Currency Misalignments and Optimal Monetary Policy: A Reexamination

Transcription

1 Currency Misalignmens and Opimal Moneary Policy: A Reexaminaion Charles Engel Universiy of Wisconsin February 9 Preliminary Draf Absrac This paper examines opimal moneary policy in an open-economy wo-counry model wih sicky prices We show ha currency misalignmens are inefficien and lower world welfare We find ha opimal policy mus arge no only inflaion and he oupu gap bu also he currency misalignmen However he ineres rae reacion funcion ha suppors his argeing rule involves only he CPI inflaion rae This resul illusraes how examinaion of insrumen rules may hide imporan rade-offs facing policymakers ha are incorporaed in argeing rules The model is a modified version of Clarida Gali and Gerler s (JME The key change is ha we allow pricing o marke or local-currency pricing and consider he policy implicaions of currency misalignmens Besides highlighing he imporance of he currency misalignmen our model also gives a raionale for argeing CPI inflaion raher han PPI inflaion as in Clarida Gali and Gerler cengel@sscwiscedu Address: Deparmen of Economics 8 Observaory Drive Madison WI I am indebed very much o Mick Devereux Jon Faus Lars Svensson and Mike Woodford for commens and suggesions ha moivaed his work I hank Chris Ken for deailed commens on an earlier draf

2 Exchange raes among he large economies have flucuaed dramaically over he pas 3 years The dollar/euro exchange rae has experienced swings of greaer han 6% and even he Canadian dollar/us dollar has risen and fallen by more han 35% in he pas decade bu inflaion raes in hese counries have differed by only a percenage poin or wo per year Should hese exchange rae movemens be a concern for policymakers? Would i no be beer for policymakers o focus on oupu and inflaion and le a freely floaing exchange rae sele a a marke deermined level? I is widely undersood ha purchasing power pariy does no hold in he shor run Empirical evidence poins o he possibiliy of local-currency pricing (LCP or pricing o marke Tha is exporing firms may price discriminae among markes and/or se prices in he buyers currencies A currency could be overvalued if he consumer price level is higher a home han abroad when compared in a common currency or undervalued if he relaive price level is lower a home Currency misalignmens can be very large even in advanced economies There is frequen public discussion of he imporance of conrolling currency misalignmens For example on November 3 8 Rober Rubin (former US Secreary of he Treasury and Jared Bernsein (of he Economic Policy Insiue co-auhored an op-ed piece in he New York Times ha argued Public policy has been seriously deficien [because of] false choices grounded in ideology One of he principles hey argue ha all should agree upon is we need o work wih oher counries oward equilibrium exchange raes Ye here is lile suppor in he modern New Keynesian lieraure on moneary policy for he noion ha cenral banks should arge exchange raes Specifically if policymakers are already opimally responding o inflaion and he oupu gap is here any reason o pay aenion o exchange-rae misalignmens? Our answer is yes In a simple familiar framework his paper draws ou he implicaions for moneary policy when currency misalignmens are possible Currency misalignmens lead o inefficien allocaions for reasons ha are analogous o he problems wih inflaion in a world of saggered price seing When here are currency misalignmens households in he Home and Foreign counries may pay differen prices for he idenical good A basic ene of economics is ha violaions of he law of one price are inefficien if he good s marginal cos is he same irrespecive of where he good is sold i is no efficien o sell he good a differen prices We Many sudies have found evidence of violaions of he law of one price for consumer prices Two prominen sudies are Engel (999 and Akeson and Bursein (8 The lieraure is voluminous hese wo papers conain many relevan ciaions

