Monetary Policy and Welfare in a Small Open Economy

Transcription

1 Moneary Policy and Welfare in a Small Open Economy Bianca De Paoli yz Absrac This paper analyzes opimal moneary policy in a small open economy feauring monopolisic compeiion and nominal rigidiies. I shows ha he uiliy-based loss funcion for his economy can be wrien as a quadraic expression of domesic inflaion, oupu gap and real exchange rae. The presence of an inernal monopolisic disorion and a erms of rade exernaliy drives opimal policy away from domesic inflaion argeing and affecs he opimal level of exchange rae volailiy. When domesic and foreign goods are close subsiues for each oher, he opimal policy rule implies lower real exchange rae volailiy han a domesic inflaion argeing regime. The reverse is rue when he elasiciy of subsiuion beween goods is low. JEL Classificaion: F4, E52, E58 and E6. Keywords: Welfare, Loss Funcion, Terms of Trade Exernaliy, Opimal Moneary Policy, Small Open Economy. Firs Draf: Ocober 2003 Curren Version: Sepember 22, 2008 I would like o hank Gianluca Benigno, Anna Lipinska and Giancarlo Corsei (he edior) for many helpful suggesions on he paper. I also hank wo anonymous referees, Kosuke Aoki, Jordi Galí, Nobu Kiyoaki, and Jens Søndergaard for valuable commens. This work was largely carried ou before he auhor joined he Bank of England, and he views expressed in his paper are hose of he auhor and do no necessarily reflec he views of his insiuion. y Bank of England, London School of Economics and CEP, London, Unied Kingdom. z Correspondence o: Bianca De Paoli, Bank of England, Threadneedle Sree, London EC2R 8AH, Unied Kingdom. bianca.depaoli@bankofengland.co.uk -

2 Inroducion A he hear of he policy debae in open economies lays he quesion of wheher cenral banks should reac solely o flucuaions in domesic oupu and inflaion, or wheher here is a role for sabilizing inernaional relaive prices. While some academic sudies show ha policy objecives in an open economy can be isomorphic o he ones in a closed economy (e.g. Clarida, Gali and Gerler (200) and Gali and Monacelli (2005)), oher conribuions have sressed ha welfare in open economies may be affeced by a erms of rade exernaliy (e.g. Obsfeld and Rogoff (998) and Corsei and Peseni (200)). The curren paper focuses on he implicaions of his exernaliy for opimal moneary policy in a small open economy seing. We derive a welfare-based loss funcion for his economy, compue he opimal moneary policy plan, and rank he performance of simple policy rules. Our analysis illusraes how he presence of such exernaliy and an inernal monopolisic disorion drives opimal policy away from producer price inflaion argeing and affecs he opimal level of exchange rae volailiy. As documened in he rade heory lieraure (see, for example, Corden (974)), a erms of rade exernaliy arises in an open economy seing when he elasiciy of demand for expor (or supply of impors) is less han infinie. This fac implies ha a social planner may wish o exploi domesic monopoly power by imposing an expor ax (or an impor ariff) and, hereby, improve is erms of rade. Such exernaliy has been exensively discussed in he inernaional finance lieraure. Corsei and Peseni (200) analyse he welfare implicaions of changes in money supply in a seing characerized by an inernal disorion - relaed o monopolisic supply in he domesic marke - and he aforemenioned exernal disorion - relaed o he counry s monopoly power in rade. In closed economies, he inernal disorion implies ha a moneary expansion can increase oupu oward is efficien level. Bu in open economies his expansion also reduces domesic consumers purchasing power inernaionally. Because of he laer effec, expansionary policies may reduce welfare. As emphasized in Tille (200), he overall impac of changes in he money supply depends on he relaive size of hese wo disorions. In a sochasic wo-counry environmen, he erms of rade exernaliy has also been shown o play a crucial role in he welfare and policy evaluaion (see, for example, Corsei and Peseni (2005) and Benigno and Benigno (2003)). Our analysis focuses on he implicaion of he erms of rade exernaliy for moneary policy in a small open economy seing, which also feaures he inernal monopolisic disorion and nominal rigidiies. The small open economy model is derived as a limiing case of a wo-counry dynamic sochasic general equilibrium framework. The model assumes no rade or financial fricions, ha is, he law of one price holds and asse markes are complee. In his seing, we follow he linear-quadraic approach developed by Suherland (2002) and Benigno and Woodford (2006), and derive a uiliy-based loss funcion for he cenral bank. We show ha his loss funcion can be approximaed by a quadraic expression in domesic inflaion, oupu gap and real exchange rae. Moreover, we obain he opimal response o exogenous shocks in he form of a argeing rule. Our framework encompasses, as special cases, a closed economy seing wih a general seadysae degree of monopolisic compeiion (as in Benigno and Woodford (2005)) and a small open Early conribuions emphasizing inernaional welfare spillovers of moneary policy shocks also include Obsfeld and Rogoff (995) and (998).

