Price Stability In Small Open Economies

Transcription

1 MPRA Munch Personal RePEc Archve Prce Sably In Small Open Economes Mkhal Dmrev and Jonahan Hoddenbagh Boson College November 212 Onlne a hp://mpra.ub.un-muenchen.de/46118/ MPRA Paper No , posed 12. Aprl 213 8:4 UTC

2 Prce Sably n Small Open Economes Mkhal Dmrev and Jonahan Hoddenbagh Frs Draf: November, 212 Ths Verson: February, 213 We sudy he conduc of moneary polcy n a connuum of small open economes. We solve he model globally n closed form whou resrcng he elascy of subsuon beween home and foregn goods o one. Usng hs global closed-form soluon, we gve an exac characerzaon of opmal moneary polcy and welfare wh and whou nernaonal polcy cooperaon. We consder he cases of nernaonally complee asse markes and fnancal auarky, producer currency prcng and local currency prcng. Under producer currency prcng, s always opmal o mmc he flexble-prce equlbrum hrough a polcy of prce sably. Under local currency prcng, polcy should fx he exchange rae. Even f subsuably dffers from one, he connuum of small open economes mples ha he share of each counry s oupu n he world consumpon baske (and herefore he mpac of he counry s monopoly power) s neglgble. Ths removes he ncenve o devae from prce sably under producer currency prcng or a fxed exchange rae under local currency prcng. There are no gans from nernaonal moneary cooperaon n all cases examned. Our resuls sand n conras o hose n he leraure on opmal moneary polcy for large open economes, where sraegc neracons drve opmal polcy away from prce sably or fxed exchange raes, and gans from cooperaon are presen, when subsuably dffers from one. Keywords: Open economy macroeconomcs; Opmal moneary polcy; Prce sably. JEL Classfcaon Numbers: E5, F41, F42. Ths paper grew ou of our shared experence n he Inernaonal Macroeconomcs course augh by Fabo Ghron. I s mpossble o express our graude o hm n words. We hank Perpaolo Bengno, Bob Kng, Eyal Dvr, Susano Basu and Peer Ireland for helpful commens. We also hank semnar parcpans a he BC/BU Green Lne Macro Meeng. Any errors are our own. Deparmen of Economcs, Boson College, Chesnu Hll MA, E-mal: dmrev@bc.edu. Deparmen of Economcs, Boson College, Chesnu Hll MA, E-mal: jon.hoddenbagh@gmal.com.

3 1 Inroducon Prce sably s wdely vewed as a benchmark moneary polcy for cenral banks, and has been found o be opmal n many closed economy sudes. 1 However, here s a longsandng debae over he desrably of prce sably n he open economy. Asymmerc shocks, curren accoun mbalances, cross-border prcng decsons, erms of rade movemens, sraegc neracon beween counres, polcy cooperaon and oher elemens unque o he open economy nroduce complcaons ha are no presen n he closed economy. These addonal ngredens presen a challenge for undersandng opmal polcy n he open economy. In he classc case of producer currency prcng (PCP), exchange rae flucuaons passhrough no mpored goods prces one-for-one. Under PCP, he frs generaon of mcrofounded models proved ha a cenral bank should mmc he flexble prce allocaon, emphaszng he somorphsm beween opmal moneary polcy n closed and open economes (Clarda, Gal and Gerler (22) and Obsfeld and Rogoff (22)). Bu he opmaly of prce sably hnged on wo crucal assumpons: unary elascy of subsuon beween home and foregn goods, and PCP. Unary elascy mples complee rsk-sharng across counres va erms-of-rade movemens, so ha households face no dosyncrac consumpon rsk n complee markes, ncomplee markes or fnancal auarky (Cole and Obsfeld (1991)). Counerfacually, under unary elascy expor revenues are consan and mmune o exchange rae flucuaons and producvy shocks. No expendure swchng occurs, so ha household consumpon does no shf o goods from cheaper counres (counres wh a deprecaed currency for example). Bengno and Bengno (23) relaxed hs assumpon, and showed ha prce sably s no longer opmal for PCP under non-unary elascy. Ther work affrmed a cenral bank s ncenve o manpulae s erms of rade, removng he somorphsm beween opmal moneary polcy n closed and open economes. 2 In addon, a number of sudes have shown ha under local currency prcng (LCP), when exchange rae flucuaons do no pass-hrough no mpored goods prces, prce sably s no opmal (Devereux and Engel (23), Corse and 1 In hs paper, we refer o prce sably as he polcy whch mplemens he flexble prce allocaon. A nonexhausve ls of papers fndng prce sably o be opmal n closed economes ncludes Goodfrend and Kng (1997, 21), Kng and Wolman (1998), and Woodford (2, 22). See Schm-Grohé and Urbe (211) for a recen survey. A counerexample s provded by mul-secor models such as ha n Aok (21). 2 Bengno and Bengno (23) hearkened back o he earler predcons of non-mcrofounded models, whch esablshed a deflaonary bas n he open economy due o a cenral bank s ncenve o apprecae s erms of rade (Rogoff (1985), Oudz and Sachs (1984), and Canzoner and Henderson (199)). Bengno and Bengno (26) laer showed ha prce sably s opmal for large open economes under non-unary elascy when cenral banks cooperae and markes are complee. Corse, Dedola and Leduc (21) also examne opmal moneary polcy under fnancal auarky, and show ha prce sably s never opmal n fnancal auarky for large open economes. Ths conrass wh our resuls for small open economes, where prce sably s opmal under PCP n all cases, even n fnancal auarky. 2