3 find ha currency misalignmens lead o a reducion in world welfare and ha opimal moneary policy rades off his currency misalignmen wih inflaion and oupu goals These currency misalignmens arise even when foreign exchange markes are efficien Tha is he currency misalignmen disorion ha concerns policymakers arises in he goods marke from price seing and no in he foreign exchange marke The model of his paper deermines he foreign exchange rae in an efficien currency marke as a funcion of fundamenal economic variables The lieraure has indeed previously considered models wih local-currency pricing Some of hese models are much richer han he model considered here To undersand he conribuion of his paper i is helpful o place i relaive o four ses of papers: Clarida Gali and Gerler ( develop wha is probably he canonical model for openeconomy moneary policy analysis in he New Keynesian framework Their paper assumes ha firms se prices in he producer s currency (PCP for producer-currency pricing Their wocounry model also assumes Home and Foreign households have idenical preferences These wo assumpions lead o he conclusion ha purchasing power pariy holds a all imes he consumpion real exchange rae is consan This paper inroduces local-currency pricing ino CGG s model We derive simple rules for moneary policy ha are similar o CGG s While he model is no rich relaive o sophisicaed models in he lieraure (models ha inroduce capial working capial capaciy uilizaion habis in preferences ec he simple model is helpful for developing inuiion because he model can be solved analyically an explici second-order approximaion o he policymaker s loss funcion can be derived explici arge crieria for policy can be derived and explici ineres rae reacion funcions can be derived As we shall quickly explain we learn a lo from hese relaionships in he simple model The paper also allows Home and Foreign households o have differen preferences They can exhibi a home bias in preferences a larger weigh on goods produced in a household s counry of residence 3 This generalizaion does no change he opimal arge crieria a all in he CGG framework bu as we now explain is helpful in developing a realisic LCP model Benigno and Benigno (3 6 are imporan conribuions ha use models similar o Clarida Gali and Gerler s bu consider opimal policy when he opimal subsidies o deal wih monopoly disorions are no presen in seady sae 3 de Paoli (9 allows for home bias in preferences in a small open economy model There is home bias in he sense ha while he counry is small he limi of he raio of expendiure share on home goods o populaion share is no equal o one Faia and Monacelli (8 examine opimal moneary policy in a small open economy model wih home bias using a Ramsey syle analysis Pappa (4 considers a wocounry model wih home bias However he second-order approximaion o he welfare funcion is

4 Devereux and Engel (3 explicily examine opimal moneary policy in a wo-counry framework wih LCP Corsei and Peseni (5 exend he analysis in several direcions However neiher of hese sudies is suied oward answering he quesion posed above: is currency misalignmen a separae concern of moneary policy or will he opimal exchange-rae behavior be achieved hrough a policy ha considers inflaion and he oupu gap? These models have a couple of crucial assumpions ha make hem unsuied o answering his quesion Firs like CGG hey assume idenical preferences in boh counries This assumpion (as we show below leads o he oucome ha currency misalignmens are he only source of CPI inflaion differences beween he wo counries in he LCP framework Eliminaing inflaion differences eliminaes currency misalignmens and vice-versa 4 Second price sickiness is he only disorion in he economy in hese papers In conras CGG inroduce cos-push shocks so ha policymakers face a radeoff beween he goals of zero inflaion and zero oupu gap In Devereux and Engel (3 he opimal moneary policy under LCP ses inflaion o zero in each counry hus eliminaing any currency misalignmen By inroducing home bias in preferences he igh link beween relaive inflaion raes and currency misalignmens is broken A more realisic model for inflaion resuls one in which relaive CPI inflaion raes depend no only on currency misalignmens bu also on he inernal relaive price of impored o domesically-produced goods Moreover we follow CGG in allowing for cos-push shocks 5 This paper also derives opimal policy in a framework ha is consonan wih he bulk of New Keynesian models of moneary policy analysis Devereux and Engel (3 and Corsei and Peseni (5 assume price seing is synchronized wih prices se one period in advance 6 Here we adop he sandard Calvo price-seing echnology which allows for asynchronized price seing This change is imporan because i emphasizes he poin ha he cos of inflaion under sicky is misaligned relaive prices Also he previous papers assumed ha he money supply expressed in erms of deviaions of consumpion from is efficien level raher han in erms of he oupu gap so he analysis is no sricly comparable o ours See Woodford (3 for a discussion of why we approximae in erms of he oupu gap raher han consumpion 4 See Duare and Obsfeld (8 who emphasize his poin 5 The conribuion of Suherland (5 meris aenion His wo-counry model allows for imperfec passhrough and for differences in Home and Foreign preferences His model is saic and he derives a welfare funcion in which he variance of he exchange rae appears However he oher erms in he welfare funcion are prices so i is no clear how his funcion relaes o sandard quadraic approximaions ha involve oupu gaps and inflaion levels Moreover Suherland does no derive opimal moneary policy in his framework 6 A sophisicaed exension of his work is he recen paper by Corsei Dedola and Leduc (7 Tha paper exends earlier work in several dimensions including saggered price seing Bu i does no direcly address he issue of wheher currency misalignmens belong in he argeing rule along wih oupu gaps and inflaion 3

