Wha Price Index Should Cenral Bank Targe? An Open Economy Analyi Miaki Maumura December 22, 2018 Click here for he lae verion Abrac There i currenly a debae abou wha price index cenral bank hould arge when economie are open and expoed o inernaional price hock Thi paper derive he opimal price index by olving he Ramey problem in a New Keyneian mall open economy model wih an arbirary number of ecor Thi approach improve on exiing heoreical benchmark becaue 1 i make an explici diincion beween he conumer price index CPI and he producer price index PPI, and 2 i allow exogenou inernaional price hock o play a role Qualiaively, I ue he analyical expreion of he opimal price index o dicu ha popular indice, uch a he PPI and he core/headline CPI, are ubopimal becaue hey ignore he heerogeneiy in price ickine and he effec of inflaion on he rade urplu Quaniaively, I calibrae a 35-ecor verion of he model for 40 counrie and how ha abilizing he opimal price index yield ignificanly higher welfare han alernaive indice JEL code: F41, E52, E58, and E61 Keyword: Price Index, Small Open Economy, Opimal Moneary Policy, Targeing I am deeply indebed o my advier David Weinein and Michael Woodford I alo hank Haan Afrouzi, Andre Drenik, Marc Giannoni, Takaohi Io, Jennifer La O, Karel Meren, Emi Nakamura, Michael Plane, Sephanie Schmi-Grohe, Jon Seinon, Marin Uribe and he paricipan of eminar for many helpful uggeion on he paper I am alo graeful o he Cener on Japanee Economy and Buine a he Columbia Univeriy for he financial uppor Thi paper wa parly wrien a he Federal Reerve Bank of Dalla, while I wa a dieraion fellow I hank he financial uppor from AEA ummer fellowhip All error are mine Email: mm4387@columbiaedu Columbia Univeriy in he Ciy of New York 1
1 Inroducion A many mall open economie SOE have hifed heir moneary policy from exchange rae peg o inflaion argeing policie, here ha been growing inere in which price index hey hould arge The heory of opimal moneary policy wih a muli-ecor economy can be ued o anwer hi queion, a in Aoki [2001] and Woodford [2010], bu uch analye o far have been limied o cloed economy eup, leaving open economy queion unanwered, uch a he effec of inernaional commodiy price and he role of rade paern Thi lack of he opimal price index heory in an open economy underlie he ongoing debae over he choice beween, for inance, he headline conumer price index CPI veru he core CPI or he CPI veru he producer price index PPI In hi paper, I derive he opimal price index for open economie o abilize by olving he problem of a cenral bank aemping o maximize houehold welfare, ie, a Ramey problem I call he derived index he Ramey price index RPI and preen i analyical formula Due o he openne of my model, he index depend on he expor hare of oupu in each ecor in addiion o he parameer udied in cloed economy model uch a he conumpion hare, price ickine and he elaiciy of ubiuion By calibraing he model o 40 counrie wih 35 ecor, I find ha 1 RPI abilizaion perform beer for all counrie in erm of welfare han headline CPI, core CPI, or PPI abilizaion and 2 he ranking of he indice oher han he RPI differ acro counrie To derive he opimal price index, I begin wih he muli-ecor DSGE model wih oupu price ickine analyzed in Woodford [2010] The ue of a muli-ecor model i neceary o anwer my reearch queion ince differen price indice arie due o he difference in weigh applied o he price in differen ecor Oupu price ickine i he key moneary fricion in my model and he workhore model in he lieraure, in keeping wih exenive empirical evidence ee Nakamura and Seinon 2008, for example Under oupu price ickine, volaile inflaion caue mipricing by firm, leading o welfare-damaging inefficien producion aciviie A he key deparure from Woodford [2010], I allow each ecor in he economy o expor a par of i oupu Thi openne allow for a difference beween CPI and PPI becaue when he economy can rade, wha i produced i no necearily conumed The choice beween he wo indice i ofen he focu of moneary policy dicuion epecially for commodiy exporer and developing counrie For inance, Frankel [2010] numerically analyze Lain American commodiy exporer and conclude ha producer price baed indice beer perform han conumer price baed indice in erm of price abiliy India changed i arge index from PPI o CPI in 2016; ee Rajan [2016] The exiing heoreical framework i no uiable o anwer hi ype of queion ince he conumpion baed weigh coincide wih he producion baed weigh 1 Anoher key feaure of my model i he ue of an SOE eup raher han a wo-counry eup Thi i o capure he noion of inernaional price movemen ha i exogenou o he economy The Bank of Japan, for example, argued ha he movemen of he inernaional oil price wa he mo imporan reaon ha i failed o achieve i inflaion arge; ee 1 The inpu-oupu rucure i anoher reaon ha he PPI and CPI can differ I focu on he difference ariing from he rade in hi paper 2
Kawamoo and Nakahama [2017] The SOE framework allow me o anwer he queion of wheher he economy hould bear uch volailiy in inflaion ha i caued by inernaional price change In hi muli-ecor New Keyneian NK SOE DSGE environmen, I olve he Ramey problem and obain he opimal price index ha remain conan in he long-run expecaion under he Ramey oluion Thi mean ha my propoed opimal price index i baed on welfare maximizaion raher han an arbirary objecive The welfare maximizaion problem i ubjec o opimizing behavior of he repreenaive houehold and firm under moneary fricion The ue of he Ramey framework alo mean ha he moneary policy conidered in hi paper i no limied o a paricular cla of moneary policy uch a he Taylor rule Depie he generaliy of he choice of moneary policy, I how ha, in he long-run, a paricular price index remain conan I explore he propery of hi RPI qualiaively and quaniaively The key rade-off beween abilizing one price index veru anoher can be underood by conidering he co of volaile inflaion rae in he ecor wih lower weigh in each price index Therefore, he reuling opimal price index ake he form of a weighed um of he price in differen ecor, where he weigh aigned o each ecor reflec he co of inflaion in each ecor In oher word, in a muli-ecor environmen, he inflaion rae of all he ecor canno be abilized imulaneouly following a hock ha lead o a relaive price change For example, when a change in world demand lower he efficien relaive