4 Pesen (25), Corse, Dedola and Leduc (21)). 3 In hs paper, we sudy he heorecal condons under whch prce sably s opmal n a connuum of small open economes. We solve he model globally n closed form whou resrcng he elascy of subsuon beween home and foregn goods o one. To he bes of our knowledge, we derve he frs closed-form soluon for an open economy model ha allows subsuably o dffer from one, enablng us o consder cases away from perfec rsk-sharng, ncludng fnancal auarky. As n Bengno and Bengno (23, 26), Corse and Pesen (21, 25), Devereux and Engel (23), Suherland (24) and ohers, we assume one-perod n advance prce seng and no home bas. Whle prevous sudes of opmal moneary polcy n small open economes focus on PCP wh non-cooperave polcymakers, we gve an exac characerzaon of opmal moneary polcy and welfare wh and whou nernaonal polcy cooperaon for PCP and LCP, n complee markes and fnancal auarky. 4 The connuum framework allows us o examne he mplcaons of nernaonal polcy cooperaon and solve for Nash equlbra, whch s no possble n he sandard small open economy seup where here s only one measure zero small open economy and one res of he world block. Combned wh our closed form soluon, we oban exac welfare gans resulng from cooperaon amongs a connuum of small, nerdependen economes. 5 Our resuls pon o he mporance of counry sze n he conduc of opmal moneary polcy. We prove ha for small open economes mplemenng he flexble prce allocaon hrough a polcy of prce sably s opmal under PCP, whle a fxed exchange rae s opmal under LCP. Boh resuls hold for any elascy of subsuon beween home and foregn goods, n complee markes and fnancal auarky, wh or whou nernaonal polcy cooperaon. We 3 Devereux and Engel (23) sudy opmal moneary polcy for wo large open economes under PCP and LCP n a complee markes model wh unary elascy and fnd ha a fxed exchange rae s he opmal polcy under LCP. Corse, Dedola and Leduc (21) examne LCP under non-unary elascy n complee markes and fnancal auarky for cooperave polcymakers, and show ha a fxed exchange rae s no opmal. We do he same exercse for small open economes and fnd ha a fxed exchange rae s opmal for LCP n all cases examned, for any elascy. 4 A non-exhausve ls of sudes of opmal moneary polcy n small open economes ncludes Gal and Monacell (25), Faa and Monacell (28), De Paol (29), Caao and Chang (212), Heva and Ncoln (212) and Monacell (212). Generally, hese papers allow for a rcher specfcaon and show varous resuls dependng on he emphass and specfc ngredens. All focus on PCP and do no examne poenal gans from cooperaon. Modelng LCP n a connuum of small open economes s que dffcul because he law of one prce and purchasng power pary no longer hold. Wh wo economes, s no hard o keep rack of exchange rae polcy, bu n he connuum hs proves more challengng. 5 To he bes of our knowledge, we are he frs o examne cooperaon whn he connuum framework. Our analyss does no face he problems encounered when conducng such an exercse n a log-lnear model. In a log-lnear model, he seady sae wll be dfferen n cooperave and non-cooperave equlbra. As such, s no clear how o make an accurae comparson of welfare beween cooperave and non-cooperave regmes. Ths s one reason why here was such an emphass on closed-form soluons n he early mcrofounded leraure on nernaonal polcy cooperaon. See Corse and Pesen (21, 25) and Obsfeld and Rogoff (21, 22). In hs paper we focus only on moneary polcy cooperaon, bu n a relaed paper we consder opmal fscal polcy and fscal cooperaon across counres (Dmrev and Hoddenbagh (213)). 3

5 also show ha nernaonal moneary cooperaon does no mprove welfare for small open economes. Under cenral bank commmen, he cooperave and non-cooperave soluons exacly concde under any combnaon of PCP, LCP, complee markes and fnancal auarky for any elascy of subsuon. 6 The drvng force behnd our resuls s he absence of sraegc neracon among polcymakers due o her small sze n proporon o he res of he world. Even when subsuably s dfferen from one, he share of each counry s oupu n he world consumpon baske, and herefore he mpac of he counry s monopoly power, s neglgble. Small open economes are analogous o frms n a model of monopolsc compeon: polcymakers have marke power bu canno nfluence he behavor of oher counres n he model. Sraegc neracons beween counres are hus absen. The absence of sraegc neracon has very dfferen mplcaons under PCP and LCP. Under PCP, he polcy decsons of small open economes have no effec on consumpon baskes n foregn counres, whch shus down he expendure swchng channel abroad. Unable o nduce expendure swchng or nfluence foregn ncome, polcymakers charge a consan erms of rade markup, ulzng monopoly power a he counry level. Monopolsc producers n our model also charge a consan markup ulzng her monopoly power a he frm level whn each counry, bu polcymakers remove hs domesc markup va producon subsdes. Snce moneary polcy can only affec he varance and no he level of hese consan markups, he opmal soluon for he polcymaker under PCP s o mmc he flexble prce allocaon. In conras, when economes are large, polcymakers mus nernalze he effec of her decsons on he res of he world. When domesc polcymakers charge hgher markups, foregn households become poorer and shf consumpon oward cheaper producs. Under PCP, expendure swchng abroad forces polcymakers n large economes o behave more cooperavely and charge lower markups. Snce he relave sze of an economy changes sochascally wh producvy shocks, he domesc polcymaker n a large open economy fnds opmal o devae from flexble prces and charge a varable markup raher han a consan one. Under LCP, prces are fxed one perod n advance n he currency of he mporer, so he proporon of ncome allocaed o counry-specfc goods s nsulaed from moneary polcy and exogenous shocks. As a resul, a consan proporon of world ncome s spen on each counry s unque varey, and expendure swchng does no occur. In addon, small open economes canno affec world ncome, so domesc oupu and labor are unaffeced by moneary polcy. The cenral bank hus maxmzes welfare by sablzng domesc consumpon va a fxed exchange rae. Wh a fxed exchange rae, labor flucuaes wh producvy shocks 6 Ths maches resuls for large economes under unary elascy (Obsfeld and Rogoff (22)), where gans from cooperaon are absen, bu conrass wh resuls for large open economes under non-unary elascy (Bengno and Bengno (26), Corse e al (21) and Suherland (24)), where gans are presen. 4

