h ener ino he pricing decisions of ( MC ( (( W (( Z ; (8 ( MC ( (( W (( Z : (9 This chrcerizion of he subsidies su ces for he purpose of eiher reducing or elimining he mrk-up disorions ssocied wih monopolisic compeiion in he goods mrkes. The Opiml ricing roblem. re-opimizing domesic rm h under LC pricing chooses domesic nd foreign price, e (h nd e (h, o mimize he epeced discouned vlue of is ne pro s, + E 8 < : M ;+ ec;+ (h + e ;+ (h e (h + MC+ + ec ;+ (h + e ;+ (h S + e (h + MC+ 9 ; ; ( where M ;+ C + C + is he sochsic discoun fcor (SD for -periods hed nominl pyo s (corresponding o he domesic represenive household, subjec o pir of demnd consrins 8
in ech goods mrke, ec ;+ (h + e ;+ (h ec ;+ (h + e ;+ (h e (h + e (h +! C + + + ; (! C + + + : ( ere, e C;+ (h nd e C ;+ (h indice he consumpion demnd for ny vriey h home nd brod respecively, given h prices e (h nd e (h remin unchnged beween ime nd +. Similrly, e ;+ (h nd e ;+ (h indice he households invesmen demnd hose sme prices. Similrly, I chrcerize he objecive of he foreign rm s, + E 8 < : M;+ ec;+ (f + e e(f ;+ (f S + + MC + + ec ;+ (f + e ;+ (f e (f + MC + 9 ; ; ( where M ;+ C + C + is he foreign SD. The demnd consrins of he foreign rm re, ec ;+ (f + e ;+ (f ec ;+ (f + e ;+ (f e (f + e (f +! C + + + ; (! C + + + ; ( given h prices e (h nd e (h remin unchnged beween ime nd +.. The Monery olicy Rule nd Governmen Budge Consrin The Tylor rule is ofen de ned s he rdemrk of modern monery policy. In h cse he policy insrumen of he domesic nd foreign monery uhoriies re he shor-erm res I nd I respecively, while I nd I re heir corresponding sedy se vlues. I ssume h he monery uhoriies of boh counries se shor-erm nominl ineres res ccording o Tylor (99 ype rules, I M (I i " I I M I i "I # i y Y ; ( Y # i ; (7 Y y Y I derive he demnd for vriey h in he home nd foreign mrkes by combining he rs-order condiions in ( (7, secion. I derive he demnd for vriey f in he home nd foreign mrkes by combining he rs-order condiions in ( (7, secion. 9
where M nd M re he (domesic nd foreign monery policy shocks or he shocks o he ineres re policy rules, nd re he (gross CI in ion res, while Y nd Y re he respecive oupu levels. inlly, nd re he sedy se (gross CI in ion res, nd Y nd Y re he respecive sedy se oupu levels. In oher words, he monery policy rules in ( (7 respond o locl CI in ion nd oupu deviions from heir respecive sedy se levels. The inde cpures boh smoohing erm nd sysemic policy componen. This inde speci cion of he Tylor rule kes more sndrd form once i is log-linerized. The scl policy in ech counry is chrcerized by blnced budges in every period, nd subsidy o he producion coss of he locl producers h is fully nnced by lump-sum on households. I summrize he governmen s budge consrins s, Z T MC Y (h dh MC Y ; (8 T MC Z Y (f df MC Y ; (9 where T nd T re he domesic nd foreign lump-sum es on households, nd nd re he domesic nd foreign subsidies epressed s frcion of he producion coss. Given he producion funcions in ( (, he pre-subsidy mrginl cos equions in ( (7 nd he fc h cpil-o-lbor rios in ( ( re equlized cross rms wihin ech counry, i is possible o wrie he governmen budge consrins s follows, Z K T MC L (h dh L MC (K (L ( (W L (Z K ; ( K T MC Z L (f df L MC (K (L ( (W L (Z K ; ( where he second equliy follows lso from he lbor mrke clering condiions (i.e. L nd L Z Z L (h dh L (f df. s hese governmen budge consrins illusre, he vlue of he subsidy cn be epressed s shre of n ggrege funcion of he lbor income, W L nd W L respecively, nd he cpil renl income, Z K nd Z K respecively. s epeced, in he limi whenever he lbor shre converges o one, i.e.!, he echnology becomes liner-in-lbor, nd he subsidy becomes equl o frcion of he lbor income for ech counry. In Mrínez-Grcí nd Søndergrd (8 his subsidy is compleely ignored, so he implici ssumpion is h nd T T. Ofen, however, hese subsidies re used o neurlize he mrk-up disorion inroduced by he ssumpion h rms produce nd sell heir vrieies under monopo-