5 was he insrumen of moneary policy This paper follows CGG and mos of he modern lieraure in assuming ha he policymakers direcly conrol he nominal ineres rae in each counry 7 3 Monacelli (5 has considered opimal moneary policy under LCP in a simple smallcounry model 8 Bu a small-counry model is no capable of addressing he global misallocaion of resources arising from violaions of he law of one price In such a model impor prices are exogenous for he Home counry and he welfare of he res of he world is ignored Hence such a framework is no designed o consider he problems of currency misalignmens 4 There are many papers ha numerically solve very rich open economy models and examine opimal policy Some of hese papers allow for local currency pricing Many of hose papers are in he framework of a small open economy and so do no specifically accoun for he global misallocaion of resources ha occurs wih currency misalignmens 9 Moreover many use ad hoc welfare crieria for he policymaker or approximaions ha are no sricly derived from household welfare One of he main conribuions of his paper is o derive he role of he currency misalignmen in he policymaker s loss funcion Some papers have considered wheher i is beneficial o augmen he ineres rae reacion funcion of cenral banks wih an exchange-rae variable They ask he quesion: if he Taylor rule has he ineres rae reacing o inflaion and he oupu gap is here any gain from adding he exchange rae? Typically hese sudies find lile or no evidence of welfare gains from adding he exchange rae o he Taylor rule The quesion posed his way is misleading To undersand his poin i is helpful firs o reurn o he opimal policy analysis in CGG Tha paper finds (recall under heir assumpion of PCP ha opimal moneary policy can be characerized by a pair of arge crieria or argeing rule : y + ξπ H = and y + ξπ F = In hese equaions y refers o he oupu gap of he Home counry he percenage difference beween he acual oupu level and is efficien level π H is he producer-price inflaion rae in he Home counry Analogously y is he Foreign 7 While he model of his paper adheres sricly o he se-up of CGG changing only he assumpions of idenical preferences and LCP insead of PCP price seing he model is very similar o ha of Benigno s (4 Woodford (7 also considers he LCP version of CGG (hough no for opimal moneary policy analysis and makes he connecion o Benigno s paper 8 See also Leih and Wren-Lewis (6 who examine a small-open economy model wih non-raded goods (bu wih PCP for expor pricing 9 See for example Kollmann ( Smes and Wouers ( Ambler Dib and Rebei (4 and Adolfson e al (8 For example Smes and Wouers ( Ambler Dib and Rebei (4 and Adolfson e al (8 In a small open economy see Kollmann ( and Leiemo and Sodersrom (5 In a wo-counry model see Wang (9 4

6 oupu gap and π F he Foreign PPI inflaion rae These equaions describe he opimal radeoff of he oupu gap and inflaion for he policymaker I will be desirable o allow inflaion o be posiive if he oupu gap is negaive for example CGG hen derive opimal ineres rae rules ha will deliver hese opimal policy radeoffs They find he opimal ineres rae reacion funcions (assuming discreionary policy are: r rr bπ H = + and F r = rr + bπ r is he Home nominal ineres rae and rr is he Wicksellian or efficien real ineres rae The response of he ineres rae o inflaion b is a funcion of model parameers 3 The key poin o be made here is ha CGG s model shows ha opimal policy mus rade off he inflaion and oupu goals of he cenral bank Bu he opimal ineres rae reacion funcion does no necessarily include he oupu gap Tha is adding he oupu gap o he ineres rae rule ha already includes inflaion will no improve welfare Focusing on he insrumen rule does no reveal he role of he oupu gap ha is apparen in he argeing rule in he erminology of Svensson (999 An analogous siuaion arises in he LCP model concerning currency misalignmens We can characerize he arge crieria in his model wih wo rules as in he CGG model The firs is y + y + ξπ ( + π = This rule a firs glance appears o be simply he sum of he wo arge crieria in he CGG model I is excep ha he inflaion raes ha appear in his radeoff ( π and crierion is π are CPI inflaion raes raher han PPI as in CGG s model The second arge q ξπ π σ + ( = where q is he real exchange rae (defined as Foreign prices relaive o Home prices expressed in a common currency q is he deviaion of he real exchange rae from is efficien level We define he parameers in his equaion below The imporan poin is ha he radeoff described here relaes real exchange raes and relaive CPI inflaion raes For example even if inflaion is low in he Home counry relaive o he Foreign counry opimal policy may under some circumsances sill call for a ighening of he moneary policy sance in he Home counry if he Home currency is sufficienly undervalued Like CGG we can derive he opimal ineres rae rules ha suppor hese argeing rules We find hese ineres rae reacion funcions are = + and r rr bπ r = rr + bπ They are idenical o he ones derived in CGG (he parameer b is he same excep hey arge CPI inflaion raher han PPI inflaion as in CGG The conclusion is ha while he arge crieria ξ is a preference parameer defined below 3 Specifically we show below ha b = ρ + ( ρσξ We define he parameers below 5