price of oil, he cenral bank need o eenially chooe one of wo opion: 1 a able oil price and an increae in non-oil price and 2 a able non-oil price and a decreae in he oil price Given hi rade-off, we hould abilize he price of he ecor wih he higher co of inflaion My fir main reul i he analyical formula for he RPI In paricular, I highligh hree leon from he formula The fir wo leon come from each of he wo componen of he formula The formula i a weighed um of differen log price in differen ecor, where he weigh repreen he welfare co of inflaion in each ecor I how ha he weigh coni of wo par, one repreening he ize of he ecor and he oher repreening he eniiviy of he producion wedge o inflaion in he ecor I alo how ha he RPI formula doe no direcly depend on inernaional price The hird leon come from wha i no in he formula The fir leon from he fir componen of he RPI i ha he ize of he ecor in he RPI weigh need o be meaured in erm of he producion ize raher han he conumpion ize Thi i becaue he co of inflaion in my model i he efficiency lo in producion If here i inefficiency in producion, i i welfare damaging eiher hrough reduced conumpion, more work or a negaive effec on he rade balance, which affec he economy hrough a igher budge conrain Therefore, regardle of wheher i oupu i conumed or expored, inefficiency in producion i coly in a ecor ha i large in erm of producion An implicaion of hi i ha he cenral bank hould abilize PPI raher han CPI if everyhing ele i conan However, here i a cavea in hi imple akeaway, a my quaniaive analyi how ha he abilizaion of PPI doe no necearily perform beer han CPI abilizaion due o he econd componen of he RPI weigh The econd componen of he RPI weigh i a combinaion of a well-known ickine parameer and le frequenly highlighed bu equally imporan parameer, repreening he 3
elaiciy of ubiuion beween differeniaed good wihin a ecor Thee wo parameer govern he eniiviy of inefficiency o inflaion in he ecor in queion The mechanim comprie wo ep Fir, volaile inflaion caue mipricing by he firm in a ecor Thi ep depend on he degree of price ickine Second, mipricing lead o deviaion of demand and producion from he efficien level Thi ep depend on he elaiciy of ubiuion The addiion of ecoral heerogeneiy in he elaiciy of ubiuion provide he econd leon ha i imporan when we dicu core inflaion argeing veru headline inflaion argeing Recall ha he difference beween he wo meaure i wheher hey include commodiy price uch a food and energy 2 While he lieraure o dae ha focued on one characeriic of commodiie, namely price flexibiliy, he high elaiciy of ubiuion i alo an imporan characeriic 3 A i andard in he convenional argumen, if we bae our deciion only on he price flexibiliy of differen ecor, we hould aign a lower weigh o commodiy ecor and hu favor he ue of core inflaion argeing However, if we focu on he laer characeriic, we hould place greaer weigh on commodiy ecor Given my analyical formula, wheher we hould place le weigh on price in commodiy ecor or no depend on he relaive ize of price flexibiliy and elaiciy The hird leon from he analyical formula i ha exogenou inernaional price do no appear in i Thi i depie he fac ha I naurally model he effec of exogenou inernaional price In my model, he firm repond o he change in he co of impored maerial caued by he change in he inernaional price of inpu The firm alo know ha a deviaion of heir expor price from hoe of heir inernaional compeior reul in a change in expor demand I how ha hee inernaional price affec he opimal price index if and only if hey affec he oupu price of domeic ecor Thi i becaue volaile inflaion caue efficiency lo in producion regardle of he caue of he volailiy, and hu, we do no need o adju he formula for he price index depending on wheher uch volailiy come from inernaional price A an implicaion, alhough we may end o hink ha cenral bank are no reponible for inflaion volailiy caued by inernaional price movemen, a cenral bank hould be concerned abou volailiy a long a i affec he RPI To underand hi poin, noe ha alhough inernaional price are exogenou, domeic price can be conrolled via change in he exchange rae Imagine an economy where all he domeic price of differen ecor are proporional o he inernaional price in hoe ecor The raio beween he vecor of inernaional price and he vecor of domeic price i he exchange rae If he cenral bank elec one domeic ecor, i i poible o abilize he domeic price of ha ecor by adjuing he exchange rae o offe inernaional price movemen Of coure, hi operaion affec all oher ecor, o he cenral bank face a rade-off beween abilizing one ecor and abilizing anoher The RPI indicae how o balance hi rade-off My econd main reul i obained from quaniaive analyi, where I compare he welfare under imple abilizaion policie for he RPI and hree convenional price indice Here, a 2 Alhough he original definiion of he core inflaion rae involve economeric model ha aemp o idenify he perien componen of he inflaion rae ee, for example, Wynne 2008, he opimal moneary policy lieraure ha pracically inerpreed he core index a an index excluding food and energy 3 See Nakamura and Seinon [2008] on price flexibiliy and Broda and Weinein [2006]on he elaiciy of ubiuion 4
imple abilizaion policy mean a policy in which he inflaion rae in erm of he price index in queion i zero in boh he hor and long run In realiy, implemening hee policie via eiher Taylor rule or exchange rae inervenion i impler han implemening he Ramey oluion ielf However, i i no obviou ha he imple abilizaion of he RPI yield higher welfare han he abilizaion of oher price indice ince he analyical reul only ae he opimaliy of long-run abilizaion of he RPI, and he Ramey oluion ielf, in general, involve hor-run deviaion from complee abilizaion Calibraing o 40 counrie wih 35 ecor, I how ha, for all counrie in my ample, RPI abilizaion perform he be among he abilizaion cheme for he four indice conidered The lo from a imple abilizaion of he RPI compared wih he Ramey oluion urn ou o be negligible and le han one-hundredh, on average, of he lo from imple abilizaion of he oher indice in erm of eady-ae conumpion Thi mean ha he RPI i uiable no only for long-run abilizaion arge bu alo for hor-run arge Anoher imporan poin from he welfare calibraion i ha here i no imple akeaway oher han he RPI Thi i becaue he ranking of oher abilizaion policie