lisic compeiion. In order o elimine his disorion, su ces o se he subsidy in ech counry o sisfy he following pir of condiions, ( ; ( : rom here i follows h he opiml subsidy for boh counries is chrcerized s, < ; ( which is funcion of he elsiciy of subsiuion cross vrieies produced wihin counry, i.e. >. The mrk-up is lso funcion solely of he elsiciy of subsiuion cross vrieies. Since he elsiciy is ime-invrin, so re he mrk-ups nd he subsidies (epressed s shre of he producion coss needed. or simpliciy, I re he subsidies s secondry policy insrumen inended eclusively o del wih he mrk-up disorion nd, herefore, I ssume from now on h he subsidy s frcion of he producion coss will be invrin over ime nd idenicl cross counries, i.e., ; ; ( where lso denoes he sedy se subsidy in boh counries. In urn, he mrk-up nd he subsidy independenly of wheher he subsidy is se o zero or neurlizes prilly or olly he mrk-up disorion only ec he dynmics of he model up o rs-order pproimion becuse hose erms ener ino he compuions of he sedy se invesmen shre. Rher hn choosing speci c vlue for he subsidy, I view is shre over he producion coss s noher srucurl prmeer of he model. The Opimliy Condiions Since he model is buil round wo mosly symmeric counries, ll he rs-order condiions repored correspond o he home counry unless oherwise noed. The Opimliy Condiions from he ouseholds roblem. Given he srucure described in ( (, he soluion o he sub-uiliy mimizion problem implies h he home nd foreign households demnds for ech vriey re given by, C (h C (f C (h C (f (h C (h ; (h ; 8h [; ] ; ( (f C (f ; (f ; 8f [; ] ; ( (h (f C ; (h C ; (f (h (f ; 8h [; ] ; ( ; 8f [; ] ; (7
while he demnds for he bundles of home nd foreign goods re simply equl o, C C ; ; (8 C C C C ; C ; C ; ; (9 ; ( ; ( where he shre of he home goods in he foreign ggregor is nd he shre of foreign goods in he foreign ggregor is. These equions deermine he demnd funcions in he model. The equilibrium condiions of he households problem include pir of lbor supply funcions (he inremporl rs-order condiions which cn be epressed s, W (C W (L s ' ; ( (C (L s ' ; ( plus he pproprie no-onzi gmes, rnsversliy condiions, he budge consrins nd he lw of moions for cpil in boh counries. urhermore, he equilibrium condiions re compleed wih se of equions h chrcerize he bond porfolio llocion of he domesic nd foreign households s well s wih pir of equions h ccoun for he cpil-invesmen decisions of ech household. The cpil-invesmen condiions, in urn, depend on he choice of he djusmen cos funcion ( in he lw of moion for cpil in (, while he bond porfolio llocions depend on he inernionl borrowing coss pid by he domesic households o rde on foreign bonds (in ( nd (. The opiml bond porfolio choices for he domesic nd foreign households cn be described wih he following se of equilibrium condiions, E " C+ E " C+ C C " C E + C + I + S + S I + I # # # ; ( + (N + ; ( ; ( where I de ne he rel ne foreign sse posiion (in unis of domesic consumpion of he domesic household s N + SB +. Combining he Euler equions of he domesic household in ( nd (, he following relionship holds rue, E " C+ C + # S+ I I (N + ; (7 S