3 economy seing wih uniary elasiciy of ineremporal and inraemporal subsiuion (as in Galí and Monacelli (2005)). 2 As a resul of he exernal disorion, when domesic and foreign goods are close subsiues, an improvemen in he erms of rade can increase welfare by inducing agens o consume more impored goods. These consumers are beer off since hey can reduce heir labor effor wihou a corresponding fall in heir consumpion levels. Using a second order approximaion of he model, we show ha a less volaile real exchange rae is associaed wih a more appreciaed exchange rae on average, or an improved erms of rade. Therefore, when domesic and foreign goods close subsiues in he uiliy and a erms of rade improvemen can enhance welfare, he volailiy of he real exchange rae is lower under he opimal rule han under a domesic inflaion argeing regime. For sufficienly large values of he elasiciy of inraemporal subsiuion, a policy ha arges he exchange rae can ouperform (ha is, lead o higher welfare) one ha sabilizes domesic prices. The conclusions are, neverheless, revered if he subsiuabiliy beween domesic and foreign goods is low and he expendiure swiching abiliy of erms of rade movemens is reduced. We should noe ha in our seing, moneary policy is affeced by an exernal disorion because he small open economy reains some marke power over is erms of rade. Oher conribuions o he lieraure, however, consider a small open economy model in which producers of radable goods are price akers. In such a seing, and in he presence of a non-radable secor in which prices are sicky, Lama and Medina (2007) find ha he goal of moneary policy should be o replicae he flexible price equilibrium. 3 Moreover, our analysis assumes a cashless economy and does no allow for any acive fiscal insrumen o operae on economic disorions. Models ha consider differen policy insrumens, such as labor or consumpion axes, and moneary fricions, driven by ransacion consrains, include Adao, Correia and Teles (2003), Correia, Nicolini and Teles (2003) and Hevia and Nicolini (2006). Our resuls, neverheless, complemen he analysis of he policy implicaions of he erms of rade exernaliy documened in he lieraure. Namely, Corsei and Peseni (2005), Devereux and Engel (2003), and Suherland (2005) have shown ha he presence of incomplee pass-hrough, arising from local currency pricing sraegies of firms, can give rise o an inernaional dimension of moneary policy. In paricular, when prices of goods are sicky in local currencies, foreign firms profis are a funcion of domesic moneary policy; and if he domesic cenral bank ignores his link, impor prices would be oo high relaive o prices of domesically produced goods. 4 Under producer currency pricing, Benigno and Benigno (2003) idenify gains from policy cooperaion 2 The framework considered in Galí and Monacelli (2005) only imposes uniary elasiciy of ineremporal and inraemporal subsiuion when deriving he objecive funcion. 3 Similar conclusions are suggesed in Aoki (200), in which a wo-secor closed economy model is applied o a small open economy ha produces differeniaed goods whose prices are sicky and impors goods whose prices are flexible. We should noe ha, given ha he model considered in Lama and Medina (2007) also feaures segmened markes, he aforemenioned resul holds when absracing from policy incenives o enhance risk sharing. 4 Devereux and Engel (2003) sugges ha opimal policy in his environmen may consis of a fixed exchange rae regime. Bu, as Corsei (2007) and Duare and Obsfeld (2005) poin ou, he fac ha he exchange rae does no move under he opimal policy can be a resul of he (symmeric) srucure of he model and no a produc of efficien sabilizaion. 2

4 across counries, as policymakers acing independenly have an incenive o affec he erms of rade. 5 Furhermore, Tille (2002) finds ha in he presence of secor specific shocks, exchange rae flucuaions ranslae ino misallocaion of resources beween differen firms wihin a secor. As a resul, he moneary auhoriy has an incenive o resric hese flucuaions. In addiion, as demonsraed in Corsei, Dedola and Leduc (2007), sraegic ineracions beween upsream and downsream firms may preven perfec sabilizaion of he domesic price of final goods. These ineracions can induce misalignmen of prices boh across firms and across secors, even when shocks affec downsream firms only. Ye, hese conribuions assume a wo-counry seing. A specific advanage of sudying a small open economy is ha one can race more direcly he macroeconomic implicaions of moneary and exchange rae policy disregarding cross-border sraegic ineracions in policy making. Finally, while our approach is o consider a simple specificaion of an open economy and derive an analyical represenaion of he moneary policy problem, a large se of he lieraure have used numerical echniques o evaluae opimal policy in open economies (see, for example, he works of Kollman (2002), Søndergaard and Cova (2004), and Bergin e. al. (2007)). Alhough hese sudies have he disadvanage of no being able o illusrae analyically he economic mechanism behind heir resuls, hey can be applied o more complex, and possibly more realisic seings. The remainder of he ex is srucured as follows. In Secion 2, we describe he model feaures and presen he small open economy s dynamics. Secion 3 is dedicaed o he derivaion of welfare and he quadraic loss funcion. In Secion 4, we derive he opimal policy plan, and he inefficiencies of he flexible price allocaion are illusraed in Subsecion 4.. In Secion 5, we examine he performance of sandard policy rules in our small open economy. Finally, concluding remarks are presened in Secion 6. 2 The Model The baseline framework consiss of a wo-counry dynamic general equilibrium model wih complee asse markes, monopolisic compeiion in producion and sicky prices. In paricular, we assume ha home price seing follows a Calvo-ype conrac. The model feaures complee passhrough, as prices are se in he producer s currency. Moreover, we absrac from moneary fricions by considering a cashless economy. 6 In he seup considered, even hough he law of one price holds, deviaions from purchasing power pariy arise from he exisence of home bias in consumpion. This bias depends on he degree of openness and he relaive size of he economy. The specificaion allows us o characerize he small open economy by aking he limi of he home economy size o zero. Prior o applying he limi, we derive he opimal equilibrium condiions for he wo-counry model. Afer he limi is aken, he wo counries, Home and Foreign, represen he small open economy and he res of he world, respecively. In his secion we presen a summary of he model s equilibrium condiions in log-linear form, while Appendix A conains he full derivaion of he model. As illusraed in Table, he small 5 A similar analysis is discussed in Pappa (2004). 6 For deails on his specificaion see Woodford (2003, chaper 2). 3