6 whle consumpon remans consan. In conras, moneary polcy n large open economes can nfluence domesc labor under LCP. Expansonary moneary polcy a home wll ncrease wages, whle goods prces reman fxed. Ths nduces home and foregn households o work more, snce domesc and foregn frms face hgher demand for her goods. Hgher real wages and an ncrease n hours worked lead o hgher domesc consumpon. Polcymakers n large open economes hus face a rade-off beween labor and consumpon volaly. In such cases, a fxed exchange rae s no longer opmal. 7 The paper proceeds as follows. We begn by layng ou he model for PCP n Secon 2. In Secon 3, we examne opmal moneary polcy n he closed economy o develop nuon, and hen move o he open economy where we solve for opmal moneary polcy under PCP, frs n complee markes and hen n fnancal auarky. Fnally, we lay ou he model for LCP n Secon 4, and solve for opmal moneary polcy n complee markes and fnancal auarky. Secon 5 concludes. 2 The Model We consder a connuum of small open economes represened by he un nerval, as popularzed n he leraure by Gal and Monacell (25). 8 Each economy consss of a represenave household and a represenave frm. All counres are dencal ex-ane: hey have he same preferences, echnology, and prce-seng. Ex-pos, economes wll dffer dependng on he realzaon of her echnology shock. Households are mmoble across counres, however goods can move freely across borders. Each economy produces one fnal good, over whch exercses a degree of monopoly power. Ths s crucally mporan: counres are able o manpulae her erms of rade even hough hey are measure zero. However, because counres are small here wll be no sraegc neracon. As explaned n he nroducon, he expendure swchng effec and he ably o nfluence foregn ncome are boh absen when economes are small. Polcymakers wll hus charge a consan erms of rade markup on her expors. We use one-perod-n-advance prce seng o nroduce nomnal rgdes. 9 Monopolsc frms se nex-perod s nomnal prces, n erms of domesc currency, pror o nex-perod s producon and consumpon decsons. These frms wll charge a consan markup n he flexprce equlbrum, ulzng her monopoly power a he frm level. Gven hs prese prce, frms supply as much oupu as demanded by households. 7 Unless labor eners he uly funcon as a lnear erm, as n Devereux and Engel (23), n whch case a fxed exchange rae s opmal. In our model, labor eners he uly funcon as a non-lnear erm bu we sll fnd ha a fxed exchange rae s opmal under LCP. 8 A smlar verson of hs model appears n Dmrev and Hoddenbagh (213) where we employ wage rgdy nsead of prce rgdy and sudy he opmal desgn of a fscal unon whn a currency unon. 9 Assumng rgd wages or prces has no mpac on he resuls: hey yeld dencal polcy mplcaons. 5

7 We lay ou a general framework below, and hen hone n on wo specfc cases: complee markes and fnancal auarky. To avod addonal noaon, we gnore me subndces unless absoluely necessary. When me subndces are absen, we are mplcly referrng o perod. Households expeced uly In each economy [, 1], here s a represenave household wh lfeme E 1 k= β k ( C 1 σ +k 1 σ ) χn +k 1 + ϕ where β < 1 s he household dscoun facor, C s he consumpon baske or ndex, and N s household labor effor (hnk of hs as hours worked). Households face a general budge consran ha ness boh complee markes and fnancal auarky; we wll dscuss he dfferences beween he wo n subsequen secons. For now, s suffcen o smply wre ou he mos general form of he budge consran: C = (1 τ ) ( W P (1) ) N + D + T. (2) The non-dsoronary ax rae on household labor ncome n counry s denoed by τ, whle T s a lump-sum ax rebae o households. Ne axes equal zero n he model, as any amoun of governmen revenue s rebaed lump-sum o households. The consumer prce ndex corresponds o P, whle he nomnal wage s W. D denoes sae-conngen porfolo paymens expressed n real consumpon uns, and can be wren n more deal as: D P = where B j s a sae-conngen paymen n currency j. 1 E j B j dj, (3) E j s he exchange rae n uns of currency per one un of currency j; an ncrease n E j sgnals a deprecaon of currency relave o currency j. When nernaonal asse markes are complee, households perform all cross-border rades n conngen clams n perod, nsurng agans all possble saes n all fuure perods. The ransveraly condon smply saes ha all perod ransacons mus be balanced: paymen for clams ssued mus equal paymen for clams receved. Leavng he deals n he appendx, we use he followng relaonshp as he ransversaly condon for complee markes: E = β C σ D =, (4) 1 Equaon (3) holds n all possble saes n all perods. Deals are provded n Appendx A.1. 6

8 whle n fnancal auarky D =. Inuvely, he ransversaly condon (4) spulaes ha he presen dscouned value of fuure earnngs should be equal o he presen dscouned value of fuure consumpon flows. Under complee markes, consumers choose a sae conngen plan for consumpon, labor supply and porfolo holdngs n perod. Consumpon and Prce Indces Households n each counry consume a baske of mpored goods. Ths consumpon baske s an aggregae of all of he varees produced by dfferen counres. The consumpon baske for a represenave small open economy, whch s common across counres, s defned as follows: [ C = cj ] dj (5) where c j s he consumpon by counry of he fnal good produced by counry j, and s he elascy of subsuon beween domesc and foregn goods (he Armngon elascy). Because here s no home bas n consumpon, counres wll expor all of he oupu of her unque varey, and mpor varees from oher counres o assemble he consumpon baske. Prces are defned as follows: upper case P j denoes he prce n counry (n currency ) of he unque fnal good produced n counry j, whle CP I s he aggregae consumer prce ndex n counry. Gven he above consumpon ndex, he consumer prce ndex wll be: [ CP I = ] 1 P 1 1 j dj. (6) Consumpon by counry of he unque varey produced by counry j s: c j = [ Pj CP I ] C. (7) We assume ha producer currency prcng (PCP) holds, and ha he law of one prce (LOP) holds, so ha he prce of he same good s equal across counres when convered no a common currency. We defne he nomnal blaeral exchange rae beween counres and j, E j, as uns of currency per one un of currency j. LOP requres ha: P j = E j P jj. (8) Gven LOP and dencal preferences across counres, PPP wll also hold for all, j counry 7

9 pars: CP I = E j CP I j, (9) The erms of rade for counry j wll be: T OT j = P Expors from j P Impors o j = P jj CP I j, (1) where T OT j s defned as he home currency prce of expors over he home currency prce of mpors. Snce he domesc consumpon baske s enrely composed of foregn npus, he domesc CPI wll be equal o he home currency prce of mpors. Now we can ake (7), and usng (8) and (9), solve for demand for counry j s unque varey: Y j = c j d = [ Pj CP I ] C d (8)+(9) = [ Pjj CP I j ] C d = T OT j C w. (11) where C w s defned as he average world consumpon across all economes, C w = C d. Producon Each economy consss of a group of nermedae goods producers, h [, 1], who exercse monopoly power over her unque varey, and a perfecly compeve fnal goods exporer who aggregaes he nermedaes n CES fashon no a fnal expor good. For smplcy, we assume ha nermedaes are non-radable. Thus, each counry wll bundle s nermedaes no one fnal expor good. 11 Fgure 1 below llusraes he bundlng of nermedaes no a fnal expor good, and he bundlng of hose expors no he household consumpon baske. We dfferenae beween he markup on nermedae domesc goods (µ = he markup on he fnal expor good (µ = o manpulae her erms of rade. Fgure 1 ε ) and ε 1 ), whch s mposed only f polcymakers decde 11 We assume non-radable nermedaes wh a fnal radable consumpon good ha aggregaes hose nermedaes for smplcy. In Gal and Monacell s (25, 28) seup, nermedae goods are radable, such ha every counry s mpor consumpon baske s made up of an nfne number of varees mpored from an nfne number of counres. Ths requres negrang over wo connuums. Whle s sraghforward for us o manan her seup, we prefer he racable alernave: a fnal goods exporer bundles he domescally produced nermedaes for expor. In hs way, each counry produces only one unque varey, and we only need negrae over one connuum. Ths assumpon does no change he resuls n any way. In boh cases he household consumpon baske n each counry s made up of mpored goods from all counres, whch are hemselves made up of nermedaes produced domescally. 8