7 include currency misalignmens he currency misalignmen is no in he opimal ineres rae reacion funcion If we focus on only he laer we miss his radeoff he policymaker faces Previous sudies have found lile welfare gain from adding an exchange rae variable o he Taylor rule Properly speaking hese sudies examine he effecs of simple argeing rules under commimen Our resuls describe he welfare funcion he arge crieria and he opimal ineres rae reacion funcions under discreionary policymaking Bu our resuls here sugges ha even if here is no role for he currency misalignmen in a simple argeing rule exchange rae concerns may sill be imporan in erms of welfare This poin is brough ou in he conex of a relaively simple model ha can be solved analyically (wih approximaions bu is obscured in larger models ha are solved numerically 4 We proceed in wo seps Afer seing ou he obecives of households and firms he producion funcions and he marke srucure we derive a global loss funcion for cooperaive moneary policymakers We can derive he period loss funcion wihou making any assumpions abou how goods prices or se or how wages are se 5 We find ha in addiion o squares and cross-producs of Home and Foreign oupu gaps and he cross-secional dispersion of goods prices wihin each counry he loss also depends on he squared currency misalignmen This loss funcion evaluaes he welfare coss arising because firms se differen prices in he Home and Foreign counry (assuming he coss of selling he good in boh counries are idenical and does no depend on wheher he price differences arise from local-currency price sickiness from price discriminaion or for some oher reason We hen follow CGG and assume a Calvo mechanism for price seing However we allow for possibiliy of local-currency pricing As in CGG we derive opimal policy under discreion We obain boh argeing rules and insrumen rules We consider only opimal cooperaive policy Our goal is o quanify he global loss from currency misalignmens which we can see by deriving he loss funcion for a policymaker ha aims o maximize he sum of uiliies of Home and Foreign households Pracically speaking inernaional agreemens ha prohibi currency manipulaion may mean ha he currency misalignmen can only be addressed 4 Coenen Lombardo Smes and Sraub (7 examine opimal moneary policy in a wo-counry model ha exhibis incomplee pass-hrough However he numerical analysis does no allow he reader o see explicily he role of currency misalignmens 5 Excep ha we do assume ha all households (which are idenical se he same wage As we noe laer his rules ou a model of saggered wage seing such as in Erceg Henderson and Levin ( hough a generalizaion o encompass ha case would be sraighforward 6