which represens vrin of he uncovered ineres re priy condiion where he risk premium ppers ied o he rel ne foreign sse posiion of he domesic household. This equilibrium condiion governs he erms of inernionl risk-shring in his environmen wih incomplee sse mrkes. Le me de ne Tobin s q s Q. Then, he domesic cpil-invesmen decisions cn be described wih he following se of ddiionl equilibrium condiions, Q E ( C+ Q C E C+ C Z+ @ ( + ; ; K + + Q + ( + V + + ; + @K + h Q + V + @( +; ;K + @ + i h i : @(; ;K V ( ; ; K + @ Under no djusmen coss (NC, he pir of condiions dded o ccoun for he cpil-invesmen decisions of he domesic households re summrized s, Q E ( C+ C Z+ + Q + ( ; (8 + Q V : (9 similr se of derivions llows me o wrie he following pir of equions for Tobin s q in he foreign counry, ( C Q E + Z + C + + Q + ( ; ( Q V : ( The Lgrnge muliplier on he lw of moion relive o he Lgrnge muliplier on he budge consrin epressed in rel erms, denoed Q nd Q respecively, hs he inerpreion of being he rel shdow price of n ddiionl uni of cpil (or Tobin s q. In he cse wihou djusmen coss nd IST shocks is well-known h Tobin s q is ecly equl o one, s hese equions show. Under cpil djusmen coss (CC, he pir of condiions dded o ccoun for he cpil-invesmen decisions of domesic households re, Q E ( C+ Q C " Z + + + Q + ( V + + K +!# + ; ( K + i: ( V h K + K K similr se of derivions llows me o wrie he following pir of equions for Tobin s q in he foreign
counry, ( C Q " E + Z + C + + Q + ( V+ +!# + K+ K+ ; ( Q h V K + K K i: ( The Lgrnge muliplier on he lw of moion relive o he Lgrnge muliplier on he budge consrin epressed in rel erms, denoed Q nd Q respecively, hve he inerpreion of being he rel shdow price of n ddiionl uni of cpil (or Tobin s q. Under invesmen djusmen coss (IC, he pir of condiions dded o ccoun for he cpil-invesmen decisions of domesic households re, Q E ( C+ Q C + E C+ C V h + Q + ( ; ( + Z+ Q + V + + + + i ; (7 similr se of derivions llows me o wrie he following pir of equions for Tobin s q in he foreign counry, ( C Q E + Z + C Q + E C + C V h + Q + ( ; (8 + Q +V+ + + + i : (9 Once gin, he Lgrnge muliplier on he lw of moion for cpil relive o he Lgrnge muliplier on he budge consrin epressed in rel erms, denoed Q nd Q respecively, hve he inerpreion of being he rel shdow prices of n ddiionl uni of cpil (or Tobin s q. The Opimliy Condiions from he irms roblem. The necessry nd su cien rs-order condiions for he domesic rm producing vriey h give me he following pir of price-seing formuls, + + E M ec;+ ;+ (h + e ;+ (h e (h M ;+ ec ;+ (h + e ;+ (h S + e (h E + MC+ + MC+ ; (7 : (7
Similrly, he rs-order condiions for he foreign rm producing vriey f give me he following price-seing formuls, + E + E " M;+ ec;+ (f + e ;+ (f e (f S + M ;+ ec ;+ (f + e ;+ (f e (f!# + MC + + MC + ; (7 : (7 Using he lw of lrge numbers nd he inheren symmery of he rms problem, he price sub-indees on domesic vrieies, nd, become, + ( e (h ; (7 + ( e (h ; (7 while he price sub-indees on foreign vrieies, nd, re compued s, + ( e (f ; (7 + ( e (f : (77 Equions (7 (77 re convenien wy o reformule (7 (. ggrege Oupu nd Relive rice Disorions. The producion funcions in ( ( cn be re-wrien s, K Y (h L (h ; 8h [; ] ; (78 L K Y (f L L (f ; 8f [; ] ; (79 since cpil-o-lbor rios re equed cross ll rms wihin counry due o fcor price equlizion (s implied by equions ( (. lso becuse of fcor price equlizion, ll households wihin counry supply he sme moun of lbor nd cpil, h is L nd K in he domesic counry nd L nd K in he foreign counry. In urn, fcor prices equlize in ech counry becuse he fcors of producion re homogeneous (hough immobile cross borders nd fcor mrkes re perfecly compeiive. owever, since he pricing decisions of rms re no synchronized, he mouns of lbor nd cpil lloced o ech individul rm will di er in every period. cross ll rms, he cpil nd lbor demnds equl heir respecive supplies. ence, oupu cn be There is mss one of idenicl households in ech counry ll of which follow he sme opiml ph in heir decisions. Then, L nd K in he domesic counry nd L nd K in he foreign counry represen simulneously he verge nd he ggrege supply of lbor nd cpil respecively. Moreover, hey lso chrcerize he individul decisions of ech household since ll households wihin counry conform o he counry verge.