10 Producon of nermedaes requres echnology, whch s common across frms whn a counry, and labor N (h), whch s unque o each frm. We do no need o assume a specfc dsrbuon for echnology, a luxury afforded by our closed form soluon. We do assume ha echnology s ndependen across me and across counres, bu s dencal across frms whn he same counry. Gven hs, he producon funcon of a represenave nermedae goods frm h n counry wll be: y (h) = n (h). (12) Because nermedae goods frms produce dfferenaed vares, hey have monopoly power over her oupu, whch leads o a markup n nermedae goods. Perfecly compeve fnal goods exporers aggregae he nermedae npu of each frm, so ha producon of he represenave fnal exporer n a specfc counry s: [ Y = ] ε y (h) ε 1 ε 1 ε dh, (13) where ε s he elascy of subsuon beween dfferen nermedaes, and µ = ε ε 1 s he markup. The prce of he fnal good n counry, P, wll be a funcon of he nomnal prce for nermedae goods, p (h): [ P = ] 1 p (h) 1 ε 1 ε dh. Cos mnmzaon by he perfecly compeve fnal goods exporer leads o he followng demand for nermedaes: [ ] ε p (h) y (h) = Y. (14) P In summary, monopoly power may be exercsed a he frm and he counry level: a he frm level because of dfferenaed domesc varees, and a he counry level because each economy produces a unque varey for expor. Polcymakers can use hs counry-level monopoly power o manpulae her erms of rade. Bu agan, because economes are small and canno affec world ncome or nduce expendure swchng abroad, polcymakers wll choose a consan erms of rade markup. Households and fnal goods exporers have no monopoly power and are perfecly compeve I s enrely plausble o shf he counry-level monopoly power from he polcymaker o he fnal goods exporer. All resuls wll be exacly he same: erms of rade markups wll be consan jus as n a model of monopolsc compeon. As such, prce sably wll reman opmal under PCP. 9

11 Inermedae goods frms wll prce her unque good one-perod-n-advance accordng o he followng condon, whch resuls from prof maxmzaon: E 1 C σ W y (h) CP I p (h) = µ C σ. (15) y E (h) 1 CP I Households maxmze uly (1) subjec o her budge consran (2). The FOC wh respec o labor wll gve he followng household labor supply condon: ( ) W χ = N ϕ CP I 1 τ Cσ. (16) Frms are dencal, so ha n equlbrum p (h) = P and y (h) = n (h) = N = Y. Usng he labor demand condon (15) and he labor supply condon (16), and he fac ha prces are prese a me 1, he labor marke clearng condon wll be: ( ) χµ E 1 N 1 = 1 τ E 1 C σ. (17) Y T OT Ths s he general labor marke clearng condon; holds n he closed economy and n he open economy for PCP. Under PCP, he demand for he unque varey (11) wll gve he followng labor marke clearng condon: ( ) χµ 1 = 1 τ E 1 E 1 N C σ Y C 1 w. (18) Takng he expecaons operaor ou of (18) wll gve he flexble prce equlbrum for PCP. We now urn our aenon o he dfference beween complee markes and fnancal auarky. 2.1 Complee Markes In hs secon, we assume ha agens n each economy rade a full se of domesc and foregn sae-conngen asses. Households n all counres wll maxmze her uly (1), choosng consumpon, lesure, money holdngs, and a complee se of sae-conngen nomnal bonds, subjec o (2). Rsk-Sharng Complee markes and PPP mply he followng rsk-sharng condon: C σ C σ +1 = C σ j C σ j+1, j (19) 1

12 whch saes ha he rao of he margnal uly of consumpon a me and + 1 mus be equal across all counres. Imporanly, hs condon does no mply ha consumpon s equal across counres. Consumpon n counry wll depend on s nal asse poson, moneary polcy, he dsrbuon of counry-specfc shocks, he covarance of global and local shocks, and oher facors. When he cenral bank n economy changes s polcy, he consumpon allocaon n counry may change as well. For example, moneary polcy can affec he covarance beween home producon and world consumpon, and hs covarance wll nfluence he level of household consumpon even under complee markes. The rsk-sharng condon (19) and he ransversaly condon (4) are boh robus o changes n moneary polcy. If we combne he wo, he resulng goods marke clearng condon wll also be robus o changes n moneary polcy. We solve for hs condon n he appendx, and smply sae he resul below. Lemma 1 When (4),(17), and (19) hold, consumpon n counry can be expressed as a funcon of world consumpon. C = E [ ] 1 =1 β Y Cw σ T OT E 1 =1 β Cw 1 σ C w. (2) Usng he fac ha s ndependen across me and and across counres, and prces are prese, (2) s equvalen o C = E 1 Y T OT = C 1 we 1 Y. (21) 2.2 Fnancal Auarky The aggregae resource consran under fnancal auarky specfes ha he nomnal value of oupu n he home counry (expors) mus equal he nomnal of consumpon n he home counry (mpors). Tha s, rade n goods mus be balanced. In a model wh cross-border lendng, bonds would also show up n hs condon, bu n fnancal auarky, hey are obvously absen. The prmary deparure from complee markes les n he household and economy-wde budge consrans. P Y Expors = CP I C Impors Usng he fac ha (11) holds under boh complee markes and fnancal auarky, and subsung hs no (22), one can show ha demand for counry s good n fnancal auarky wll be C = C 1 wy (22). (23) 11