8 in a cooperaive environmen 6 Tha is i seems likely ha if cenral banks are going o move oward policies ha explicily arge exchange raes hey will do so cooperaively The Model The model we examine is nearly idenical o CGG s We consider wo counries of equal size while CGG allow he populaion of he counries o be differen Since he populaion size plays no real role in heir analysis we simplify along his dimension Bu we make wo significan generalizaions Firs we allow for differen preferences in he wo counries Home agens may pu a higher weigh in uiliy on goods produced in he Home counry Home ν ν households pu a weigh of on Home goods and on Foreign goods (and vice-versa for Foreign households This is a popular assumpion in he open-economy macroeconomics lieraure and can be considered as a shor-cu way of modeling openness Tha is a less open counry pus less weigh on consumpion of impored goods and in he limi he economy becomes closed if i impors no goods The second maor change we allow as already noed is we allow for goods o be sold a differen prices in he Home and Foreign counries The model assumes wo counries each inhabied wih a coninuum of households normalized o a oal of one in each counry Households have uiliy over consumpion of goods and disuiliy from provision of labor services In each counry here is a coninuum of goods produced each by a monopolis Households supply labor o firms locaed wihin heir own counry and ge uiliy from all goods produced in boh counries Each household is a monopolisic supplier of a unique ype of labor o firms wihin is counry We assume ha here is rade in a complee se of nominally-denominaed coningen claims Monopolisic firms produce oupu using only labor subec o echnology shocks In his secion we will no make any assumpions on how wages are se by monopolisic households or prices are se by monopolisic firms In paricular prices and wages may be sicky and here may be LCP or PCP for firms We derive he period loss funcion for he policymaker which expresses he loss (relaive o he efficien oucome in erms of wihin-counry and inernaional price misalignmens and oupu gaps This loss funcion applies under various assumpions abou how prices are acually se and so is more general han he policy rules we subsequenly derive which depend on he specifics of price and wage seing 6 For example he IMF Aricles of Agreemen sae ha member counries shall avoid manipulaing exchange raes o preven effecive balance of paymens adusmen or o gain an unfair compeiive advanage over oher members See Saiger and Sykes (8 7

9 All households wihin a counry are idenical We will assume ha in each period heir labor supplies are idenical This assumpion rules ou saggered wage seing as in Erceg Henderson and Levin ( because in ha model here will be dispersion in labor inpu across households ha arises from he dispersion in wages se Our se-up is consisen wih sicky wages bu no wage dispersion However i is enirely sraighforward o generalize he loss funcions we derive o allow for wage dispersion following he seps in Erceg Levin and Henderson We do no do ha because we wan our model o be more direcly comparable o CGG a Households The represenaive household in he home counry maximizes ( σ + φ U( h =Ε β C+ ( h N+ ( h = σ + φ σ > φ C ( h is he consumpion aggregae We assume Cobb-Douglas preferences: ν ν ( ( ( H F C h = C h C h ν ( ( ( If ν = Home and Foreign preferences are idenical as in CGG There is home bias in preferences when ν > each counry: (3 In urn CH ( h and CF ( h are CES aggregaes over a coninuum of goods produced in ξ ξ ξ ξ CH ( h = C ( H h f df and ξ ξ ξ ξ CF ( h = C ( F h f df N ( h is an aggregae of he labor services ha he household sells o each of a coninuum of firms locaed in he home counry: (4 N ( h = N ( h f df Households receive wage income W ( h N ( h aggregae profis from home firms They pay lump-sum axes each period T Each household can rade in a complee marke in coningen claims (arbirarily denominaed in he home currency The budge consrain is given by: + + (5 PC ( h + Z( D( h = W( h N( h +Γ T + D( h s Ω Γ 8

10 where Dh ( represens household h s payoffs on sae-coningen claims for sae Z + ( is he price of a claim ha pays one dollar in sae occurring a ime In his equaion P is he exac price index for consumpion given by: + condiional on sae (6 P = k P P ν / ( ν / H F k ν ν ( ν / ν / = ( ( / ( / P H is he Home-currency price of he Home aggregae good and P F is he Home currency price of he Foreign aggregae good Equaion (6 follows from cos minimizaion Also from cos minimizaion P H and P F are he usual CES aggregaes over prices of individual varieies f: (7 ( ξ ξ P = P ( f df and ( P P ( f df H H F = F ξ ξ Foreign households have analogous preferences and face an analogous budge consrain Because all Home households are idenical we can drop he index for he household and use he fac ha aggregae per capia consumpion of each good is equal o he consumpion of each good by each household The firs-order condiions for consumpion are given by: ν (8 PHCH = PC ν (9 PC F F = PC ( ξ PH ( f CH ( f = C PH ( β ( H and + σ + + ξ PF ( f CF( f = C PF C( / C( ( P / P = Z ( In equaion ( we explicily use an index for he sae a ime for he purpose of clariy Z + ( is he normalized price of he sae coningen claim Tha is i is defined as Z + ( divided by he probabiliy of sae F + condiional on sae Noe ha he sum of Z across all possible saes a ime + mus equal / R + ( where R denoes he gross nominal yield on a one-period non-sae-coningen bond Therefore aking a probabiliy-weighed sum across all saes of equaion ( we have he familiar Euler equaion: ( β + ( σ RΕ C( / C( ( P / P+ = 9