dded up o be epressed s funcion of he common (ggrege produciviy shock s well s he ggrege cpil nd lbor supplied by he locl households, i.e. Y Y Z Z Y (h dh (K (L ; (8 Y (f df (K (L ; (8 where, by lbor mrke clering, i follows h L Equions ( Z L (h dh nd L Z L (f df. ( deermine he demnd funcion for ech vriey produced home nd brod. Equions ( (7 coupled wih he mrke clering condiions he vriey level llow me o ggrege oupu demnd s follows, Y Z Z [C (h + (h + C (h + (h] dh " (h C + + (h C + # dh Y e Z Z! C + +! e C [C (f + (f + C (f + (f] df " (f C + + (f + ; (8 C + # df e! C + +! e C + ; (8 where, e e e e Z Z Z Z ( (h dh ( (h dh ( (f df ( (f df e + ( e (h ; (8 e + ( e (h ; (8 e + ( e (f ; (8 e + ( e (f : (87 Similrly, equions ( ( coupled wih he mrke clering condiions he vriey level llow me o
ggrege oupu s follows, Z Y [C (h + (h + C (h + (h] dh " Z (h " Z dh# (C + + (h dh#! e! e (C + + Y Z [C (f + (f + C (f + (f] df " Z (f " Z df# (C + + (f df#! e! e (C + + (C + (C + ; (88 (C + (C + ; (89 where he shre of he home goods in he foreign ggregor is nd he shre of foreign goods in he foreign ggregor is. Equions (88 o ggrege consumpion, ggrege invesmen s well s relive prices. Rel Renl Res of Cpil nd he irm s E ciency Condiions. (89 ie he ggrege oupu demnd in boh counries Combining he implied producion equions in (8 (8 wih he e ciency condiions for rms in ( ( nd he lbor supply equions from he households problem (in equions ( (, I cn epress he rel renl re of cpil in erms of produciviy shocks, ggrege consumpion, oupu nd physicl cpil, i.e. Z L W ( +' (C (Y +' (K ( +( ' ; (9 K Z W L K ( +' (C (Y +' (K ( +( ' : (9 These wo equions summrize he e ciency condiion h requires he cpil-o-lbor rios o be proporionl o he fcor price rios. This chrcerizion is lso convenien becuse i implies h I do no need o keep rck of wges or lbor in he simulion of his equilibrium. Mnipuling he sme se of equilibrium condiions lile bi more lso llows me o re-wrie he rel wges in erms of he rel renl re of cpil, he produciviy shock, ggrege consumpion, oupu nd physicl cpil, i.e. W W ( ' +( ' ( ( +( ' ' ( ' +( ' +( ' (C (Y ' +( ' (C +( ' (Y ' Z +( ' ' +( ' Z ( ' +( ' ; (9 ( ' +( ' : (9 These wo equions su ce for he purpose of replcing rel wges ou of he mrginl cos equions. 7
The Resource Consrin. equilibrium (under equliy s, The budge consrin in ( of he domesic household cn be epressed in (C + + T + B + + I I S B+ + I S B + B + S B + W L + Z K + r : Using he domesic bond mrke clering condiion in (, he budge consrin cn be reduced s follows, C + + T + I S B + + I S B+ S B + W L + Z K + r : The per-period pro s of he domesic rms disribued o he domesic households cn be clculed s follows (including he lbor subsidies, r Z Z [ (h (C (h + (h + S (h (C (h + (h] dh ( W L ( Z K [ (h (C (h + (h + S (h (C (h + (h] dh + MC Y W L Z K : Therefore, replcing he epression for per-period pro s inside he budge consrin of he domesic household I ge h, C + + T + S B + I Z " S B+ + S B+ # [ (h (C (h + (h + S (h (C (h + (h] dh + MC Y : Using he domesic governmen blnced-budge rule in (8, I obin h, C + + I S B + " S B+ + Z S B + # [ (h (C (h + (h + S (h (C (h + (h] dh : Then, I cn use he demnd equions in ( (7 nd (8 ( o derive he following epression for he resource consrin, I S B + + SB + SB SB + Z " (h (h + Z Z S (h (h (C + +S (h dh (C + + dh (h (C + #dh (C + (C + (C + ; 8
Using he price indees in (7 ( I infer h he relive price erms h ener ino he resource consrin mus be equl o one, i.e. Z (h dh ; Z (h dh ; nd so I cn re-wrie he resource consrin iself s, I S B + + SB SB + SB + S + (C + + (C + + RS (C + (C + (C + (C + ; (9 where I hve lso mde use of he de niion of he rel echnge re in (. De ning now he rel ne foreign sse posiion of he domesic household s N + SB +, I cn re-wrie equion (9 more convenienly s, I h N + + (N + i S S N + (C + + RS (C + (C + : Equion (9 is he resource consrin equion h is needed o close down he model wih incomplee sse mrkes. (9 The Deerminisic, Zero-In ion Sedy