varie acro counrie depending on he combinaion of rade paern and price ickine Tha i, CPI argeing perform beer han PPI argeing for ome counrie while headline CPI perform beer han core CPI argeing for oher counrie, depending on he combinaion of price ickine, he elaiciy of ubiuion, and rade paern The only reul common o all counrie in my ample i ha RPI abilizaion perform beer han he abilizaion of he oher indice 11 Relaed lieraure Thi paper i an open economy exenion of he mehod o derive he opimal price index from he Ramey problem developed in Woodford [2010] The price index in Woodford [2010] can be obained a a pecial cae of he RPI propoed in hi paper by leing he expor in each ecor be zero and requiring he elaiciy of ubiuion o be homogeneou acro ecor However, he oher direcion, ie, deriving he RPI from Woodford index, i no raighforward Thi i becaue he ize of each ecor in Woodford [2010] can be inerpreed eiher a he ize of conumpion or he ize of producion, and one migh ugge differen open economy exenion of he index depending on he inerpreaion My analyi and he reuling formula for he RPI how ha he correc inerpreaion i he ize of producion Thi paper i he fir o heoreically how ha he ize of ecor in he abilizaion objecive hould be meaured by producion ize raher han conumpion ize in a muliecor SOE environmen A imilar feaure can be een in he reul of Gali and Monacelli [2005], who demonrae he opimaliy of oupu price abilizaion in a model wih only one producion ecor However, having muliple ecor i key o anwering he queion of which price index o arge ince hi creae he crucial rade-off beween abilizing one ecor veru anoher when he fir-be allocaion canno be achieved In paricular, heir analyi canno ell wheher he reul i coming from he aumpion ha here i only one ecor wih icky price or he aumpion ha he economy produce in only one ecor Thi make i difficul o generalize heir model o variou rade paern commonly oberved in he real world uch a he commodiy imporing cae My general formula enable me 5
o eparaely dicu he effec of producion and ickine and can be applied no only o he pecial cae of Gali and Monacelli [2005] bu alo o he oppoie polar cae commodiy exporer and he inermediae cae There i a lieraure ha analyze he opimal moneary policy in wo-counry model ee Corei e al 2010 and Engel 2011, for example and he model of a moneary union ee Gali and Monacelli [2008] and Kekre [2018], for example Thi paper differ from hi lieraure in wo ene Fir, alhough, imilarly o Woodford [2010] and hi paper, hee paper ofen idenify he cenral bank rade-off depending on price ickine, hey do no derive he price index ha balance he rade-off excep for pecial cae ha achieve he fir-be allocaion Second, he wo-counry eup of hee paper are eenially cloed ince he wo counrie or he counrie in he union do no rade wih he re of he world Therefore, heir framework canno anwer he queion of how o deal wih inernaional price movemen In hi paper, I ue he erm opimal price index, bu he derived price index doe no necearily coincide wih he opimal indice in he lieraure on index heory: ee Diewer e al [2009], for example Thi i becaue he purpoe of he index are differen In index heory, Diewer e al [2009] among oher aemp o obain an accurae meaure of he co of living while my aim i o obain he index for he cenral bank abilizaion arge By olving he houehold opimizaion condiion in he parial equilibrium ene, we can ee ha he CPI i he opimal price index in he ene of he co of living in my model However, my analyi how ha he opimal price index for he cenral bank abilizaion arge i differen from he CPI I i naural o obain differen opimal price indice for differen purpoe From a echnical poin of view, he open economy exenion in hi paper involve wo innovaion ha are alo applicable o oher SOE problem The fir i he definiion of he Ramey problem, which i conien wih he aumpion of he iming of ae marke Specifically, he Ramey planner need o recognize ha ome of he effec of i policy will be offe by he inurance effec of he ae marke In hi way, I can compare he cenral bank econd-be problem wih he planner fir-be problem and offer inuiive dicuion comparing he wo The definiion of he Ramey problem i in line wih he Ramey axaion lieraure, bu he previou NK SOE lieraure ha defined he Ramey problem in a differen way, and hence, he fir-be allocaion canno erve a a benchmark for he analyi The definiion of he Ramey problem in hi paper can implify and clarify he analyi by De Paoli [2009], for example, of he cae of he inefficien eady ae The econd innovaion of hi paper i differenial ax rae ha depend on he place of conumpion, which allow me o implify he analyi under erm of rade exernaliie wihou relying on exreme aumpion on parameer value Thi i anoher feaure ha diinguihe my paper from Gali and Monacelli [2005], who impoe a ubidy ha parially offe eady-ae inefficiency and eliminae he re of inefficiency by eing a parameer value uch ha he value of expor doe no repond following any hock I believe my novel implificaion i ueful for moneary policy dicuion under erm of rade exernaliie The remainder of he paper proceed a follow In Secion 2, I fir explain he SOE NK DSGE model wih which I define he Ramey problem In Secion 3, I explain my analyical reul I fir ae he key aumpion on ax rae ha make he analyi imple before approximaing he Ramey problem The main heorem ae ha he RPI i abilized 6
in he long run, which i he juificaion for my propoal of RPI abilizaion Secion 4 dicue he quaniaive welfare comparion Secion 5 conclude he paper 2 Mehod I derive he RPI by olving he Ramey problem of a cenral bank aemping o maximize he welfare of a repreenaive houehold given marke conrain in an SOE NK DSGE model Thi ecion decribe hee marke conrain and define he Ramey problem The economy feaure an arbirary number of ecor wih heerogeneou oupu price ickine a la Calvo [1983] There i no domeic inpu-oupu rucure, bu he producion require labor and impored inermediae good The oupu can eiher be expored or domeically conumed When expored, he price i icky in he producer currency Specifically, I denoe he number of ecor by S N, wihin each of which, a coninuum of firm produce differeniaed good The differeniaed good are aggregaed wihin each ecor The economy i mall and open in he ene ha inernaional condiion are exogenou The co of impored maerial are given by he exogenou inernaional price