5 open economy s log-linearized equilibrium dynamics can be summarized by an aggregae supply (AS), an aggregae demand (AD) and a risk sharing equaion (RS). [Inser Table abou here] [Inser Table 2 abou here] The variables ^C and ^C denoe domesic and foreign consumpion, ^Y denoes domesic oupu, ^Q denoes he real exchange rae and ^ H represens domesic (producer price) inflaion. The sochasic environmen is characerized by hree srucural shocks. Governmen spending, ^g, is assumed o be compleely financed by lump-sum axes and is reaed as an exogenous shock. Monopolisic compeiion in producion leads o a wedge beween marginal uiliy of consumpion and marginal disuiliy of producion. We allow for flucuaions in his wedge by assuming a ime-varying proporional ax. We refer o hese flucuaions as mark-up shocks, ^. Produciviy shocks are represened by ^". The parameers of he model are described in Table 2. 7 Equaion (AS) represens he small open economy s Phillips curve. Noe ha he flexible price allocaion is idenical o he equilibrium allocaion ha would prevail were policymakers o arge domesic inflaion (ha is, he case in which! 0 and, herefore, k! ; is equivalen o he case in which ^ H = 0, 8). Equaion (AD) illusraes how he demand for he small open economy s producs depends on foreign and domesic (privae and public) consumpion. Equaion (RS) is derived from he complee marke assumpion, and represens he opimal risk sharing agreemen beween agens in he small economy and agens in he res of he world. Given domesic exogenous variables, ^", ^g, ^ ; and exernal condiions, ^C, he small open economy sysem of equilibrium condiions is closed by specifying he moneary policy rule. In Secions 4 and 5, we examine differen specificaions for his rule. Apar from analyzing he opimal moneary policy regime in he form of a argeing rule, we evaluae he performance of alernaive sandard policy rules such as an exchange rae peg, and boh consumer price index (CPI) and producer price index (PPI) inflaion argeing. Foreign dynamics are governed by foreign supply and demand condiions (AS and AD ): [Inser Table 3 abou here] The specificaion of he foreign policy rule complees he sysem of equilibrium condiions which deermine he evoluion of foreign inflaion ^, foreign oupu ^Y and foreign consumpion ^C. The economic dynamics of he res of he world are independen of he dynamics of he real exchange rae or any variable in small open economy. Therefore, he policymaker of he small open economy can rea ^C as exogenous. The policy choice of he res of he world may modify he way in which foreign srucural shocks affec ^C, bu i does no influence how he laer affecs he small open economy. 3 Welfare In a microfounded model, welfare and he policymaker s objecive funcion can be precisely derived from households uiliy. Following he linear-quadraic approach developed by Suher- 7 Noe ha, in our framework, he parameer deermining he degree of openness also governs he degree of home bias. In paricular, he level of home bias is given by. See Appendix A for deails. 4

6 land (2002) and Benigno and Woodford (2006), we obain he objecive funcion from a secondorder approximaion of his uiliy (or, equivalenly, a second-order Taylor expansion of Equaion (A.) in Appendix A). Moreover, we eliminae he linear erms in he Taylor expansion, using a second-order approximaion of he model s equilibrium condiions. As a resul, we obain a purely quadraic approximaion for he objecive funcion. 8 Alernaive approaches o welfare evaluaion include he compuaional mehods described in Collard and Juillard (200), Kim e al. (2003) and Schmi-Grohé and Uribe (2004). Such echniques are based on perurbaion mehods and deliver a numerical evaluaion of he opimal policy problem. In conras, he mehodology adoped here delivers an analyical represenaion of he policy problem, which is similar o he one used in he radiional lieraure on moneary policy evaluaion. A second-order Taylor expansion of he uiliy funcion implies he following approximaion for welfare W o = U c CE0 X " ^C ^Y + 2 ( (+) 2 ( ^Y ) ^C 2 (+)^" ) 2 2 k ^H 2 # + :i:p + O(jjjj 3 ); () where is he seady sae level of markup, O(jjjj 3 ) represens erms of order higher han wo, and he expression :i:p sands for erms independen of policy. 9 Using he mehod described above, we can re-wrie he above expression as a quadraic funcion of oupu, real exchange rae and domesic inflaion. Thus, as shown in Appendix B, he loss funcion for he small open economy can be expressed as X L o = U cce0 2 y (^y ) q (^q ) (^ H ) 2 +:i:p + O(jjjj 3 ); (2) We define he welfare-relevan gaps as ^y = ( ^Y ^Y T ) and bq = ( ^Q ^QT ), where ^Y T and ^Q T are policy arges ha depend on he various shocks. 0 The weighs of inflaion, oupu and he real exchange rae gap in welfare losses, ; y ; and q, are funcions of he srucural parameers of he model. Expressions for policy arges and weighs are shown in Appendix B. The welfare losses represened in Equaion (2) are affeced by wo economic disorions which are common o he closed economy framework. These disorions are driven by he presence of price sickiness and monopolisic compeiion in domesic producion. Price sickiness inroduces an incenive o reduce inflaion flucuaions, and monopolisic compeiion inroduces an incenive o reduce seady-sae producion inefficiencies. These wo facors jusify he presence of (^ H ) 2 and (^y ) 2 in Equaion (2). In fac, if we characerize a closed economy (by seing = 0), Equaion 8 As formalized in Judd (996), and discussed in Benigno and Woodford (2006), in order o obain an approximaion of he objecive funcion ha is fully accurae o second-order, he effec of second momens on he mean of he variables should be aken ino accoun. The linear-quadraic approach adoped in his paper incorporaes hese effecs by obaining a purely quadraic approximaion for he policy objecive. 9 These erms depend solely on exogenous shocks and, herefore, are no affeced by he policy choice. 0 The erminology policy arge is ofen used in works which consider ha moneary policy follows argeing rules (see, for example, Svensson (2002)). I refers o variables ha are composies of differen shocks, and, hus, i does no refer o wha one could consider as observable policy arges, such as acual inflaion or exchange raes. 5