13 Complee markes and auarky dffer only by goods marke clearng. In complee markes consumpon s equal o expeced domesc oupu expressed n consumpon baskes; n auarky consumpon s equal o realzed domesc oupu expressed n consumpon baskes. 3 Opmal Moneary Polcy Now ha we ve lad ou he model n deal, for boh complee markes and fnancal auarky, we consder opmal polcy for a varey of scenaros. Whou loss of generaly, we assume a cashless lmng economy. 13 Cenral banks wll opmze by choosng labor nsead of money supply or an neres rae rule, bu all hree are equvalen n hs model: money supply wll deermne he neres rae, whch wll n urn deermne labor. One can easly wre down a money supply rule or neres rae rule ha exacly mplemens he allocaon resulng from opmzaon over labor. Bu for he sake of smplcy, we assume he cenral bank opmzes over labor. The mng of he model s descrbed n Fgure 2 below. Before any shocks are realzed, naonal cenral banks declare her polcy for all saes of he world. Wh hs knowledge n hand, households lay ou a sae-conngen plan for consumpon, labor, money and asse holdngs. Afer ha, shocks h he economy. Noe ha under fnancal auarky, no nernaonal asse radng wll occur. -1 Fgure 2 1 2, 3,..., -1 Cenral bank declares moneary polcy Household makes sae-conngen plan Perod 1 shocks are realzed Perod shocks are realzed We begn wh an analyss of opmal moneary polcy n a closed economy verson of our model, and hen proceed o he open economy. In all cases, we consder opmal polcy under commmen. 3.1 Closed Economy The Flexble Prce Allocaon To solve for he flexble prce allocaon n he closed economy, smply ake expecaons ou of (17), se T OT = 1, and use goods marke clearng (Y = C = N ). Ths wll gve us a 13 Bengno and Bengno (23) descrbe a cashless-lmng economy n deal n her appendx, pp

14 sysem of wo equaons n wo unknowns, N and C : 1 = ( ) χµ N 1 τ C 1 σ, C = N. The soluon o hs wo equaon sysem s he flexble prce allocaon for he closed economy: ( ) 1 C = 1 τ χµ. Opmal Moneary Polcy The cenral bank wll choose labor o maxmze he expeced uly of he represenave agen, gven he closed economy labor marke and goods marke clearng consrans. C 1 σ max E 1 N 1 σ χ N (1 + ϕ) (24) s.. ( ) χµ E 1 N 1 = 1 τ E 1 C 1 σ (25) C = N (26) Alhough s sandard pracce o use a welfare-loss funcon for opmal polcy evaluaon, we can smply use he household uly funcon because of our global, closed-form soluon. 14 Proposon 2 In he closed economy under ex ane commmen, he cenral bank wll maxmze (24) subjec o (25) and (26). The soluon o hs problem s: C =. ( ) 1 Prce sably s he opmal polcy, as he cenral bank exacly replcaes he flexble prce allocaon. 1 τ χµ Proof See Appendx A. Opmal Polcy Under A Socal Planner Proposon 3 In he closed economy, he socal planner wll maxmze (24) subjec o (26), ( ) 1 gnorng he labor condon (25). The soluon o hs problem s: C =. The socal planner wll replcae he flexble prce allocaon and remove he monopolsc markup, yeldng he Pareo effcen allocaon. 1 χ Proof See Appendx A. 14 The only reason o use a welfare-loss funcon s f he model n queson mus be approxmaed around a seady sae. Here, no such approxmaon s requred, and hus a welfare-loss funcon s no needed. 13

15 In comparng he soluons descrbed n Proposon 1 and 2, noce ha he socal planner mmcs he flexble prce allocaon whle elmnang he monopolsc markup µ (Proposon 2), whle he markup and labor ax reman when we only consder opmal moneary polcy (Proposon 1). In he case of ex ane commmen n Proposon 1, a fscal auhory may nroduce subsdes o exacly offse he markup and replcae he socal planner equlbrum. I s sraghforwad o show ha τ = 1 µ wll ge rd of he monopolsc dsoron on labor npus and gve he Pareo effcen allocaon. We ve suded opmal polcy n he closed economy, and proved he opmaly of prce sably. The dvne concdence holds, a well known resul n he closed economy. One already sees he lnk beween sable monopolsc markups and a desre o mmc he flexble prce allocaon. We now urn our aenon o he open economy, where we prove ha opmal moneary polcy n closed and small open economes s somorphc n boh complee markes and fnancal auarky. 3.2 Global Socal Planner The global socal planner s a benevolen dcaor ha dsrbues goods across counres n order o maxmze aggregae world uly. Ths scenaro s analogous o perfec cooperaon among he socal planner s of all counres. The global socal planner soluon defnes he Pareo effcen allocaon. Snce he economes n our model are dencal ex-ane, he global socal planner wll maxmze a weghed uly funcon over all counres, [ C 1 σ 1 σ χ N (1 + ϕ) subjec o he consumpon baske and he aggregae resource consran: [ C = Y = N = cj ] d, (27) ] dj, (28) c j dj. (29) Proposon 4 The global socal planner wll maxmze uly weghed over all counres 14

16 (27), subjec o (28) and (29). The soluon o hs opmzaon problem s: EU = C 1 σ ( 1 C = χ ( 1 N = χ ( 1 Y = χ ( w = ( 1 1 σ 1 ), 1 + ϕ ) 1 w, ) 1 (1 σ)() ()() w ) 1 (1 σ)() ()() w ()(), (), ) ()() d. The global socal planner soluon exacly replcaes he flexble prce allocaon whle elmnang monopolsc markups, yeldng he Pareo opmal allocaon. Proof See Appendx A. The global socal planner allocaon above s a radonal benchmark for he evaluaon of dfferen polcy regmes. Because hs s he Pareo effcen allocaon, provdes a naural marker wh whch o compare varous polces under commmen. Noce ha here are no markups n he Pareo effcen allocaon: he benevolen global socal planner has elmnaed he markup on nermedae goods µ, and ressed he empaon o manpulae he erms of rade. In he nex secons we wll look closely a opmal moneary polcy for cenral banks and see wha condons are necessary o replcae he global socal planner allocaon. 3.3 Complee Markes In hs secon we examne he opmal moneary polcy for cooperave and non-cooperave cenral banks n complee markes. The objecve funcons for cooperave and non-cooperave cenral banks are below. C 1 σ max E 1 N 1 σ max N C 1 σ 1 σ χn 1 + ϕ χn 1 + ϕ d (3a) (3b) Non-cooperave cenral banks wll choose he opmal amoun of labor o maxmze her domesc welfare (3a), whle cooperave cenral banks wll choose he opmal amoun of labor n order o maxmze he welfare of all economes (3b). In boh cases, polcymakers wll maxmze he approprae objecve funcon subjec o labor marke clearng (31) and 15