ime he endogenou exchange rae The price of expor i compared wih he exogenou prevailing price in he inernaional marke, o which he foreign demand for he counry expor repond The economy alo ake he ae price in complee inernaional ae marke a given The moneary auhoriy aemp o maximize he welfare of he repreenaive domeic houehold, which conume good from all he ecor and provide labor The moneary auhoriy ake he opimizaion behavior of he houehold and firm under aggered price eing a given I alo ake exogenou inernaional marke condiion a given I aume he imele perpecive following Woodford [2003] 21 Marke condiion Secor are heerogeneou in price ickine and he elaiciy of ubiuion acro differeniaed good wihin a ecor The former i already idenified a key o obaining he opimal price index in he cloed economy lieraure Alhough heerogeneiy in he elaiciy ha no been highlighed in he lieraure, i i quaniaively imporan and inuiive Tha i, a high elaiciy of ubiuion implie ha a mall mipricing lead o a remendou wing in demand and i hu coly o welfare For he model o be applicable o differen counrie wih differen rade paern, I ue a general producion echnology and a general rade paern By adjuing he parameer of he producion echnology of my model, one can conider a counry uch a Japan imporing commodiie, ie, good wih flexible price and high elaiciie of ubiuion, and exporing differeniaed good or a counry uch a Ruia doing he oppoie Compared o he common SOE framework feauring radable good and non-radable good or ha wih home good and foreign good, he decripion of he producion ecor i enriched uch ha any impored good goe hrough he domeic ecor before being 7
conumed by he houehold Thi allow me o rea differen ecor uniformly depie he generaliy My model encompae he common framework in he lieraure a pecial cae 211 The repreenaive houehold In any period of ime [0, ], he repreenaive houehold conume good from each of he S ecor denoed by C for S and upplie labor, denoed by L, o each of he S ecor I aume ha he amoun of conumpion from differen ecor are aggregaed in a Cobb-Dougla funcion wih he exponenial facor ψ for ecor S umming up o one S ψ = 1 C = S C ψ 1 Thi implie ha elaiciy of ubiuion acro ecor i one Thi i he andard aumpion ued in muli-ecor NK model; ee, for example, Aoki [2001] and Euepi e al [2011] 4 For he labor upply, I imply aume homogeneou labor ha can be ummed Thi mean ha he diuiliy from labor depend only on he aggregae amoun of work, no in he diribuion of where he houehold work L = S L 2 An alernaive would be o aume increaing diuiliy from labor upplied o each firm in each ecor Thi would increae he efficiency co of price diperion relaive o my cae Given price {P } S, W, profi {E } S, a lump um ranfer T, all denominaed in he local currency, he pricing kernel in he inernaional ae marke M, he exchange rae E, and he price Λ of iniial deb D 0, where he uni i in he uiliy in he pre-pecified inurance conrac over differen policie, he houehold maximize [ ] C 1 σ max E 0 β D 0,{C,L } S, [0, ] =0 1 σ L1+φ + ΛD 0, 1 + φ ubjec o M 0 E 0 W L + E + T P C D 0 3 =0 E S S S The fir-order condiion are a follow: β C 1 σ ψ C = M 0 λp E β L φ = M 0 λw Λ = λ The fir erm 1 σ S C ψ / 1 σ in he objecive funcion repreen he inananeou uiliy from conumpion from each ecor {C } S aggregaed according o C = 4 Thi doe no mean ha he aumpion i wihou lo of generaliy Benigno and Benigno [2003], for example, demonrae ha relaxing he aumpion of a uniary elaiciy beween a home good and a foreign good may change he deirabiliy of he flexible price allocaion 8 E
S C ψ The econd erm in he objecive funcion repreen he diuiliy from labor upply o each ecor {L } S From he expendiure minimizaion problem, he CPI conien wih hi conumpion aggregaor i ψ P 4 P = S Uing hi, inra-emporal condiion for he houehold opimizaion are expreed a follow: ψ C = P C, S 5 P L φ C σ = W 6 P I aume ha he houehold rade in he inernaional ae marke before he moneary auhoriy chooe i policy Wih hi iming convenion, he marginal uiliy for he houehold of having le deb D 0 i fixed a he exogenou level Λ acro differen poible moneary policie The conan Λ repreen he hadow price of he iniial deb in he ae marke Thi allow me o ubequenly derive an inernaional rik haring condiion ha i invarian acro policie The policy-invarian rik haring condiion i andard in he lieraure, bu how o conienly derive he condiion in a DSGE eup ha no been fully explored For furher dicuion, ee Senay and Suherland [2007] The level of conumpion i deermined by he ighne of he lifeime budge conrain Denoing he aggregae conumpion of a foreign counry and i price by C and P, we can conider he ochaic dicoun facor o be equaed o he raio of marginal uiliie of he conumer in ha foreign counry beween any wo ae of he world In paricular, if we le M 0, = τ=1 M τ be he dicoun facor from period 0, or he planning period, o period in he fuure, hen, auming he ame uiliy funcion for he foreign conumer conuming C a price P, we can inerpre he ochaic dicoun facor a M 0, = β C σ /P C0 σ 7 /P0 under he aumpion ha he foreign conumer alo ha acce o he ame complee ae marke Gali and Monacelli [2005] alo inerpre he ochaic dicoun facor in hi way Combining hi wih he iner-emporal condiion of he houehold, we have β C σ /P C0 σ /P0 ψ Λ = β C σ E P 1 Thu, we can obain he inernaional rik haring condiion C = ξc Q 1 σ, 8 where Q = E P /P i he real exchange rae and ξ = ΛP0 1 σ /C 0 i a conan For hi SOE, foreign conumpion C and he foreign conumpion price level P are exogenou, o i he ochaic dicoun facor M Noe ha if we do no aume he ae marke ha inure acro differen policie, we need o allow Λ o vary acro policie and hence he coefficien of he rik haring condiion alo varie acro policie 9