Se I posule zero-in ion sedy se where he nominl echnge re equls one, S, nd where bsorpion in boh counries di ers only by fcor reled o he sedy se rel echnge re, i.e. C + RS C +. This normlizion fcilies he log-linerizion of he equilibrium condiions of he model. owever, he normlizion reling he bsorpion of boh counries o he rel echnge re imposes ddiionl resricions on he sedy se ggrege produciviy cross counries h need o be eplored in greer deil. Moreover, i does no necessrily imply h consumpion nd invesmen equlize cross counries in sedy se. or he purpose of his pper, su ces o consider he implicions of his normlizion when lso dding cpil nd non-zero rel ne foreign sses in he sedy se. irs Sep. I look he sedy se invesmen-o-cpil rio. The domesic cpil ccumulion equion in ( (nd is foreign counerpr require he following relionships o hold in sedy se, K ( K + V ; ; K ; (9 K ( K + V ; ; K : (97 9
Opering on he sedy se cpil ccumulion equions i follows h, V ; ; K K V ; ; K K : (98 The wo equliies in (98 simplify o becuse I ssume h he sedy se IST shock is K K normlized o one in boh counries, i.e. V V, nd becuse I know h: ( under he speci cion of he cpil djusmen cos (CC funcion, i holds h ( ; nd (b under he K K speci cion of he invesmen djusmen cos (IC funcion, i holds h (. Therefore, hese djusmen cos funcions hve no sedy se implicions, nd invesmen purely replces he deprecied sock of cpil (i.e. K nd K. Second Sep. I look he nominl nd rel ineres res s well s he rel renl re of cpil nd Tobin s q. Using he fc h he invesmen-o-cpil rio is pined down by he depreciion re (from (98 nd he properies of he djusmen cos funcion ( (in (9 or (, I cn wrie he sedy se household e ciency condiions in eiher ( ( or ( (9 h summrize Tobin s q nd he rel renl re of cpil s follows, Z Q + Q ( ; (99 " Q Z # + Q ( ; ( Q Q ; ( where I eplicily used he fc h ( nd (, K K s well s he sedy se normlizion of he IST shocks o V V. These equions re sis ed rivilly in he NC model wihou djusmen coss. Nurlly, he sedy se rel renl re of cpil is, Z Z ( : ( The sedy se ineres res given by equions ( nd ( imply h, I ; ( I : ( Nominl nd rel ineres res mus be equl in his (deerminisic zero-in ion sedy se. These resuls sy h Tobin s q is equl o in sedy se, nd h he rel renl res on cpil mus be equl o he rel reurns on bonds (where he rel reurns on bonds re I I s given by ( king ino ccoun he e ec of he cpil depreciion,. Moreover, from equion ( on inernionl risk-shring, I ge h, I + B! ( fer ; (
where I hve implicily used h he sedy se nominl echnge re is S. Whenever > nd ( is sis ed, his epression holds rue only if N SB. Nurlly, given he foreign bond SB mrke clering condiion in (7, i mus follow h SB. In oher words, in sedy se here re no coss ssocied wih rding in foreign bonds becuse he domesic households se heir rel ne foreign sse posiion he reference level of. Equions ( nd (7 give me he speci cion of he monery policy rules in he cone of his model. In sedy se, i mus follow h, which immediely reduces o, I M I i "I # i y Y ; ( Y! Y! y I M I i I Y i ; (7 M ; (8 M : (9 This gives convenionl normlizion for he sedy se monery policy shocks in boh counries. Third Sep. I look he sedy se invesmen shres y nd Y y. Using he fc noed Y in (98 h in sedy se he invesmen-o-cpil rio is deermined by he depreciion re,, I cn wrie he invesmen shres s proporionl o he cpil-o-oupu rios in ech counry, rom he Cobb-Dougls speci cion of he rm producion funcions in ( y K Y ; ( y K Y : ( (, he e ciency condiions on he rm s opimizion problem in ( (, nd he ggrege producion funcions derived in (8 (8, I ge h he cpil-o-oupu rio cn be epressed s funcion of he produciviy shocks nd he fcor price rios in ech counry. ence, i follows h, y y W Z W Z " # Z W ; (! Z! W! ; ( where nd re he sedy se domesic nd foreign produciviy shocks. I lso know h in deerminisic zero-in ion sedy se ll rms chrge he sme price (he Clvo prmeer is irrelevn, he lw of one price holds nd he sndrd price seing formul under monopolisic