17 goods marke clearng (32) consrans, and producon (33) and aggregae world consumpon (34): ( ) χµ 1 = 1 τ E 1 C = C 1 we 1 Y E 1 N C σ, (31) Y C 1 w, (32) Y = N, (33) ( ) C w = Y. (34) Proposon 5 In complee markes, non-cooperave cenral banks wll maxmze (3a) and cooperave cenral banks wll maxmze (3b), subjec o (31), (32), (33) and (34). The soluon under commmen for boh cooperave and non-cooperave cenral banks n complee markes s: EU = C 1 σ ( 1 τ C = χµ ( 1 τ N = χµ ( 1 τ Y = χµ ( w = ( 1 1 σ 1 τ µ(1 + ϕ) ) 1 w ) 1 (1 σ)() ()() w ) 1 (1 σ)() ()() ()() w ) () ) ()() d The resulng equlbrum allocaon exacly concdes wh he flexble prce allocaon. Mmckng he flexble prce allocaon s hus he domnan sraegy for non-cooperave cenral banks n complee markes, and s he opmal polcy under cooperaon. Proof See Appendx A. There are a few mporan pons o noe from hs exercse. Frs of all, noe ha consumpon n he opmal allocaon s no subjec o dosyncrac echnology shocks. Because we are n complee markes, consumpon s smply a funcon of average world echnology. Second, noe ha cooperave and non-cooperave equlbra are dencal: boh yeld he flexble prce allocaon. In complee markes, we ve shown ha small open economy cenral banks wll mmc he flexble prce allocaon under boh cooperave and non-cooperave regmes, for any elascy 16

18 of subsuon beween home and foregn goods. In addon, we ve demonsraed ha nernaonal moneary cooperaon has no mpac on welfare. Ths s because moneary polcy has no power agans non-conngen dsorons lke markups, and can only address he conngen prce rgdy dsoron. In he nex secon, we conduc he same exercse for fnancal auarky. 3.4 Fnancal Auarky In fnancal auarky, he objecve funcons for cooperave and non-cooperave cenral banks wll be dencal o hose n complee markes. The only dfference n he opmzaon problem wll be n he goods marke consran. In complee markes home consumpon s a funcon of expeced oupu (32), whle n auarky home consumpon s a funcon of acual oupu (36). ( ) χµ 1 = 1 τ E 1 C = C 1 w,y E 1 N 1+η C σ Y C 1 w, (35) (36) Y = N (37) ( ) C w = Y. (38) Proposon 6 In fnancal auarky, non-cooperave cenral banks wll maxmze (3a) and cooperave cenral banks wll maxmze (3b), subjec o (35), (36), (37) and (38). The soluon under commmen for boh cooperave and non-cooperave cenral banks n fnancal auarky s: ( 1 EU = C 1 σ 1 τ µ(1 + ϕ) ( ) 1 ( 1 τ C = w χµ ( ) 1 ( 1 τ N = w χµ ( ) 1 ( 1 τ Y = w χµ ( ()() 1 σ+() w = ), ) 1 σ+(ϕ+σ), ) 1 σ+() ()() d. ) 1 σ 1 σ+(ϕ+σ), ) 1 σ 1 σ+(ϕ+σ), The resulng equlbrum allocaon exacly concdes wh he flexble prce allocaon. Mmckng he flexble prce allocaon s hus he domnan sraegy for non-cooperave cenral banks n fnancal auarky, and s he opmal polcy under cooperaon. Proof See Appendx A. 17

19 Noe ha n fnancal auarky, consumpon s a funcon of dosyncrac echnology, reflecng he lack of nernaonal rsk-sharng. As n complee markes, he opmal polcy for cooperave and non-cooperave cenral banks s o mmc he flexble prce equlbrum. We hus demonsrae he somorphsm beween opmal moneary polcy n closed and small open economes for boh complee markes and fnancal auarky. As we ve saed before, he key o hs somorphsm s he absence of sraegc neracon when economes are small. Unable o nfluence foregn consumpon or foregn ncome, polcymakers n small open economes charge a consan erms of rade markup on her expors. Combned wh monopolsc frms chargng a consan markup, he opmal polcy response s o remove prce rgdes and reurn o he flexble prce allocaon. Smlarly, when elascy s unary and economes are large, as n Clarda, Gal and Gerler (22) and Obsfeld and Rogoff (22), sraegc neracons beween counres are nullfed by a lack of expendure swchng and complee rsk-sharng va erms of rade movemens, and prce sably s opmal. 4 Local Currency Prcng We now urn our aenon o he case of LCP. Modelng LCP n a connuum of small open economes s dffcul because he law of one prce and purchasng power pary no longer hold. In wo economy models, keepng rack of exchange rae polcy s rval, bu n he connuum hs becomes challengng. Alhough we wll gloss over many of he mehodologcal nuances necessary o deal wh LCP n he connuum, all modelng deals can be found n Appendx B. To smplfy expressons, we assume log uly (σ = 1), bu allow ϕ o vary as before. Under LCP, frms prce her expor good one-perod-n-advance n he currency of he mporng counry. As such, here s zero exchange rae pass-hrough no mpor prces. Each counry s consumer prce ndex wll hus be fxed one-perod-n-advance, as exchange rae movemens wll have no mpac on mpor prces. When frms prce n hs way, he labor marke and goods marke clearng consrans wll dffer from hose under PCP. In complee markes and fnancal auarky, goods and labor marke clearng condons for LCP are: ( ) χµ 1 = 1 τ C = E 1 N E 1 C 1, (39) Y T OT P CP I E Y. (4) To keep rack of exchange rae polcy n he connuum, we nroduce he concep of a numerare currency, whch we assume s he currency of counry. Thus, E s he exchange rae beween counry and he numerare counry, and P s he mpor prce of he numerare counry s good n currency. 18