212 The individual firm echnology and aggregaion The producion echnology for firm i in ecor S i given by Y i + Yi = Z, Mi αm L α l i Y i and Y i are he oupu of firm i in ecor a ime hipped for domeic ue and expored o foreign, repecively, Z, i he ochaic ecor-pecific produciviy, M i i he impored good, and L i i labor Noe ha he Cobb-Dougla parameer α m and α l are allowed o vary acro ecor I aume ha he echnology i linear, ha i, α m + α l = 1 for all S When α m = 0, hi reduce o he producion echnology aumed in Gali and Monacelli [2005] The linear echnology aumpion make he following calculaion impler by making he marginal co independen of he amoun produced If one inead aume decreaing reurn o cale, he efficiency co of price diperion will be larger For impliciy, I alo aume α l > 0 for all S Thi mean ha all ecor ue a lea ome amoun of labor Thi i empirically rue Some counrie, on he oher hand, may impor nohing in ome ecor Therefore, I do no impoe α m > 0 By eing α m 1 and α l 0, I can conider a counry imporing in ecor Alernaively, by eing α m 0 and α l 1, I can conider a counry being killed a producing good in ecor, and depending on he demand from foreign, i i likely ha he counry expor in ecor in equilibrium There i an aggregaion firm in each ecor wih aggregaion echnologie Y = Y θ 1 θ θ 1 θ i di and Y = Y θ θ 1 θ 1 θ i di, 9 ha operae compeiively The elaiciy of ubiuion parameer θ can be heerogeneou acro ecor The co minimizaion problem of he aggregaor give he demand chedule Y i = Pi P θ Y and Y i = and he price index conien wih he aggregaion P i P θ Y, 10 P = 1 Pi 1 θ 1 θ di and P = P 1 1 θ 1 θ i di 11 Noe ha he oupu for domeic ue and foreign expor are he ame good bu labeled and priced differenly 213 The individual firm pricing deciion Aume ha in each ecor S, a randomly eleced fracion 1 λ of he firm can ree he price The price ickine parameer λ can alo vary acro ecor An individual firm in ecor ake wage W, impor price E Q, he demand funcion in equaion 10, producion funcion and ax τ a given The uni co of impored good E Q i given by he produc of he endogenou exchange rae E and exogenou and ochaic inernaional 10
price Q The price of i oupu are e by he individual firm o maximize i expeced profi Pi 0, P i 0 = arg max P,P τ=0 { 1 τ P + 1 τ λ τ E [ E E +τ M,+τ E+τ Q αm,+τ W+τ α m α l E+τ P Q αm,+τ W+τ α m α l αl Z 1,+τ αl Z 1,+τ θ P Y,+τ P,+τ P P,+τ θ Y,+τ }] 12 The realized profi E i i aggregaed wihin and acro ecor E = E i di and immediaely paid ou o he houehold Noe ha he firm are axed differenly acro ecor and beween deinaion The rae for profi earned domeically i τ and he rae for profi from foreign i τ Following he uual procedure, he opimal pricing condiion can be aggregaed o P, P P, P = P, 1 1 1 + 1 1 θ 1 F, P 1 Π λ λ K, = P, 1 1 1 + 1 1 F θ 1, P 1 Π λ λ K, where F,, K,, F,, K, are defined a follow: F, = C σ P, Y, + λ βe Π,+1 θ 1 F,+1 15 P K, = 1 τ 1 θ Q Q αm αl θ 1 C σ, W Z 1 α m P, α l P Y, + λ βe Π,+1 θ K,+1 16 F, = C σ P, Y P K, = 1 τ 1 θ 1 1 θ 1 13 14 θ 1, + λ βe Π,+1 F,+1 17 1 θ Q Q αm αl θ 1 C σ, W Z 1 α m P, α l P, + λ βe Π θ,+1 K,+1 18 Noe ha he nominal exchange rae i ubiued ou uing he definiion of he real exchange rae Q = E P /P E = Q P /P, and I defined CPI inflaion rae a Π = P /P 1 and ecoral inflaion rae a Π, = P /P 1, Π, = P /P 1 For he derivaion, ee Appendix A1 Equaion 13 and 14 govern he dynamic of ecoral inflaion Noe ha he ecoral inflaion rae Π, and he inflaion in erm of he CPI Π are relaed hrough he change in he relaive price P /P Thu, he equaion ae ha ecoral inflaion i a funcion of expeced fuure ecoral inflaion F, and he expeced fuure marginal co K, The ecoral inflaion rae i he weighed um of one and he raio F, / K,, where he weigh on one become larger a he price become ickier λ 1 When he price i compleely icky λ = 1, hen ecoral inflaion become one, meaning ha he nominal ecoral price i fixed a he previou level, and only he relaive price may move if he CPI P move A 11
he oher exreme, when he price i fully flexible λ 0, hee equaion hold by having F = K In hi cae, he expecaion erm in F and K alo diappear, reoring he flexible price equilibrium pricing rule P, P = 1 τ 1 θ θ 1 214 Reource conrain Q Q αm αl, W Z 1 α m P, α l P The marke clearing condiion are L i di = L, S M i di = M, C = Y, = Y Uing he facor demand from individual firm, hee reduce o marke clearing condiion in aggregae variable C = Y and = Y 19 and he reource conrain in aggregae variable Z L = αl Q Q /P α m W /P αm C + and M = α m α l W /P L Q Q /P, 20 where = θ P i P di 1 and = Pi P evolve according o θ di 1 are he producion wedge ha θ P P, = λ, 1 + 1 λ f, Π ; P θ, 1 21 P 1 P P 1 P θ = λ P 1, 1 + 1 λ where he funcion f i defined a f x, y; z = 1 1 λ 1 λ xy z P f,, Π ; P θ, 1, 22 P P 1 θ 1 1 θ 1 For he derivaion, ee Appendix A2 Equaion 20 combined wih he dynamic 21 and 22 are he key equaion capuring he co of inflaion in ecor Fir, a we can ee from he dynamic, ecoral inflaion or deflaion Π = P /P 1 caue larger wedge, When ecoral inflaion i zero, ie, Π = 1, he wedge decay a he rae λ o he eady ae of = 1 When he inflaion rae deviae from one, i enlarge he deviaion of he wedge from one 5 The effec of inflaion on he wedge i larger 5 Thi happen regardle of inflaion or deflaion The fir erm i increaing in Π = P /P 1, bu he econd erm i decreaing in Π = P, /P Π / P, 1 /P 1 The overall erm behave like he fir erm when Π 1 and like he econd erm when Π 1 12
when he price i icky, repreened by a larger λ, and when he differeniaed good are more ubiuable, repreened by a larger θ Price ickine limi he abiliy of firm o e a uniform price acro differeniaed good A higher elaiciy induce a larger repone of demand and hu producion o he price differenial among imilar good wihin he ecor Second, he aggregae reource conrain 20 ae ha he wedge, creae a gap beween he inpu L and he oupu C, in effecive uni, which i he ulimae ource of welfare lo in my model Even if he producion funcion in each firm i no affeced by he inflaion rae, he diribuion of producion wihin he ecor i affeced by inflaion, a explained in he previou paragraph Since uneven oupu are ranlaed ino a lower effecive oupu under he love of variey aumpion repreened by he CES aggregaor 9, ecoral inflaion caue he producion wedge 215 Small open economy aumpion Finally, I aume ha foreign demand i price elaic = P E P θ, 23 where i he exogenou oal inernaional demand for ecor and P i i aggregae price index ha i alo exogenouly given Thi aumpion can be derived from he co minimizaion condiion of a foreign buyer who aggregae he compoie good of ecor from differen counrie wih a conan elaiciy of ubiuion θ aggregaor 22 The Ramey problem The moneary auhoriy problem i defined a follow Definiion 1 The opimal moneary policy i he oluion o he following problem Given random hock Q /P, P/P, Z, S, C, ax τ, τ =0, and iniial ae variable P 1, E 1, S, 1,, 1 he cenral bank chooe a coningen plan of all he endogenou variable C, L, S C, L, P /P, P /P, Y, Y,, M, W S P, Q, Π,, K,, F,, K,, F,, D 0 o olve S [ ] C 1 σ max E 0 β =0 1 σ L1+φ + ΛD 0 1 + φ ubjec o equaion 1, 2, 4-6, 8, 13-23 and [ E 0 M 0,P =0 S P Q M Q P P ] = D 0 The la condiion i equivalen o he houehold lifeime budge conrain 3 under he aumpion ha all he profi goe o he houehold a E and he balanced governmen 13 S,