compeiion deermines he pricing decision for rms. In oher words, prices of he sme goods in diferen mrkes re equlized when epressed in unis of he sme currency, nd prices mus be equl o mrk-up over mrginl coss. Then, he pricing equions in (7 nd (7 in conjuncion wih he mrginl cos funcions in ( (7 reduce o, ( W Z ( W Z ; ( ; ( where he domesic rms subsidy in sedy se,, nd he foreign rms subsidy in sedy se,, re lso incorpored. I ssume h he sedy se subsidy is he sme in boh counries, i.e.. Dividing hese epressions by nd respecively nd hen re-rrnging erms, I ge h rel wges re linked o rel renl res on cpil by, W! W " ( # Z ; ( (! Z : (7 If I replce he epressions in ( nd (7 inside he formul for he domesic nd foreign invesmen shres obined before in ( nd (, I ge h, " Z y ( "! ( y Z! @ ( ( @!# Z # Z ; (8!! Z! Z : (9
Then, using he sedy se rel renl res on cpil derived in (, I cn sy h, y y ( ( ( 7 ( 7! ; (! : ( In oher words, he domesic nd foreign invesmen shres in sedy se depend direcly on he depreciion re of cpil,, nd he cpil shre in he producion funcion,, nd depend indirecly on he mrkup,, he governmen subsidy, nd, nd he rel renl re on cpil, (. The domesic invesmen shre lso depends on he relive price, while he foreign invesmen shre depends on he relive price. ence, he sedy se invesmen shres will no di er cross counries, i.e., y K Y ( ( 7 K Y y ; only if he sedy se governmen subsidies re he sme for ll counries (i.e. nd prices re equlized for ll bundles of goods (i.e. nd. Oherwise, he sedy se invesmen shres cn only depend on he sedy se rel ne foreign sse posiion of he domesic counry hrough is impc on he sedy se relive prices nd. ourh Sep. I look he sedy se price indees nd sub-indees s well s he rel echnge re. I wrie he sedy se price sub-indees in equions (7 ( s follows, e (h ; e (f ; ( e (h ; e (f ; ( nd, S; S; ( since he lw of one price holds in he deerminisic zero-in ion sedy se. These resuls re derived looking he pricing equions in (7 (7 s well. Using he consumpion price indees in equions
( ( evlued heir sedy se, I lso cn infer h he rel echnge re should be equl o, RS S S + + which is funcion of he sedy se erms of rde 7 + S + 7 ; (. The sedy se rel echnge re will be equl o one if i holds h. Moreover, given he de niions of he consumpion price indees in ( (, I cn lso noe h, + + + S 7 S ; ( + S 7 + 7 ; (7 which ies boh relive price epressions (which ener ino he derivion of he invesmen shres y nd y o he erms of rde. Nurlly, since +, hen nd if. ifh Sep. in (88 I look ggrege oupu demnd nd he resource consrin. The ggrege oupu demnds (89 implies in sedy se h,! Y C + +! C + + RS Y! C + +! C + + RS! C +! C + ; (8! C +! C + : (9
Moreover, he resource consrin in (9 cn be chrcerized s follows, I + +! C + + RS! C + + RS! C +! C + C + C + ; ( nd, lernively, re-wrien s, I! " I! Y C + # ; ( where I hve lredy used he domesic ggrege oupu demnd equion obined in (8. Obviously, if he sedy se rel ne foreign sse posiion of he domesic household is equl o zero (i.e., hen I obin he sndrd resul implying h Y C +. In urn, Y C + holds when i lso hppens o be rue h he lw of one price holds nd. In he more generl cse h I m eploring here, however, he relionship in ( implies h, Y " C +! # + I I ; ( where he rel ne foreign sse posiion of he domesic household,, is epressed in unis of domesic consumpion. Then, he resource consrin in ( cn be wrien s, I! I @! C + + RS! C + RS! C +! C + ; ( where I hve used he de niion of he domesic consumpion price inde in (, i.e., + : The epression for he foreign ggrege oupu demnd in (9 cn be epressed now s, Y! h C + + RS C + i : ( ence, combining his foreign ggrege demnd equion wih he second version of he resource consrin