20 Proposon 7 Under LCP, non-cooperave cenral banks wll maxmze (3a) subjec o (39) and (4). A fxed exchange rae wll be he Nash equlbrum polcy for a non-cooperave cenral bank n boh complee markes and fnancal auarky under LCP. Proof See Appendx B. E 1 C σ χµ E 1 N Y E = E 1 C j, (41) 1 τ E 1 E j C j C = E C j E j dj. (42) Proposon 8 Under LCP, cooperave cenral banks wll maxmze (3b) subjec o (41) and (42). The opmal polcy for cooperave cenral banks n boh complee markes and fnancal auarky wll be a fxed exchange rae. Proof See Appendx B. Prce sably s no longer opmal under LCP. Insead, cenral banks should fx he exchange rae. Why s a fxed exchange rae opmal under LCP? As we dscussed n he nroducon, LCP nsulaes household consumpon baskes n each counry from exchange rae flucuaons, guaraneeng a consan proporon of world ncome spen on each unque mpor good because exchange rae flucuaons do no pass-hrough no mpored goods prces. Polcymakers n small open economes are also unable o nfluence world ncome. Therefore domesc oupu and labor are unaffeced by moneary polcy. Sablzng consumpon becomes he cenral bank s goal, and he polcy mechansm o acheve hs goal s a fxed exchange rae. Wh a fxed exchange rae, labor flucuaes wh producvy shocks bu consumpon remans consan. In conras, when economes are large, moneary polcy can nfluence domesc labor under LCP. In such cases, a fxed exchange rae s no longer opmal as shown by Corse, Dedola and Leduc (21). The reasons for hs are oulned n he nroducon. 5 Concluson We derve he frs closed-form soluon for an open economy model ha does no resrc subsuably beween home and foregn goods o one. Dfferen from he sandard modelng framework n he leraure, we assume a connuum of small open economes neracng n general equlbrum, raher han wo large open economes. The racably of our framework requres smplfyng assumpons along oher dmensons. Prces are se one perod n advance, and each counry expors all of s producon and mpors varees from all oher counres o aggregae no a fnal consumpon baske. Usng hs seup, we answer he followng queson: 19

21 Wha are he heorecal condons under whch s opmal for a cenral bank o mmc he flexble prce allocaon? We prove ha for PCP, small open economy cenral banks should mmc he flexble prce allocaon for any elascy of subsuon beween home and foregn goods, n cooperave and non-cooperave equlbra, n boh complee markes and fnancal auarky. Polcymakers should focus solely on elmnang domesc prce rgdes and gnore exernal facors such as exchange rae movemens, even n he presence of monopolsc dsorons n goods markes and erms of rade dsorons whch are no nernalzed by domesc producers. Under LCP, we show ha small open economy cenral banks should fx her exchange rae. Our resuls conras wh hose n he leraure for large open economes, where s no opmal o mmc he flexble prce allocaon under PCP or o fx he exchange rae under LCP when subsuably dffers from one. We also calculae he welfare gans from cooperaon among naonal moneary auhores. We show ha for any elascy n PCP and LCP, n boh complee markes and fnancal auarky, nernaonal moneary cooperaon has no mpac on welfare. Prce sably s he domnan sraegy for cenral banks under PCP, and s hus a Nash Equlbrum n boh noncooperave and cooperave scenaros, whle a fxed exchange rae s a Nash Equlbrum under LCP n boh non-cooperave and cooperave scenaros. As a resul, no benefs arse from cross-counry moneary polcy cooperaon. Our goal n hs paper has been o provde a unfyng framework for he analyss of opmal moneary polcy n small open economes. We focus on he classc benchmark cases, PCP and LCP n boh complee markes and fnancal auarky, o ry and gan nuon and undersandng abou wha dfferenaes polcymakng n small and large open economes, parcularly when rsk sharng s no provded by erms of rade movemens. 2

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24 [26] Alan Suherland. Inernaonal moneary polcy coordnaon and fnancal marke negraon. CEPR Dscusson Papers 4251, C.E.P.R. Dscusson Papers, February 24. [27] Mchael Woodford. Moneary polcy n a world whou money. Inernaonal Fnance, 3(2):229 6, July 2. 23

25 A Appendx A.1 Rsk-sharng The household n counry wll maxmze lfeme uly (1), subjec o he followng budge consran and ransversaly condon: C (s )P (s ) = W (s )N (s ) + =1 s E j (s )B j (s )dj, q j (s )B j (s )dj =. (A.1) (A.2) B j (s ) denoes he sae-conngen bond ha pays n currency j n sae s ; q j (s ) s he prce of ha bond n perod (when all radng occurs), q j (s ) s arbrary up o a consan. Household n perod cares abou relave prce of clams across saes and currences. The ransveraly condon spulaes ha all perod ransacons mus be balanced: paymen for clams ssued mus equal paymen for clams receved. The household Lagrangan s: L = [ β P r(s ) U (C(s )) V (N(s )) + λ (s ) W (s )N (s ) + P (s ) =1 λ =1 s q j (s )B j (s )dj, Now ake he FOC wh respec o sae conngen bonds B j (s ): L B j (s ) = λ q j (s ) + β λ (s )P r(s )E j (s ) P (s ) whch gves prce of he sae-conngen bond, The analogous FOC for counry j, E j B j (s )(s )dj C (s )P (s ) (A.3) =, (A.4) q j (s ) = β λ (s )P r(s )E j (s ). (A.5) λ P (s ) L j B jj (s ) = wll yeld: q j (s ) = β λ j(s )P r(s )E jj (s ). (A.6) λ j P j (s ) Usng E jj (s ) = 1 and seng (A.5) equal o (A.6), we ge he rsk-sharng condon ] λ (s ) λ j (s ) = λ λ j P (s ) P j (s )E j (s ). (A.7) 24