budge Thi condiion i imporan for binding he planner wih he ame rade-off beween conumpion and labor a ha faced by he decenralized economy Alhough he iniial level of deb D 0 i mahemaically expreed a a choice variable, hi doe no mean ha he cenral bank can freely chooe i Recall ha I aumed in he previou ub-ecion ha he ae marke operae before he moneary auhoriy chooe i policy Thu, he moneary auhoriy ake ino accoun he change or lack hereof in he iniial level of deb D 0 when i chooe i policy In hi ene, he moneary auhoriy indirecly chooe he iniial level of deb 3 Analyical Reul In hi ecion, I derive he formula for he RPI and dicu he inuiion behind he index The juificaion of he index i given in a heorem ha ae ha RPI need o remain conan in long-run expecaion for he economy o achieve he Ramey opimal allocaion I ar by howing wo lemma ha help u underand he rade-off faced by he cenral bank The fir lemma concern he eady-ae propery ha make he analyi racable The econd lemma how how he Ramey problem can be approximaed around he eady ae A udied in Benigno and Woodford [2012], he oluion o he approximaed problem approximae he oluion o he original Ramey problem under regulariy condiion Then, I ae he heorem on he opimaliy of abilizing he RPI The formula for RPI can be inerpreed a a weighed um of price in differen ecor, where he weigh depend on oupu hare of he ecor, price ickine and he elaiciy of ubiuion wihin he ecor I dicu wo poin on he formula Fir, compared wih he CPI, he RPI i cloer o PPI ince PPI include price of expor However, he PPI i no alway beer han CPI due o he oher wo facor: price ickine and he elaiciy of ubiuion Second, inernaional price do no direcly appear in he formula Thi mean ha he cenral bank hould be concerned abou inernaional price if and only if hey affec oupu price ha appear in he RPI formula 31 Term of rade exernaliy and he efficiency of he eady ae To focu on he moneary fricion in he analyi, i i convenien o aume ha he ax rae are e o offe any real diorion ha arie under he flexible price equilibrium There are wo ype of real diorion in hi economy: monopoliic diorion and erm of rade exernaliy I i widely known wha ax rae offe he former ince i alo arie in he cloed economy eup Regarding he laer, however, no paper ha explicily defined he diorion and offe i uing a ax In hi ubecion, I how ha hee diorion can be offe by axe if we aume differen ax rae beween domeic conumpion and expor, a I do in my model The diorion are defined a wedge beween he ocial planner allocaion and he flexible price equilibrium The planner problem i defined a he maximizaion of he houehold welfare ubjec only o he reource and echnology conrain and he condiion in inernaional 14
marke The flexible price equilibrium i defined a uual Monopoliic compeiion lead o monopoliic markup in he price ha appear a diorion in he allocaion The erm of rade exernaliy, on he oher hand, come from he inabiliy of he individual firm o exploi monopoliic compeiion in he inernaional marke I define he fir-be planner problem a follow Q Definiion 2 Given, P P, P S M 0,, Λ, he planner olve max E 0 β D 0,C,M,,L S =0 =0 ubjec o he echnology conrain =0 S C ψ 1 σ 1 σ Z, M αm L α l = C + S and he iner-emporal rade balance condiion [ θ 1 θ E 0 =0 M 0,P S In defining he planner problem, I ue = P E P 1 θ P P θ P E P S L 1+φ + ΛD 0, 1 + φ ] Q M P = D 0 = 1 θ o eliminae price The objecive funcion i he ame a he welfare of he houehold in he Ramey problem in Definiion 1 The fir-be planner i conrained only by he aggregae producion echnology in each ecor and he iner-emporal rade balance condiion In building he aggregae producion funcion, I already impoed uniform producion wihin a ecor Y i = Y and o forh, a he opimaliy condiion The iner-emporal rade balance condiion doe no necearily require balanced rade in each period, bu any rade defici i financed in he inernaional ae marke, and any rade urplu i inveed in he inernaional ae marke uch ha he dicouned um of he rade urplu equal he iniial level of he exernal deb D 0 Appendix B1 how ha he planner oluion i characerized by he following: ψ α m L φ P αm C α l Z, C α l C σ Q Q = Lφ C σ S 24 and D 0 = E 0 θ 1 P θ Q P 1 θ = C = ξc Q 1 σ, M 0, P =0 Z, S Z, α m α l α m α l L φ C σ L φ C σ P Q Q P Q Q αm L C 1 θ C ψ C S 25 αm L C θ 1 θ 15 1 θ P P αm L φ α l C σ L Q 26 27
To compare hi wih he flexible price allocaion, I define he flexible price allocaion a he oluion o equaion 1, 2, 4-6, 8, 13, 14, 15-18 under λ = 0 for all S, and 19-23, and he houehold budge conrain Appendix B2 how ha he equilibrium i characerized by he following: C ψ C α l Z, α m α l θ 1 P θ Q P 1 θ C = ξc Q 1 σ L φ C σ = ν 1 P Q Q αm = χ 1 Z, α m α l L φ C σ L φ C σ P Q Q S 28 αm L C 1 θ C ψ C S 29 30 and D 0 = E 0 M 0, P =0 Z, S α m α l L φ C σ P Q Q αm L C θ 1 θ 1 θ P P αm α l L φ L Q C σ, 31 where he real wedge χ, ν are defined a χ = 1 τ 1 θ, ν = 1 τ θ θ 1 1 τ θ 1 We can ee ha he characerizaion of allocaion are equivalen excep for he wedge χ and ν The wedge χ for all repreen diorion coming from domeic monopoliic compeiion The wedge ν for all repreen diorion coming from he inabiliy of he domeic firm o exer heir monopoliic power in he inernaional marke, which I call he erm of rade exernaliy Thu, he following lemma hold Lemma 3 The flexible price allocaion i efficien if and only if χ = ν = 1 for all S Tha i, 1 τ = θ θ θ 1, 1 τ 1 = 1 τ = θ θ 1 θ 1 θ 1 θ 1 There are wo ype of inefficiency ha he ax need o addre To ee hi, noe ha even if he ax in each ecor offe he monopoliic markup in each ecor by eing 1 τ = θ / θ 1, inefficiency remain due o he difference θ θ 1 θ 1 θ 1 beween θ and θ To achieve he efficien allocaion, he ax need o offe boh inernal diorion due o domeic monopoliic compeiion and exernal diorion due o no uilizing inernaional monopoliic compeiion The exernal diorion arie when he elaiciy of foreign demand i finie and hence θ / θ 1 > 1 In hi cae, he equilibrium conumpion of expor ecor good i oo low The planner can improve welfare by exporing le while imulaneouly improving he 16