7 (2) becomes: X L c o = U cce0 2 c y (^y c ) c (^ H ) 2 + :i:p + O(jjjj 3 ); (3) where he subscrip c denoes a closed economy. The policymaker s problem in a closed economy can be illusraed by he relaive weigh of inflaion wih respec o oupu, = y ; and by he difference beween ^Y T and ^Y F lex (where ^Y F lex represens he flexible price allocaion for oupu). The soluion o hese erms is: and ^Y T;c c c y = = ^" + d^g ( + ) ^Y F lex;c = ^" + ^g ( + ) k( + ) ; (4) d( )( + )^ ( + )( + ) ; (5) ^ ( + ) ; (6) where d = ( + )( + + ( )) T;c F lex;c. As he above expressions show, ^Y 6= ^Y, so a policy of sric inflaion argeing, which mimics he flexible price allocaion, does no close he welfare-relevan oupu gap. Hence, here is a rade-off beween sabilizing inflaion and oupu. As shown in Benigno and Woodford (2003), his rade-off disappears only when he seady-sae level of producion is efficien (i.e. = ) and here are no mark-up flucuaions (i.e. ^ = 0; 8), resuling here in ^Y T = ^Y F lex. In our small open economy, apar from nominal rigidiies and he inernal monopolisic disorion, here is an exernal disorion ha gives rise o a erms of rade exernaliy (as described in Corsei and Peseni (200)). 2 Such disorion comes from he fac ha, because impored goods are no perfec subsiues o goods produced domesically, a social planner in an open economy may wish o exploi a cerain degree of monopoly power. We can illusrae some of he welfare implicaions of his exernal disorion by inspecion of he linear erms in he welfare equaion (). As shown in secion B.6 of he Appendix, hese erms can be wrien as a funcion of E[( ; ) ^Q ]; where ; is decreasing in and increasing in : 3 The larger he degree of inernal monopolisic disorion - given by he mark-up - he larger are he gains from having a more depreciaed real exchange rae on average, as his induces a higher demand for domesic goods and higher oupu. Bu he presence of he erms of rade exernaliy implies ha, when domesic and foreign goods are close subsiues, an appreciaed Benigno and Woodford (2003) consider furher specificaions, such as a non-zero seady sae governmen consumpion, which are no addressed in his paper. 2 Alhough he source of he exernal disorion presen in he curren framework is he same as he one in hese references, he policy analysis is differen. In Corsei and Peseni (200), moneary policy is characerised by unanicipaed movemens in he money supply which can affec he level of producion and he erms of rade (ex-pos). In he curren framework, moneary policy has an effec on he volailiy of economic variables, which, as emphasised in Obsfeld and Rogoff (998) and Henderson and Kim (999), affecs he expeced (or ex-ane) value of variables. 3 Noe ha Gali and Monacelli (2005) eliminae his erm by assuming ha = and = =. Bu in general, using a firs order approximaion of demand equaion and he risk sharing condiion, we can wrie ^C ^Y as a funcion of E[( ; ) ^Q ]. However, he full welfare implicaions of he linear erm can only be accessed when his erm is approximaed o second order. This is derived in secion B.5 of he Appendix. 6

8 real exchange rae (or an improvemen in he erms of rade), on average, can enhance welfare by decreasing he disuiliy of producing a home wihou an equivalen reducion in he uiliy of consumpion. 4 This is because he appreciaion would be associaed wih lower consumpion of domesic goods, while leading o larger consumpion of foreign goods. In his case, he small open economy, as a monopolis over he goods i produces, can gain from resricing he supply and increasing he price of is goods. The larger is he degree of subsiuabiliy beween goods (in paricular, he larger ); he larger is he expendiure swiching effec of changes in erms of rade, and, hence, he larger are he gains from an improvemen in he erms of rade. An appreciaion ceases o be welfare improving when hese elasiciies are small, and he erms of rade canno diver consumpion owards foreign goods. In his case, a more depreciaed real exchange rae on average can be welfare improving. The reason is ha, when domesic and foreign producs are complemens in he uiliy, he marginal uiliy from consuming domesic goods is increasing in he consumpion of foreign goods, and vice versa. An average real depreciaion is associaed wih a an increase in domesic oupu and a higher level of consumpion of domesic goods, which, in urns leads o an increase in he marginal uiliy from impors (and vice versa). 5 As a resul, he increase in overall uiliy of consumpion ouweighs he increase in he disuiliy of labour effor implied by he higher level of domesic producion. A noe on he role of risk sharing The implicaions of he aforemenioned erms of rade exernaliy crucially depend on he fac ha agens in he small open economy have an opimal risk sharing arrangemen wih he res of he world. Such arrangemen prevens agens in he small economy from experiencing negaive income effecs if hey were o reduce heir producion levels. Under financial auarky his would no be he case, as domesic agens borrowing consrains imply a igh link beween domesic consumpion and income. As shown in he secion B.6 of he Appendix, under his specificaion he linear erms in he welfare equaion () can be wrien as a funcion of E[( 0 ; ) ^Q ]; where 0 ; is a decreasing, raher han increasing, funcion of he elasiciy of subsiuion beween home and foreign goods (). Tha is, only when his elasiciy is sufficienly low an appreciaed exchange rae can improve welfare, as oupu would fall lile relaive he movemen in real exchange rae, and he effec of he appreciaion on agens purchasing power would ouweigh he reducion in oupu. 6 4 Opimal Moneary Policy In he previous secion we illusrae how welfare is affeced by he exisence of a erms of rade exernaliy. In he curren secion we analyze how his exernaliy affecs opimal policy. Following our linear-quadraic approach, he opimal moneary policy can be obained by minimizing he 4 We should noe ha in he presen framework, given ha all goods are raded, erms of rade and real exchange raes are direcly relaed. In paricular, from he definiion of he real exchange rae and he price index (see Appendix for deails), we can wrie Q = ( )(P F;=P H;) +. 5 I hank Giancarlo Corsei for suggesing his explanaion. 6 This resul is consisen wih he findings of Suherland (2004) who shows ha gains from cooperaion beween wo counries, which produce subsiue goods, are larger when here is perfec risk sharing across such counries. 7