in (, i follows h, Y! C +! + RS C +!! + RS C + I! Then, using he de niion of he foreign consumpion price inde in (, i.e., + ; I : ( he implicions of he lw of one price, nd he de niion of he rel echnge re in ( (i.e. RS S, i follows h, nd, Y S! S! S + S RS C + I! I RS RS C + I! ; Y RS I C + I! I! C +! RS I! : ( Obviously, if he sedy se rel ne foreign sse posiion of he domesic household is equl o zero (i.e., hen I obin he sndrd resul implying h Y C +. In urn, Y C + holds only whenever i lso holds rue h. In he more generl cse h I m eploring here, however, he relionship in ( implies h, Y " C + I I I! # : (7 RS This epression di ers slighly from he epression I derived for he domesic oupu demnd in ( becuse he rel ne foreign sse posiion of he domesic household,, is epressed in unis of domesic consumpion. ence, he rel echnge re needs o be inroduced in he formul in (7 in order o epress everyhing in consisen (comprble unis. Using he second version of he resource consrin in ( gin, I ge sedy se epression in erms
of domesic nd foreign bsorpion,! C + RS! C + I! : (8 This implies h C + RS C + holds only if he rel ne foreign sse posiion of he domesic household becomes equl o zero in sedy se, i.e.. In urn, C + C + holds if i is lso he cse h prices equlize, i.e. I nd. urhermore, if prices nd bsorpion equlize cross counries, hen he consumpion shres will lso be equl cross counries becuse he invesmen shres mus be idenicl (s discussed before for ( (. In summry, I hve obined he following hree key equions in (, (7, nd (8, i.e. Y C + ( ; (9 Y! C + + ( ; ( RS! C + RS! C + + ( ; ( fer replcing he sedy se rel ineres re derived in ( nd ( (i.e. I I. In more generl seing where he rel ne foreign sse posiion is poenilly di eren from zero (i.e., hen he ssumpion h bsorpion in boh counries di ers only by fcor reled o he sedy se rel echnge re (i.e. C + RS C + requires he following condiion o be sis ed,! C +! C + + ( ; ( which follows from imposing he ssumpion on bsorpion o equion (. The rel ne foreign sse posiion of he domesic household cn be wrien s frcion of he domesic bsorpion,, nd lernively s frcion of he domesic oupu, y, i.e., C + ; ( y Y : ( I hve chrcerized he domesic oupu demnd, Y, in (9, so i is possible o re-wrie he rel ne foreign sse posiion of he domesic household relive o domesic oupu demnd, y, in he following erms, y Y C + C + Y C + C + C + (! ( ; ( 7
where follows, is gin funcion of he erms of rde rio y ( +. In fc, given ( I cn wrie y s 7 : ( ence, if I choose he rel ne foreign sse posiion relive o domesic bsorpion,, o (uniquely chrcerize he sedy se erms of rde, hen I cn pin down he rel ne foreign sse posiion relive o domesic oupu demnd, y, from (. I chrcerize he shre relive o domesic bsorpion,, s he reference level of he rel ne foreign sse posiion in he model, bu I will use he lernive shre represenion, y, whenever i is more pproprie (or convenien o describe he sedy se or o de ne he prmeerizion of he model. Condiion ( bove which links domesic nd foreign bsorpion ogeher under he proposed normlizion of he sedy se cn be epressed more compcly combined wih ( s,!! + ( ; (7 or simply,! + I concenre on his epression in order o derive he erms of rde rio The price rio cn be epressed s funcion of he erms of rde opering on equion (8 I immediely obin h,! + @ + ( : (8! ( ; s funcion of. s given in (, so or simply,! ( ( + ( : (9 In he ereme cse in which whenever, hen i mus follow from his condiion h, ( : ( Insnces in which is non-zero cn be chrcerized hrough he condiion bove in (9. In urn, h would imply h he erms of rde re di eren from one, i.e., nd so is he rel echnge re, i.e. RS. In fc, fer lile bi of lgebr wih (9, I obin closed form soluion for he erms of rde 8