erm of rade The marke equilibrium canno achieve hi ince each expor ecor ake he oal demand for he expor a given, bu he planner can raegically increae he ecoral price of expor a a whole o affec he erm of rade and foreign demand To achieve hi allocaion in a decenralized manner, he fical auhoriy need o impoe differen ax rae depending on he deinaion of good In he following analyi, I aume uch efficien ax rae o focu my analyi on moneary fricion If I do no aume hi efficien level of axaion, he moneary auhoriy will have an incenive o ue differenial inflaion rae acro ecor o correc he diored real allocaion If hi force i added o he moneary rade-off ha I analyze below, he analyi become oo complicaed A he fir ep, I believe hi implificaion i beneficial in underanding he opimal price index 32 Approximaion of he Ramey problem Thi ubecion derive he approximaion o he Ramey problem around he opimal eady ae defined in Appendix B3 I denoe he log deviaion from he eady ae by he lowercae leer of he correponding ymbol of he variable All domeic nominal variable are expreed relaive o domeic CPI P All inernaional nominal variable are expreed in relaive erm o foreign CPI P I how ha when he eady ae i efficien in he ene defined in he previou ecion, he econd-order approximaion of he welfare funcion, ie, he objecive funcion of he Ramey problem, become purely quadraic wihou uilizing he econd-order approximaion of he pricing equaion Therefore, under regulariy condiion, we can obain an accurae fir-order approximaion o he oluion of he non-linear Ramey problem defined in Definiion 1 by olving he approximaed Ramey problem ha maximize quadraically approximaed welfare ubjec o linearly approximaed conrain Noe he difference beween he opimal eady ae and he efficien allocaion A mahemaically defined in Appendix B3, he opimal eady ae i opimal in he econdbe ene, where he moneary auhoriy problem ake icky pricing mechanim and marke condiion a given Therefore, he opimal eady ae need no be an efficien allocaion in he fir-be ene The appendix alo how ha he opimal eady ae can be characerized by he equaion for flexible price allocaion under conan exogenou variable and hu i efficien when he aumpion of Lemma 3 i aified Denoe he houehold welfare by W and i eady ae level by W Define he vecor of endogenou real variable a v = [c, x ] where c = [c 1,, c S ] and x = [x 1,, x S ] are he vecor of conumpion and expor of all he ecor Furhermore, define he vecor of exogenou variable a ξ = [c, x, p, q, z ], where x = [x 1,, x S], p = [p 1,, p S], q = [q 1,, q S], and z = [z 1,, z S ] 17
are he vecor of foreign demand for expor, inernaional price of expor, inernaional price of impor, and produciviy hock Before auming he efficien ax rae, by uing he marke condiion excep for he pricing equaion, I how in Appendix B5 ha he approximaed welfare can be wrien a W W = β E L 1+φ 0 φ ld α l 1 d φ c d χ 1 I c =0 + β E 0 φ ld α l 1 d φ x d χ 1 d ν 1 I x =0 + 1 [ β E 0 v Nξ Γ v2 v Nξ + [ ]] φ l θ φ c π 2 θ, + φ x π 2 2 =0 S α l κ κ, + ip where L i he eady-ae level of aggregae labor upply, φ c = C / C + i he eadyae conumpion hare of oupu in ecor, φ l = L /L i he eady-ae labor uage hare of ecor, φ x = 1 φ c i he eady-ae expor hare of oupu and d i he diagonal marix of he vecor inide he parenhee The 2S by 4S + 1 marix N define he naural level Nξ of he endogenou variable defined in he appendix The fir wo line are linear in he endogenou variable, bu when he eady ae i efficien χ = ν = 1 for all S, all of he linear erm diappear Therefore, under he efficien eady ae, we can obain a purely quadraic econd-order approximaion of welfare Appendix B6 how ha under he efficien eady ae, he naural level of he endogenou variable coincide wih he flexible price equilibrium denoed by F ξ wih a 2S by 4S + 1 marix F In he following, I denoe he log deviaion from he flexible price equilibrium by ṽ := v F ξ Furhermore, from he following relaionhip obained in Appendix B4 ψ = χ 1 φ c φ l α l S χ 1 φ c φ, l α l we can ee ha he coefficien of he inflaion rae can be implified o [ ] φ l θ φ c π 2 θ, + φ x π 2 S α l κ κ, = φ l φ c [ θ ψ π, 2 + φ ] x π 2 α S l S κ φ, c Therefore, I obain he following lemma Lemma 4 If he eady ae i efficien, approximaed opimal moneary policy can be obained by olving he linear-quadraic problem Given iniial condiion v 1 and precommimen, he cenral bank chooe { } ṽ, π, π, π o minimize =0 [ [ β E 0 ṽ Γ θ v2 ṽ + Γ π ψ π, 2 + φ ]] x π 2 κ φ, c =0 ubjec o 1 he Phillip curve S d κ 1 π βe [π +1 ] = γ P v ṽ and d κ 1 π βe [ π +1 ] = γ P v ṽ, 18
where κ = 1 λ 1 βλ /λ, and 2 he ideniie linking inflaion rae and relaive price π = 1 S 1 π + γ I v ṽ ṽ 1 + ɛ I ɛ I 1 and π Proof See Appendix B7 = 1 S 1 π + γ I v ṽ ṽ 1 + ɛ I ɛ I 1 The coefficien marice Γ v2, γv P, γv, P γv I and γv I, he calar Γ π and he reidual ɛ I and ɛ I are given in Appendix B7 The choice variable are he vecor of conumpion of each ecor c and he vecor of expor from each ecor x conained in he vecor of endogenou variable v, he vecor of inflaion rae π = [π 1,, π S ], π = [ π 1,, π S and CPI inflaion π The reaon for having CPI inflaion here i ha nominal variable are normalized by CPI inflaion One can alernaively wrie he equaion wih differen normalizaion and ill obain he ame reul for he opimal price index A i uual in cloed economy analyi, we have wo par in he objecive funcion The fir par i he quadraic erm in he gap in real variable from heir repecive naural level The econd par i he nominal par repreening he co of volaile inflaion The nominal fricion i larger when he ecor ue more labor, he price i icky, or he elaiciy of ubiuion i high Thi i inuiive becaue if he inflaion rae i volaile in a ecor, he price diperion of he ecor increae Thi mean ha o produce a cerain effecive oupu in he ecor, he ecor require more labor inpu and impored maerial, cauing diuiliy for he houehold hrough more labor or a igher inernaional budge The overall effec will be larger if he ecor ue more labor a he eady ae Inflaion volailiy lead o higher price diperion when he price i ickier Given he ame diribuion of individual price wihin a ecor, he degree of price diperion, become higher if he elaiciy of ubiuion θ i higher In he conrain, here are in oal 2S Phillip curve for domeic price and expor price in each ecor The la wo equaion in he conrain are ideniie linking ecoral inflaion rae π, π and CPI inflaion π Thi mean ha here i only one degree of freedom lef in hi problem Alhough here are differen inflaion rae for differen ecor, hey canno be freely choen ince relaive inflaion rae beween wo ecor deermine he evoluion of he relaive price of he wo ecor 33 Ramey price index Thi ubecion ae he main reul of hi paper If we define a price index uing he coefficien on he inflaion rae in he lo funcion derived in he previou ubecion, he price index ay conan in he long-run expecaion under he opimal moneary policy Thi implie ha if he cenral bank doe no abilize hi price index in he long-run, i policy i necearily ub-opimal Specifically, Appendix B8 how he following Theorem 5 Define he price index a log P = Φ 1 θ ψ S κ 19 ] log P + φ x φ c log P