9 quadraic loss funcion (2) subjec o he linear consrains specified in Table. Alernaively, we can depic hese consrains in erms of he welfare-relevan oupu and exchange rae gaps, [Inser Table 4 abou here] where u and are linear combinaion of he srucural shocks (hese are shown in Appendix B). 7 Equaion (C) shows ha he welfare-relevan gaps, ^y and ^q ; are no closed when prices are perfecly flexible. Tha is, he policy arges do no coincide wih he allocaions ha would prevail if = 0 (and consequenly k! ). Moreover, Equaion (C2) shows ha closing he oupu gap fails o eliminae he real exchange rae gap. We characerize he opimal plan assuming ha policymakers can commi o maximizing he economy s welfare. The policy problem consiss of choosing a pah for f^ H, ^y, ^q g o minimize (2), subjec o he consrains shown in Table 3, and given he iniial condiion ^ 0. In effec, he consrain on ^ 0 imposes ha he firs order condiions o he problem are ime invarian. This mehod follows Woodford s (999) imeless perspecive approach, and hereby ensures ha he policy prescripion does no consiue a ime-inconsisen problem. The mulipliers associaed wih (C) and (C2) are, respecively, ' and ' 2. Thus, he firs order condiions wih respec o ^ H ; ^y and ^q are given by: (' ; ' ; ) = k ^ H ; (7) ' 2; ' ; = Y ^y ; (8) and ( + l)' 2; + ' ; = ( ) q ^q : (9) Combining equaions (7), (8), and (9), we obain he following expression: 0 y^y + 0 q^q + 0 ^ H = 0; (0) where denoes he firs difference operaor, 0 y = ( + l) Y, 0 q = ( ) q and 0 = ( + ( + l))k. The above expression characerizes he small open economy opimal argeing rule and sipulaes how moneary policy should respond o differen shocks (according o he composiion of ^Y T and ^Q T ). 8 So wha are he implicaions of he erms of rade exernaliy for opimal policy? The policy rule described in equaion (0) prescribes responding o movemens in inflaion, oupu and he real exchange rae. Noe ha even if we express Equaion (0) as a funcion of consumer price index (CPI) inflaion insead of producer price index (PPI) inflaion b H, he argeing rule sill includes he erm ^q. So, when following his policy rule, he cenral bank may allow some variabiliy in inflaion in order o respond o cosly movemens in oupu and real exchange raes 7 Noe ha he small open economy represenaion eliminaes he need o formally characerize he open economy policy problem as a Nash equilibrium. For examples of how his is done in a wo-counry world, see Benigno and Benigno (2003) and Pappa (2004). 8 Moneary policy is, herefore, modelled as a argeing rule and no as an insrumen rule (or an ineres rae rule). See Svensson (2002, 2003) and McCallum and Nelson (2004) for a discussion on hese policy specificaions. 8

10 (and, hus, erms of rade). 9 As a maer of fac, he opimal policy rule may imply a smaller or larger volailiy in he real exchange rae relaive o a domesic inflaion argeing regime. As illusraed in Figure, under he parameer specificaion shown in Table 5, he volailiy of real exchange raes under he opimal policy rule is lower (higher) han under a policy of domesic inflaion arge when domesic and foreign goods are subsiues (complemens) o one anoher. [Inser Figure abou here] [Inser Table 5 abou here] In order o undersand his resul i is useful o examine how he mean and variance of he real exchange rae are relaed in our model. In general, by using a second order approximaion of he equilibrium condiions shown in Appendix B, i is possible o express he uncondiional mean of he real exchange rae as a funcion of second momens of he differen macro variables. 20 Moreover, if, for illusraion purposes, we consider he case in which prices are perfecly flexible (i.e. = 0), he uncondiional mean of he real exchange rae can acually be expressed solely as a funcion of he real exchange rae volailiy. 2 Tha is, in his case we can wrie E[ ^Q ] = c q var[ ^Q ] () Figures 2 and 3 illusrae he relaionship beween he mean and variance of he real exchange rae (represened by c q ) for differen values of he elasiciy of subsiuion () and degree of openness (). As he figures show, regardless of he value of and, he higher he level of he exchange rae volailiy, he more depreciaed is he real exchange rae on average (i.e. c q is always posiive). Moreover, he effec of he real exchange rae volailiy on is mean is increasing on he degree of openness () and he elasiciy of subsiuion beween home and foreign-produced goods (). [Inser Figure 2 abou here] [Inser Figure 3 abou here] So, in our framework, a less volaile exchange rae is associaed wih a more appreciaed exchange rae on average. Therefore, when a more appreciaed exchange rae is welfare improving (and as shown in secion 3 his is he when is large) opimal policy prescribes a lower volailiy of he exchange rae relaive o a domesic inflaion argeing regime. In his case, he policy maker only parially sabilizes oupu relaive o is flexible-price level ha is, oupu will be on average below he one ha would prevail if prices were perfecly flexible. This ranslaes ino a fall in he variabiliy of he real exchange rae relaive o he case of producer price inflaion argeing. The conrary is rue when is low. Under his configuraion, he policy maker finds i opimal o over-sabilize oupu relaive o he flexible price allocaion. Tha is, on average, he policy maker engineers a depreciaion in real exchange rae which produces an exernal demand for domesic goods. In his case, he exchange rae is more volaile relaive o he case of domesic price sabiliy. 9 Noe ha, as shown in foonoe 5, he erms of rade and he real exchange rae are direcly relaed in our framework. Bu, as emphasised in Corsei, Dedola and Leduc (2007), in a differen seing a policy ha sabilizes he erms of rade, does no necessarily conains he volailiy of he real exchange rae. 20 For anoher sudy examining he effec of exchange rae uncerainy on he level of exchange raes, see Obsfeld and Rogoff (998). 2 In he case of sicky prices, he real exchange rae mean would be wrien as a funcion of inflaion and real exchange rae volailiy. 9

11 4. Price sabiliy in a small open economy When is domesic price sabiliy opimal in a small open economy? As illusraed by he argeing rule expression (0), in our general specificaion, opimal moneary policy deviaes from pure domesic inflaion argeing, or, equivalenly, he flexible price allocaion is inefficien. In he curren secion, we examine he special case in which his ceases o be he case. And, o undersand he inefficiency of he flexible price allocaion in he general case, we sep back from he linear-quadraic analysis and compare he social planer s non-linear equilibrium condiions wih hose under flexible prices. 22 This analysis allows us o evaluae how he opimaliy of domesic inflaion argeing depends on srucural parameers and shocks. The planner s objecive is o maximize agens uiliy subjec o he small open economy s demand, risk sharing and relaive price equaions (Equaions (A.), (A.7) and (A.6) in he appendix). 23 Moreover, he planner in a small open economy akes exernal condiions as given. Hence, he firs order condiion can be wrien as p e H;U c (C e ) = s e V y (Y e ; " ) ; (2) where he relaive price of domesic goods is denoed by p H; = P H; =P, s e = [ + ( ) (Q e ) + (Qe ) ], and he superscrip e denoes he efficien allocaion. A full characerizaion of he efficien allocaion can be obained by combining he above equaion wih he consrains of he policy problem. On he oher hand, in he decenralized problem, he equilibrium condiion implied by monopolisic compeiion and price sickiness is given by he price seing condiion (Equaion (A.5) in he Appendix). If we assume, however, ha prices are flexible, his condiion becomes p F lex H; U c (C F lex ) = V y (Y F lex ; " ); (3) Comparing condiions (2) and (3), i is clear ha even wih perfecly flexible prices, mark-up shocks and movemens in he real exchange rae generae inefficien flucuaions in he raio of marginal disuiliy of producion and marginal uiliy of consumpion. This comparison presens an alernaive demonsraion of he fac ha, in general, a policy of domesic price sabilizaion ha mimics he flexible price allocaion does no implemen he efficien allocaion. This is only he case under cerain assumpions concerning parameer values, bu i also depends on he sources of shocks hiing he economy. We now illusrae hese special cases. If we impose ha = ; he efficiency condiion (2) and he decenralized flexible price allocaion (3) can be wrien as and ( ) (Y e (Y F lex G ) = " G ) = " (Y e ) ; (4) Y F lex : (5) 22 In a subsequen work, Faia and Monacelli (2008) follow a similar approach and characerize he opimal Ramsey allocaion. Their analysis focuses he role of home bias. 23 Noe ha here he social planner is no consrained by he pricing decisions of firms. When compuing opimal policy his furher consrain has o be included, as done in he Appendix of Benigno and Benigno (2003). 0