26 ( σ P When PPP holds, (s ) = 1, and he rsk-sharng condon smplfes o λ (s ) = C (s ) P j (s )E j (s ) λ j (s ) C j (s )) = λ λ j. When he consumpon rao s consan across counres, C = A C w. In order o solve for A, we subsue (A.5) no he ransversaly condon. =1 s q j (s )B j (s )dj = = (A.1) = = =1 s β λ (s )P r(s )E j (s ) B j (s )dj λ P (s ) β λ (s )P r(s ) λ =1 s P (s ) β λ (s )P r(s ) λ =1 s P (s ) ( E j (s )B j (s )dj P (s )C (s ) W (s )N (s ) ) We subsue C (s ) = A C w (s ) no he above equaon, and solve for A. A = = = = =1 s β W (s )N(s ) P (s ) λ (s )P r(s ) =1 s β C w (s )λ (s )P r(s ) =1 β W E N 1 P λ (s ) =1 β E 1 C w λ (s ) =1 β E 1 Y C 1 wλ (s ) =1 β E 1 C w λ (s ) =1 β E 1 Y C 1 wcw σ =1 β E 1 Cw C σ w where we used λ (s ) = A σ Cw σ (s ). Ths gves us he defnon of complee markes from he ex, equaon (2). A.2 Closed Economy Proposon 2 In he closed economy under ex ane commmen, he cenral bank wll maxmze (24) subjec o (25) and (26). The soluon o hs problem s: C =. ( ) 1 Prce sably s he opmal polcy, as he cenral bank exacly replcaes he flexble prce allocaon. Proof: The flexble prce allocaon s obaned from solvng hs wo equaon sysem n 1 τ χµ 25

27 wo unknowns (C, N): The soluon s C = ( ) 1 1 τ µχ 1 = µχ 1 τ C = N. N C 1 σ,. Now, le s reformulae he cenral bank s problem by subsung he labor marke clearng condon (25) and he goods marke clearng condon (26) no he objecve funcon. s.. C 1 σ max E 1 C 1 = µχ E 1 1 τ 1 ϕ χc 1 σ (1 + ϕ) C E 1 C 1 σ () The Lagrangan for hs consraned opmzaon problem s C 1 σ L = E 1 1 ϕ χc 1 σ (1 + ϕ). ( + λ E 1 C 1 σ µχ 1 τ E 1 The frs order condon wh respec o consumpon s 15 C ) (). (A.8) ( L = (1 + λ(1 σ)) C σ χ 1 + λµ ) C ϕ 1 ϕ =. (A.9) C 1 τ whch s equvalen o C 1 σ = χ ( ) 1 + λµ 1 τ N =. (A.1) 1 + λ(1 σ) Gven ha λ s a consan and no a varable, he frs order condon and he budge consran are sasfed only under he flexble prce equlbrum. Thus, he cenral bank s opmal polcy s o mmc he flexble prce equlbrum n he closed economy. Proposon 3 In he closed economy, he socal planner wll maxmze (24) subjec o (26), ( ) 1 gnorng he labor condon (25). The soluon o hs problem s: C =. The socal planner wll replcae he flexble prce allocaon and remove he monopolsc markup, yeldng he Pareo effcen allocaon. 1 χ Proof: Inser he aggregae goods marke clearng consran (26) drecly no he objecve 15 One can easly carry ou he same exercse by opmzng wh respec o labor. 26

28 funcon, replacng N +1, and maxmze hs objecve funcon. max C +1 [ C 1 σ +1 1 σ χ 1 + ϕ ( The soluon o hs opmzaon problem s C = A.3 Global Socal Planner ( C+1 1 χ +1 ) 1 ) ]. Proposon 4 The global socal planner wll maxmze uly weghed over all counres (27), subjec o (28) and (29). The soluon o hs opmzaon problem s: EU = C 1 σ ( 1 C = χ ( 1 N = χ ( 1 Y = χ ( w = ( 1 1 σ 1 ), 1 + ϕ ) 1 w, ) 1 (1 σ)() ()() w ) 1 (1 σ)() ()() w ()(), (), ) ()() d. The global socal planner soluon exacly replcaes he flexble prce allocaon whle elmnang monopolsc markups, yeldng he Pareo opmal allocaon. Proof: If we subsue (28) and (29) drecly no he objecve funcon (27), hen we can reformulae he problem as follows: max c j [ c j dj 1 σ ] (1 σ) [ χ 1 c ] jdj 1 + ϕ d. (A.11) Rearrangng, we have 1 max c j 1 σ [ cj ] (1 σ) dj d χ 1 + ϕ [ ] 1 c jdj d. (A.12) 27

29 The FOC wh respec o c j s [ = cj dj [ ] ] ϕ (1 σ) 1 1 c c 1 jdj j χ j. (A.13) Ths s equvalen o = ( [ 1 σ = C cj dj ] ) 1 σ 1 σ =C c 1 j χ N ϕ j j, c 1 j [ χ c jdj j ] ϕ =N ϕ j 1 j, and solvng for c j we have: c j = j C1 σ χ N ϕ. (A.14) j The consumpon baske n counry (C ) can hen be expressed as: ( C = = cj ) dj, [ ] j C1 σ χ N ϕ j dj, = ( ) ( 1 1 σ χ [ j N ϕ j ] () dj) 1 σ(). (A.15) So C does no depend on s own echnology. Now, le s solve for labor (N ) and oupu (Y ). N = Y = = c jdj from (29) [ ] dj = χ N ϕ C1 σ j χ N ϕ from goods marke clearng from (A.14) C 1 σ j dj (A.16) 28

30 From (A.15), we know ha C = C j = C for all, j. So we can ake C j ousde of he negral n (A.16) and solve for N : N = C 1 σ j χ N ϕ N = ( C 1 σ χ ) 1. (A.17) Smlarly, oupu wll be: Y = ( () χ C 1 σ ) 1. (A.18) Subsue (A.17) and (A.18) back no he defnon of he consumpon baske (A.15), and solve for he consumpon baske C n each counry, whch wll be dencal: C ()σ C = ( ) 1 1 σ χ = ( 1 χ C = C = ) 1 σ ( j C 1 σ χ ( ) ()ϕ j χ ( ) 1 ( 1 χ ()() j ) 1 j dj ()ϕ 1 σ(), j dj 1 σ() ) ()() dj. (A.19) Solve for labor and oupu by subsung (A.19) no (A.17) and (A.18) respecvely: ( 1 N = χ ( 1 Y = χ ) 1 ( ) 1 ( ()() j ()() j ) 1 σ ()() dj ) 1 σ ()() dj, (A.2) (). (A.21) Ths s he Pareo effcen allocaon. When, he flexble prce allocaon and he global socal planner allocaon become dencal. Consumpon s dencal o he frs order beween socal planner and flexble prce allocaon. However, s no rue for labor. A.4 Complee Markes and Fnancal Auarky, Proposon 5 In complee markes, non-cooperave cenral banks wll maxmze (3a) and cooperave cenral banks wll maxmze (3b), subjec o (31), (32), (33) and (34). The soluon under commmen for boh cooperave and non-cooperave cenral banks n complee markes 29