11 Analogous equaions hold for Foreign households Since coningen claims are (arbirarily denominaed in Home currency he firs-order condiion for Foreign households ha is analogous o equaion ( is: C ( / C ( ( E P / E P Z ( + σ = (3 β ( Here E refers o he home currency price of foreign currency exchange rae We offer an apology o he reader here We wan o sick o CGG s noaion who use e for he log of he nominal exchange rae Consisency requires us o use exchange rae so we have used he disinc bu similar noaion E o refer o he level of he nominal Ε o be he condiional expecaion operaor As noed above we will assume a his sage ha labor inpu of all households is he same so N = N ( h b Firms Each Home good Y ( f is made according o a producion funcion ha is linear in he labor inpu These are given by: (4 Y( f = AN ( f Noe ha he produciviy shock A is common o all firms in he Home counry N ( f is a CES composie of individual home-counry household labor given by: (5 η η η η N( f = N( h f dh where he echnology parameer η is sochasic and common o all Home firms Profis are given by: (6 Γ = + ( f PH ( f CH ( f E PH ( f CH ( f ( τ W N ( f In his equaion PH ( f is he home-currency price of he good when i is sold in he Home counry P H ( f is he foreign-currency price of he good when i is sold in he Foreign counry CH ( f is aggregae sales of he good in he home counry: (7 C ( f = C ( h f dh H H C H ( f is defined analogously I follows ha Y f = C f + C f ( H( H(

12 There are analogous equaions for A he foreign echnology parameer shock given by c Equilibrium Y ( f wih he foreign produciviy shock given by η and foreign subsidy given by τ Goods marke clearing condiions in he Home and Foreign counry are given by: (8 (9 ν PC ν P C ν ( ν / ν ν / Y = CH + CH = + = k S ( C + S C PH PH ν PC ν P C ν ( ν / ν ν / Y = CF + CF = + = k ( S C + S C PF PF We have used S and Home and Foreign counries respecively: ( S = P / P ( F H S = P / P H F S o represen he price of impored o locally-produced goods in he Equaions ( and (3 give us he familiar condiion ha arises in open-economy models wih a complee se of sae-coningen claims when PPP does no hold: ( σ C EP EPH / ( / ( S ν S ν C P PH = = Toal employmen is deermined by oupu in each indusry: (3 N ( N ( ( f df A = = Y f df = A CHVH + CHVH where (4 V H PH ( f PH ξ df and V H ξ PH ( f df P H Log-linearized Model In his secion we presen some log-linear approximaions o he models presened above The full se of log-linearized equaions appears in Appendix A Our approach o he opimal policy decision is o consider a second-order approximaion of he welfare funcion around he efficien seady sae The derivaion of he loss funcion iself requires a second-order approximaion of he uiliy funcion iself bu in he course of he derivaion will acually require second-order approximaions o some of he equaions of he model However for many purposes he firs-order approximaions are useful: he consrains in he opimizaion problem need only be approximaed o he firs order; he opimaliy condiions for moneary policy he

13 arge crieria are linear; and we can analyze he dynamics under he opimal policy in he linearized model In our noaion lower case leers refer o he log of he corresponding upper case leer less is deviaion from seady sae If firms se he same price for Home and Foreign consumers hen s = s We will assume o a firs order s = s even if firms se differen prices in he wo counries Tha is for he aggregae price indexes p p = p p so relaive prices are he same in he Home F H F H and Foreign counries This relaionship will urn ou o hold in our Calvo pricing model under LCP when he frequency of price adusmen is idenical in he wo counries We define he log of he deviaion from he law of one price as: (5 Δ e + p p H H Because s = s we have ha he law of one price deviaion is he same for boh goods: Δ e + p p F F The marke-clearing condiions (8 and (9 are approximaed as: ν ( ν ν ν (6 y = s + c + c ν ( ν ν ν (7 y = s + c + c The condiion arising from complee markes ha equaes he marginal uiliy of nominal wealh for Home and Foreign households equaion ( is given by: (8 σc σc =Δ + ( ν s (9 (3 (3 where We can express c c and s in erms of y D+ ν D ( ν ν( ν c = y + y + Δ D D D D+ ν D ( ν ν( ν c = y + y Δ D D D σ ( ν s = ( y y Δ D D D σν ν ν ( + ( y and Δ : The model is closed and soluions for he endogenous variables can be derived once policy rules are deermined We urn o consideraion of opimal moneary policy