12 The above expressions illusrae ha he seady-sae flexible price allocaion is inefficien (i.e. Y flex 6= Y e ) unless = =( ): I also shows ha, he flexible price allocaion and he efficien allocaion move proporionally o one anoher following produciviy shocks. Therefore, if he small open economy is subjec solely o produciviy shocks, a policy ha mimics he flexible price allocaion (ha is, a producer price inflaion argeing regime) is opimal when =. The same resul holds for foreign shocks. 24 In he case of exogenous flucuaions in governmen expendiure, Y flex and Y e do no move proporionally, unless he seady-sae level of oupu is efficien. Finally, mark-up shocks generae disproporional movemens in Y flex and Y e regardless of he seady-sae characerizaion. This resul is consisen wih he findings of Benigno and Woodford (2004) in a closed economy seing. Therefore, he assumpions needed in order o have a producer price inflaion argeing as he opimal plan are: = ; and, in he case of fiscal shocks, ^ = 0; 8; = =( Under his specificaion, he weighs on he loss funcion are: ): and and he arge for oupu is y = ( + ); (6) ( ) q = 0; (7) ( ) = k ; (8) ^Y T = ^Y F lex = ( + ) f" Y; + g g : (9) The relaive weighs specified in equaions (6) and (8) are analogous o hose in he closed economy, and he policy arge coincides wih he flexible price allocaion. When =, inernaional relaive price movemens have income and subsiuion effecs ha offse each oher, and hus, do no have expendiure swiching implicaions. Under his assumpion, and provided ha he seady-sae is efficien (which is guaraneed by assuming ha = = ( )), welfare is no longer affeced by he erms of rade exernaliy or he inernal monopolisic disorion. Thus, since nominal rigidiy is he only relevan economic disorion, domesic price sabiliy is opimal. Moreover, he opimal plan does no respond o exernal shocks. Under his specificaion, opimal moneary policy in a small open economy is isomorphic o opimal policy in a closed economy. This resul is consisen wih he findings of Galí and Monacelli (2005), who characerized he loss funcion for he case in which = = and he seady-sae level of oupu is efficien (he auhors also assume ha = = ( )). Bu we should noe ha heir framework only includes produciviy shocks, and as he above resul suggess, considering oher sources of shocks can have implicaions for he efficiency of he flexible price allocaion. 24 Noe ha, when = ; he flexible price allocaion and he efficien allocaion of oupu are insular o foreign shocks.

13 5 Ranking sandard policy rules In secion 4, we presen he opimal moneary policy in he form of a argeing rule. Bu he implemenaion of such a rule may no be sraighforward, eiher because is arges are difficul o monior ( ^Y T and ^Q T depend on unobservable shocks) or because is weighs are complex funcions of srucural parameers. For his reason, i is useful o compare he performances of simple policy rules and analyze how hese are affeced by he erms of rade exernaliy. We compue a ranking, based on our welfare measure, of policies ha arge domesic prices (or he producer price index - PPI), consumer prices (or CPI) and a fixed exchange rae regime (or PEG). This is done for differen values of and. Table 6 shows he policy rule ha leads o he highes level of welfare when he economy is subjec o all shocks. 25 The exercise shows ha while PPI argeing leads o higher welfare when he inraemporal and ineremporal elasiciies are low, for sufficienly high values of and ; a fixed exchange rae regime is he one associaed wih he highes level of welfare. 26 In line wih he discussion in secion 4, he ranking of policy rules illusraed in Table 6 is, by and large, driven by he amoun of sabilizaion of oupu relaive o he flexible price allocaion implied by he differen policy regimes. When domesic and foreign producs are good subsiues o one anoher (corresponding o he lower righ corner of Table 6), he welfare gains from a erms of rade improvemen (or a real exchange rae appreciaion) ouweigh he gains from sabilizing domesic prices. As a resul, a policy of fixed exchange raes, which ies policymakers hands, only parially sabilizes he oupu relaive o he flexible price allocaion, and is associaed wih a more appreciaed real exchange rae on average, leads o a higher welfare han a policy of domesic inflaion argeing. On he oher hand, when he inraemporal and ineremporal elasiciies are low (corresponding o he upper lef corner of he able), a policy ha oversabilize oupu and (i.e. leads o a higher level of oupu and a more depreciaed real exchange rae relaive o he flexible price allocaion) is welfare improving. So, in his case a PPI inflaion argeing regime ouperforms he exchange raed peg regime. For an inermediae value of and, a policy of CPI argeing is he preferred regime. [Inser Table 6 abou here] To assess he relevance of he above resuls, we evaluae he welfare coss associaed wih adoping differen rules. Table 7 illusraes he welfare gains or losses of adoping PPI inflaion argeing raher han an exchange rae peg, when he economy is faced by all he differen shocks. The numbers represen he percenage difference in seady-sae consumpion beween a PPI and a PEG. 27 The resuls show ha hese gains (or losses) are beween 0.032% o %. We have also conduced some sensiiviy analysis and compued he welfare losses following each shock individually. We found ha he losses were significanly larger following exernal shocks. 25 The sochasic environmen is parameerized o characerize a small indusrialized economy. See Table 5 for a full descripion of he parameer values used in his exercise. 26 We should, however, qualify ha our seing presens a sylized specificaion for he small open economy, which does no allow for liquidiy runs which could compromise he performance of exchange rae pegs. 27 P P I;P EG In paricular, we compue Wd = W P P I W P EG U c(c) uiliy of he represenaive agen. = P P I 2( )(U0 U0 P EG ) U c(c), where U 0 is he expeced life-ime 2