s funcion of s follows, Terms of rde + ( ( : ( re ied o he rel ne foreign sse posiion of he domesic households s shre of domesic bsorpion,, s well s o oher srucurl prmeers of he model (i.e. he ineremporl discoun fcor,, he shre of he home goods in he domesic ggregors,, nd he shre of foreign goods,. Replcing epression ( ino he epression for he rel ne foreign sse posiion of he domesic households s shre of domesic oupu demnd in (, I obin h, y ( + + ( ( ( ( ( + ( ( ( : ( In oher words, here is very srighforwrd mpping beween he domesic rel ne foreign sse shre on domesic bsorpion nd on domesic oupu demnd, nd y respecively. Similrly, I cn use he formul for sedy se erms of rde se epressions for RS, nd RS S + obined in (, (, (7 s follows, + + + 7 + + ( + + ( in ( o re-wrie he sedy ( ( ( ; ( 7 + + ( ( [ ( ] ; ( + ( 7 ( + + ( + ( ( + + ( ( : ( Ineresingly, non-zero domesic rel ne foreign sse posiion in sedy se (i.e. does no imply h he rel echnge re mus inevibly di er from one. In he specil cse where he consumpion bskes re idenicl cross counries (i.e., hen I ge h RS independenly of. Therefore, if he consumpion bskes re ruly idenicl, hen he ssumpion h bsorpion cross counries mus be reled o he rel echnge re in sedy se (i.e. C + RS C + reduces o he convenionl ssumpion h bsorpion equlizes cross counries (i.e. C + C +. 9
Sih Sep. I revisi he cross-counry invesmen shres. I de ne he domesic nd foreign sedy se invesmen shres relive o ggrege oupu (i.e. y in ( nd y in (, while he invesmen nd consumpion shres relive o bsorpion cn be de ned s, C + Y ( Y C + ( ( 7! Y C + " # ( 7 C + ( ; ( C + C c C + ; (7 C + Y Y C + ( 7! Y! C + ( ( 7 ( C + + ( C + RS ; (8 c C C + ; (9 fer hving used he ggrege oupu equions derived in (9 nd (. rom here, using he de niion of he rel ne foreign sse posiion of he domesic household s shre of domesic bsorpion found in ( nd he ssumpion h in sedy se i mus hold h C + RS C +, I cn rgue h he consumpion nd invesmen shres relive o bsorpion mus be equl o, C + ( ( 7 ( ( ; ( C c C + ; ( C + ( 7 + ( + ( ( ; ( ( c C C + : (
Le me denoe he consumpion nd invesmen shres in he specil cse wih zero ne foreign sse posiion, i.e. wih, s follows, ( ( ( 7 ; c ; ( ( 7 ; c : ( Then, when he sedy se rel ne foreign sse posiion is zero nd he sedy se subsidies re equl cross counries, he sedy se consumpion nd invesmen shres in ( ( reduce o he convenionl cse where he bsorpion shres re equl cross counries nd equl o he shres epressed relive o oupu, i.e. if nd hen y y nd c c y c y c c c. Sevenh Sep. I look he pricing equions from he rms opimizion problem. I cn wrie he monopolisic compeiion price-seing rule for he domesic rms wih heir mrginl cos funcion combining ( nd (7 nd he monopolisic compeiion price-seing rule for he foreign rms wih heir mrginl cos funcion combining (7 nd (7 s in ( nd (, i.e., W ( W ( W ( W ( Z (!! Z! ( ; ( ; (7 where I hve divided ech equion by he consumpion price level of is respecive counry nd I hve ppropriely replced he rel renl re on cpil derived before in (. The prices of ll domesic vrieies re equlized, nd so re he prices of ll foreign vrieies. rom he lbor supply equions in ( nd (, I obin h, W W C L s ' ; (8 C L s ' : (9 Given he rm producion funcions in ( nd ( nd he fc h cpil-o-lbor rios re equlized cross rms wihin counry, i is possible o wrie he following sedy se domesic nd foreign ggrege
oupu demnd equions from (8 nd (8, 7 i.e., nd, mos impornly, I cn esblish h, K Y K Y Y K L ; (7 Y K L ; (7 L y L ; (7 Y Y! L! y L! ; (7 Y where he second equliy uses he fc h he cpil-o-oupu rio is proporionl o he invesmen shre over oupu in ech counry s noed in ( nd (, i.e. y K Y nd y K Y. Using he lbor mrke clering condiions, i.e. L s L nd L s L, I cn epress he domesic nd foreign lbor supply equions s, W W C Y ' ' L C Y " ' y Y C Y ' L! ' C Y ' nd so he domesic nd foreign price-seing rules in ( nd (7 become simply, Y " Y # ' y ; (7 # ' ; (7 ( ( +' y ( ' ( +' y ( ' ( C Y ' C ; (7 Y ' (77 : Then, ppropriely replcing he invesmen shres y nd y derived in ( nd ( i is possible o 7 Given h he prices of ll vrieies equlize wihin counry in sedy se, hen he wedges in (?? (?? mus be equl o one in sedy se, i.e.. Therefore, he supply of he ggrege oupu bundle (of vrieies in ech counry obined in (?? (?? mus be equivlen o he ggregion of ech individul vriey in (8 (8. Then, i mus be he cse h Y Y nd Y Y nd h here is no oupu loss due o price dispersion. owever, he sedy se cn sill be disored by he mrk-up chrged by ech rm opering under monopolisic compeiion unless he subsidy fully elimines his disorion s i would hppen in (.