14 3 Loss Funcions and Opimal Policy We derive he loss funcion for he cooperaive moneary policy problem The loss funcion is derived from a second-order approximaion o households uiliy funcions Loss is measured relaive o he efficien allocaions The policymaker wishes o minimize (3 β Χ+ = Ε This loss funcion is derived from household s uiliy given in Equaion ( The perperiod loss Χ + represens he difference beween he maximum uiliy achievable under efficien allocaions and he uiliy of he marke-deermined levels of consumpion and leisure We derive he loss funcion for he cooperaive policymaker which is he relevan crierion for evaluaing world welfare We aim o highligh he global inefficiency ha arises from currency misalignmens For ha reason we se aside he difficul issues involved wih deriving he loss funcion for a non-cooperaive policymaker and defining a non-cooperaive policy game We noe for fuure work ha here are hree echnical problems ha arise if we ry o examine non-cooperaive policy One problem is discussed in deail in Devereux and Engel (3 To consider policies in he non-cooperaive framework we wan o look a he effecs of one counry changing is policies holding he oher counries policies consan In a complee markes world o evaluae all possible alernaive policies we need o calculae prices of saeconingen claims under alernaive policies In paricular we canno assume equaion ( which was derived assuming equal iniial Home and Foreign wealh holds under all alernaive policies ha he compeiive policymaker considers Policies can change sae-coningen prices and herefore change he wealh disribuion I is no uncommon for sudies of opimal policy in open economies o rea equaion ( as if i is independen of he policy choices bu i is no There is a special case in which i holds in all saes which is he se-up in CGG When he law of one price holds when Home and Foreign households have idenical preferences and here are no preference shocks and when preferences over Home and Foreign aggregaes are Cobb- Douglas equaion ( holds in all saes This well-known oucome arises because in all saes of he world he erms of rade change in such a way as o leave Home/Foreign wealh unchanged 3

15 Even if we could surmoun his echnical challenge 7 here are a couple of oher echnical challenges ha appear in our framework ha did no plague CGG Firs in he LCP model we do no have S = S ha he relaive price of Foreign o Home goods is idenically he same in boh counries This relaionship holds up o a firs-order log-linear approximaion in he LCP model (as long as we assume equal speeds of price adusmen for all goods 8 bu only up o a firs-order Messy firs-order relaive price erms ( s and s appear in he obecive funcion of he non-cooperaive policymaker However hose wash ou in he obecive funcion under cooperaion Second CGG nealy dichoomize he choice variables in heir model he Home policymaker ses Home PPI inflaion and he Home oupu gap aking he Foreign policy choices as given and vice-versa for he Foreign policymaker Such a nea dichoomy is no possible in he model wih currency misalignmens we canno us assign he exchange rae o one of he policymakers In a sense all of hese echnical problems are relaed o he real world reason why i is more reasonable o examine policy in he cooperaive framework when currency misalignmens are possible The non-cooperaive model assumes ha cenral banks are willing and able o manipulae currencies o achieve beer oucomes However in pracice WTO and IMF rules as well as implici rules of neighborliness may prohibi his ype of policy 9 Maor cenral banks have generally been unwilling o announce explici arges for exchange raes wihou full cooperaion of heir parners Even if he cooperaive policy analysis is no a realisic descripion of acual policy decision-making he welfare funcion is a measure of wha could be achieved under cooperaion Appendix B shows he seps for deriving he loss funcion when here are no currency misalignmens bu wih home bias in preferences I is worh poining ou one aspec of he derivaion In closed economy models wih no invesmen or governmen consumpion equals oupu Tha is an exac relaionship and herefore he deviaion of consumpion from he efficien level equals he deviaion of oupu from he efficien level o any order of approximaion: c = y In he open economy he relaionship is no as simple When preferences of Home and Foreign agens are idenical and markes are complee hen he consumpion aggregaes in Home and Foreign are always equal (up o a consan of proporionaliy equal o relaive wealh Bu ha is no rue when preferences are no he same 7 The Appendix of Devereux and Engel (3 demonsraes how his problem can be handled 8 See Benigno (4 and Woodford (7 on his poin 9 See Saiger and Sykes (8 for a discussion of how some have inerpreed WTO rules as prohibiing currency manipulaion and he relaionship beween IMF rules prohibiing exchange rae manipulaion and WTO rules prohibiing cerain expor subsidies 4