14 [Inser Table 7 abou here] 6 Conclusion In his paper we formalize a small open economy model as a limiing case of he wo-counry general equilibrium framework; characerize is uiliy-based loss funcion; and derive he opimal moneary plan. The model developed in his work encompasses, as special cases, a small open economy wih efficien levels of seady-sae oupu and a closed economy framework. The uiliy-based loss funcion for a small open economy can be wrien as a quadraic expression of domesic inflaion, oupu gap and real exchange rae. In our framework, when he economy experiences produciviy and foreign shocks exclusively, domesic inflaion argeing is opimal only under a paricular specificaion for preferences. If fiscal disurbances are also presen, opimal price sabiliy addiionally requires a producion subsidy. We have also shown ha a policy of domesic inflaion argeing leads o higher welfare han CPI or exchange rae argeing if domesic and foreign goods are no close subsiues for each oher. Conversely, when his subsiuabiliy is high, he opimal rule implies lower real exchange rae volailiy han a PPI regime. The ools developed in his paper can be applied o differen economic environmens. The model presened here assumes ha asse markes are complee. Relaxing his assumpion can lead o a more realisic framework, and, as hined in our analysis, can change he resuls obained. Also, his paper characerizes a seing in which he moneary auhoriy has one policy insrumen bu faces hree policy incenives (driven by monopolisic compeiion, price rigidiies and he erms of rade exernaliy). Previous works have demonsraed ha seady-sae inefficiencies generaed by monopolisic compeiion can someimes be resolved wih a producion subsidy. 28 An ineresing exension of he presen paper could assess wheher he inroducion of furher policy insrumens would implemen he firs bes. 29 Oher ineresing avenue for fuure research may include he inroducion of differen secors See Woodford (2003) and Benigno and Woodford (2005). 29 In paricular, his exension could follow he approach proposed by Adao, Correia and Teles and Correia, Nicolini, and Teles (2003). 30 Some of his analysis can already be found in Lipińska (2008), who exends he curren work o includes a nonradable secor. 3

15 A The Model Preferences We consider wo counries, H (Home) and F (Foreign). The world economy is populaed wih a coninuum of agens of uni mass, where he populaion in he segmen [0; n) belongs o counry H and he populaion in he segmen (n; ] belongs o counry F. The uiliy funcion of a consumer j in counry H is given by: 3 U j = E X s U(Cs) j V (y s (j); " s ) : (A.) s= Households obain uiliy from consumpion U(C j ) and conribue o he producion of a differeniaed good y(j) aaining disuiliy V (y(j); "): Produciviy shocks are denoed by " s. C is a Dixi-Sigliz aggregaor of home and foreign goods, defined by C = v C H + ( v) C F ; (A.2) where > 0 is he inraemporal elasiciy of subsiuion and C H and C F are consumpion subindices ha refer o he consumpion of home-produced and foreign-produced goods, respecively. The parameer deermining home consumers preferences for foreign goods, ( v); is a funcion of he relaive size of he foreign economy, n; and of he degree of openness, ; more specifically, ( v) = ( n): Similar preferences are specified for he res of he world, C = v C H + ( v ) C F ; (A.3) wih v = n. Tha is, foreign consumers preferences for home goods depend on he relaive size of he home economy and he degree of openness. Noe ha, for <, he specificaion of v and v generaes a home bias in consumpion, as in Suherland (2005). The sub-indices C H (CH ) and C F (CF ) are Home (Foreign) consumpion of he differeniaed producs produced in counries H and F. These are defined as follows: " C H = n " CH = n Z n 0 Z n 0 c (z) c (z) dz # ; C F = " n Z # " dz ; CF = n n Z n c (z) c (z) dz # ; (A.4) dz # ; (A.5) where > is he elasiciy of subsiuion across he differeniaed producs. The consumpionbased price indices ha correspond o he above specificaions of preferences are given by h P = vp H + ( v) (P F ) i ; (A.6) 3 In he subsequen secions, we assume he following isoelasic funcional forms: U(C ) = C and V (y ; " Y;) = " Y; y+ ; where is he coefficien of relaive risk aversion and is equivalen o he inverse of he Frisch elasiciy of + labor supply. 4

16 and h P = v P H + ( v ) (PF ) i ; (A.7) where P H (PH ) is he price sub-index for home-produced goods expressed in he domesic (foreign) currency and P F (PF ) is he price sub-index for foreign produced goods expressed in he domesic (foreign) currency: P H = Z n p (z) n 0 Z dz ; PF = p (z) n n Z n Z PH = p (z) dz ; P n F = p (z) 0 n n We assume ha he law of one price holds, so P H; p(h) = Sp (h) and p(f) = Sp (f); P dz ; (A.8) dz : (A.9) (A.0) where he nominal exchange rae, S ; denoes he price of foreign currency in erms of domesic currency. Equaions (A.6) and (A.7), ogeher wih condiion (A.0), imply ha P H = SPH and P F = SPF. However, as Equaions (A.8) and (A.9) illusrae, he home bias specificaion leads o deviaions from purchasing power pariy; ha is, P 6= SP : For his reason, we define he real exchange rae as Q = SP P : From consumers preferences, we can derive he oal demand for a generic good h, produced in counry H, and he demand for a good f; produced in counry F: ( y d p (h) PH; " # ) (h) = vc + v ( n) C + G ; (A.) n ( y d p (f) PF; (f) = P F; P Q " ( v) n n C + ( v ) Q C # + G ) ; (A.2) where G and G are counry-specific governmen shocks. We assume ha he public secor in he Home (Foreign) economy only consumes Home (Foreign) goods and has preferences for differeniaed goods analogous o he ones of he privae secor (given by Equaions (A.4) and (A.5)). Moreover, we consider he case in which flucuaions in governmen spending, G (G ), or proporional axes, ( ), are exogenous and compleely financed by lump-sum ransfers, T r (T r ); made in he form of domesic (foreign) goods. Finally, o porray our small open economy, we use he definiion of v and v and ake he limi for n! 0. Consequenly, condiions (A.) and (A.2) can be rewrien as ( y d p (h) PH; " # ) (h) = ( )C + C + G ; (A.3) y d (f) = " P H; p (f) P F; P # ( P F; P C + G ) Q : (A.4) Equaions (A.3) and (A.4) show ha exernal changes in consumpion affec he small open economy, bu he opposie is no rue. Moreover, movemens in he real exchange rae do no affec he oal demand for goods produced in he res of he world. 5