wih Φ = θ ψ 1 + φ x S κ φ c Then, under he oluion o he Ramey problem, lim E log P T = Φ 1 log P T I call hi price index P he RPI ince i abilizaion i deirable a he oluion o he Ramey problem The calar Φ i ued o normalize he coefficien o um o one Thi heorem ae ha he long-run abilizaion of he RPI can be obained a a neceary condiion of he oluion of he Ramey problem The heorem moivae he cenral bank policy ha abilize he inflaion rae meaured in hi index ince if hi price index i no abilized in he long run under ome policy, he policy mu be ub-opimal The convere i no necearily rue Tha i, complee abilizaion of hi price index doe no necearily guaranee ha he economy follow he opimal pah conien wih he fir-order condiion Alhough i i generally poible o derive he if-and-only-if condiion uing he mehod of Giannoni and Woodford [2010], he condiion i generally complicaed To keep my dicuion imple and inuiive, I propoe he ue of a imple policy rule ha alway abilize he RPI The welfare analyi in Secion 4 how ha he welfare lo from imple RPI abilizaion policy i negligible compared o he opimal moneary policy and ha i perform beer han he abilizaion of headline CPI, core CPI, and PPI The RPI i a weighed um of price in differen ecor, where he weigh depend on conumpion hare ψ, he elaiciy of ubiuion θ, he Phillip curve lope κ ha conain he informaion of he price ickine λ and he rade paern φ x /φ c The weigh reflec he rade-off ha he moneary auhoriy face A he derivaion indicae, he weigh ake he form of he coefficien on inflaion rae in he lo funcion of he Ramey problem repreening he co of inflaion in differen ecor If he volailiy of he inflaion rae in a ecor i relaively more coly o welfare han ha in oher ecor, he RPI aign higher weigh o he former ecor Noe ha hi price index will remain conan even if here i a uni-roo proce in he exogenou variable ha may reul in a permanen change in he naural level of endogenou variable Thi fac hould be noed ince if all exogenou variable are aionary, price level under any price index will evenually coincide afer all raniory hock die ou 331 Comparion wih CPI and PPI To underand he relaionhip beween he RPI and he convenional price indice, le u conider he weigh on ecor under log P = log P Recalling ha φ c + φ x = 1, he weigh on he price in ecor become θ ψ 1 + φ x = θ CP I {}}{ 1 ψ κ φ c κ φ }{{ c } P P I From hi expreion, we can ee ha weighing under he RPI can be een a ha under PPI muliplied by he eniiviy of he wedge o inflaion θ /κ The PPI weigh i 20
relevan becaue he co of inflaion appear a he wedge in producion; ee equaion 20 Therefore, he relevan ize of he ecor i he producion ize raher han conumpion ize However, he quaniaive reul in he nex ecion how ha he eniiviy of he wedge o inflaion θ /κ i imporan in he ene ha PPI argeing omeime perform wore han CPI argeing The reaon for he incluion of hi addiional facor i ha a given inflaion volailiy caue differen wedge ize depending on price ickine, ummarized by κ, and he elaiciy of ubiuion, capured by θ Compared o he CPI weigh, ψ, he PPI weigh i higher for exporing ecor Thi i becaue when ome of he oupu i expored, he conumpion weigh on he ecor i maller han he opimal weigh In uch a cae, we can obain he correc ize of he ecor by inflaing he conumpion weigh ψ by he oupu-o-conumpion raio 1/φ c We can alo obain he price index derived in Woodford [2010] a a pecial cae by auming no rade φ c = 1 and a homogeneou elaiciy of ubiuion θ = θ In hi pecial cae, he weigh aigned o ecor i 6 ψ /κ The previou lieraure ha argued for core inflaion abilizaion baed on he obervaion ha he non-core ecor have higher degree of flexibiliy or higher value of κ, reuling in diproporionaely maller weigh on hoe ecor The RPI adju for he elaiciy of ubiuion θ and rade 1/φ c The former ha he effec of placing a higher weigh on ecor wih higher ubiuabiliy wihin he ecor Thi i imporan ince ome non-core ecor do have higher value of he elaiciy of ubiuion The laer ha he effec of placing a higher weigh on expor ecor Thi may hif he opimal weigh away from he core weigh and cloer o he headline weigh for commodiy exporing counrie 332 Role of inernaional commodiy price Anoher leon ha we can learn from he formula for RPI i ha inernaional commodiy price P, Q do no appear direcly in he index Tha i, he formula for RPI in Theorem 5 i a weighed um of price e by domeic firm P and P Even if hoe price are influenced by inernaional price, he formula doe no adju for or offe he influence of exernal facor Noe ha hi i depie he fac ha I naurally model he effec of exogenou inernaional price A in he pricing equaion 13-18, he inernaional price of inpu Q affec he firm pricing behavior hrough heir marginal co A in he expor demand equaion 23, price of inernaional compeior P affec expor demand The former ha a fir-order impac on ecoral price, and he laer ha a fir-order impac on he rade balance and a econd-order impac on ecoral price We can oberve from he formula in Theorem 5 ha hee inernaional price affec he opimal price index if and only if hey affec he oupu price of domeic ecor Thi i becaue volaile inflaion caue efficiency lo in producion regardle of he caue of he volailiy, and hu, we do no need o adju he formula for he price index depending on wheher uch volailiy come from inernaional price In oher word, oupu price in he formula are ufficien aiic in he meaure of he mo welfare-relevan inflaion rae 6 Thi i no exacly he ame a he expreion in Woodford 2010 ince I am implifying he analyi in one dimenion, namely, heerogeneiy in he labor Thi will affec he expreion for he κ reflecing he increaing diuiliy from uneven labor upply 21