17 Price-seing mechanism Prices follow a parial adjusmen rule à la Calvo (983). Producers of differeniaed goods know he form of heir individual demand funcions (given by Equaions (A.3) and (A.4)), and maximize profis aking overall marke prices and producs as given. In each period a fracion, 2 [0; ); of randomly chosen producers is no allowed o change he nominal price of he goods hey produce. The remaining fracion of firms, given by ( ); chooses prices opimally by maximizing he expeced discouned value of profis. The opimal choice of producers ha can se heir price ~p (j) a ime T is, herefore: ( X() T E ~p (j) U c (C T ) Y H;T P H;T ) ~p (j) P H;T V y (~y ;T (j); " Y;T ) = 0: P H;T P T ( T )( )U c (C T ) (A.5) Monopolisic compeiion in producion leads o a wedge beween marginal uiliy of consumpion and marginal disuiliy of producion, represened by ( )( ) : We allow for flucuaions in his wedge by assuming a ime-varying proporional ax. Hereafer, we refer o hese flucuaions as mark-up shocks, where = ( )( ). In he presen model, only affecs he pricing mark-up and has no furher fiscal consequences. So we name hese disurbances mark-up shocks, bu hey are essenially driven by labor ax flucuaions. Given he Calvo-ype seup, he price index evolves according o he following law of moion, (P H; ) = P H; + ( ) (~p (h)) : (A.6) The res of he world has an analogous price seing mechanism. The asse marke srucure We assume ha, as in Chari e al. (2002), markes are complee boh domesically and inernaionally. In his seing, agens have access o a complee se of sae coningen securiies, and he ineremporal marginal rae of subsiuion is equalized across counries, U C C+ S P U C (C ) S + P+ = U C (C + ) U C (C ) P P + : (A.7) Using he definiion of he real exchange rae and assuming symmeric iniial condiions across counries, i follows ha U C (C ) = U C (C ) Q : (A.8) The presence of home bias leads o differen shadow prices of consumpion across counries, or, equivalenly, differen consumpion based price indexes. As inuiively explained in Corsei, Dedola and Leduc (2008), efficien risk sharing requires ha "he marginal benefi of an exra uni of foreign consumpion should be equal o is marginal cos, given by he domesic marginal uiliy of consumpion imes Q, i.e. he relaive price of C in erms of C ". Tha is, efficien risk sharing does no imply real risk sharing, as real marginal uiliies are no equalized across counries. 6

18 B Welfare Approximaions In his Appendix, we derive a second order approximaion of he equilibrium condiions of he model and he uiliy funcion under he assumpions ha Q = ; C = C and G = G = 0. Neverheless, we allow for an inefficien level of seady sae oupu (and he degree of his inefficiency is given by ). Our derivaion follows closely he approach presened in Benigno and Benigno (2003, 2006). B. Demand The second order approximaion of he demand condiion in he small open economy (Equaion (A.3)) can be wrien as: X d 0yy + 2 y0d y y + y 0D e e + :i:p + O(jjjj 3 ) = 0; (B.9) where y = d 0 y = Dy 0 = h i h i ^Y ^C ^p H; ^Q ; e = ^" ^ ^g ^C h i h i ; d 0 e = 0 0 ; ( ) 0 ( ) ; 0 0 ( ) ( ) D0 e = ( ) 0 2 ( ) 0 0 ( ) Noe ha ^g is defined as log(g =Y ) in order o allow for a zero seady sae level of governmen expendiure: B.2 Risk Sharing Equaion Given our uiliy funcion specificaion, Equaion (A.8) gives rise o a exac log linear expression, and he firs and second-order approximaions are herefore idenical. In marix noaion, we have: X E c 0 yy + 2 y0 C y y + y 0 C e e = 0; (B.20) c 0 y = h 0 0 C 0 y = 0; C 0 e = 0: i h ; c 0 e = i ; B.3 The Real Exchange Rae Using he definiion of he real exchange rae, we can wrie he second-order approximaion of Equaion (A.6) as: X E f 0 yy + 2 y0 F y y + y 0 F e e + :i:p + O(jjjj 3 ) = 0; (B.2) 7

19 where f 0 y = h Fy 0 = ( ) ( ) 2 i h ; fe 0 = ( =( )) i ; ; F e 0 = : B.4 Price Seing The second order approximaion of he price seing equaion (A.5), given (A.6), can be wrien in he following way: Q o = X E a 0 yy + 2 y0 A y y + ya 0 e e + 2 a^^ 2 + :i:p + O(jjjj 3 ); (B.22) where a 0 y = A 0 y = h i h 0 ; a 0 e = 2 (2 + ) ; A0 e = 6 4 i ; a^ = ( + ) k ; ( + ) : B.5 Welfare The second-order approximaion of he uiliy funcion, U ; can be wrien as: X W o = U cce0 wyy 0 2 y0 W y y yw 0 e e 2 w^^ 2 + :i:p + O(jjjj 3 ); (B.23) where w 0^ = h i k ; w0 y = = 0 0 ; (+) Wy 0 0 ( ) 0 0 = ; W 0 e = : In addiion, using he second-order approximaion of he equilibrium condiion derived in Secions B. o B.4, we can eliminae he erm wyy 0 from Equaion (B.23). In order o do so, we derive he vecor Lx; such ha h i a y d y f y c y Lx = w y ; 8