16 Equaion ( shows ha we do no have C = C under complee markes even if he law of one price holds for boh goods Because of his we do no have c + c equal o y + y excep o a firs-order approximaion Since we are using a second-order approximaion of he uiliy funcion we need o accoun for he effec of differen preferences (or he effecs of he erms of rade in ranslaing consumpion gaps ino oupu gaps We find: ( ( νν σ σ σ+ φ ξ σ σ H F 4D (33 Χ = ( y y ( y + y ( + p p The period loss depends on he squared oupu gap in each counry as well as he squared difference in he oupu gaps The erms σ and σ represen he cross-secional variance of ph pf prices of Home goods and Foreign goods respecively (Recall D σν ν ν ( + ( Appendix B also shows he derivaion of he loss funcion in he more general case in which currency misalignmens are possible Two aspecs of he derivaion meri aenion Firs in examining he firs-order dynamics of he model we can make use of he firs-order approximaion s = s Tha is an exac equaion when he law of one price holds bu we do no require ha his relaionship hold o a second-order approximaion The derivaion of he loss funcion mus ake his ino accoun The second poin o noe is ha as is sandard in his class of models price dispersion leads o inefficien use of labor Bu o a second-order approximaion his loss depends only on he cross-secion variances of p H p p H F and no heir comovemens (which would play a role in a hird-order approximaion (34 The loss funcion Χ is deermined by: νν ( σ( σ σ+ φ ν( ν Χ = ( y y ( y + y Δ 4D 4D ξ ν ν ν ν σ ph+ σ ph + σ pf + σ pf and I is imporan o recognize ha he loss funcion derived here as well as he loss funcion derived previously (equaion (33 do no depend on how prices are se indeed wheher prices or sicky or no The loss funcion here generalizes (33 o he case in which here are deviaions from he law of one price so ha Δ This can be seen by direcly comparing (33 o (34 In (34 σ is he cross-secional variance of Home goods prices in he Home counry σ is he ph cross-secional variance of Home goods prices in he Foreign counry ec If here is no currency ph p F misalignmen hen Δ = so p f = p f + e and H ( H ( p f = p f + e for each firm f In F( F( 5

17 H H ha case σ p = σ p and σ p = σ p because he exchange rae does no affec he crosssecional variance of prices If we have Δ = o (33 F F σ = σ and σ ph ph = σ hen (34 reduces pf pf Why does he currency misalignmen appear in he loss funcion? Tha is if boh Home and Foreign oupu gaps are zero and all inflaion raes are zero wha problem does a misaligned currency cause? From equaion (3 if he currency is misaligned hen inernal relaive prices ( s mus also differ from heir efficien level if he oupu gap is zero The Home and Foreign counries could achieve full employmen bu he disribuion of he oupu beween Home and Foreign households is inefficien For example suppose Δ > which from (3 implies we mus have s < if boh oupu gaps are eliminaed On he one hand Δ > ends o lead o overall consumpion in Home o be high relaive o Foreign consumpion (equaions (9-(3 Tha occurs because financial markes pay off o Home residens when heir currency is weak Bu Home residens have a home bias for Home goods Tha would lead o overproducion in he Home counry were i no for relaive price adusmens which is why s < We highligh he fac ha he loss funcions are derived wihou specific assumpions abou price seing no o give a false paina of generaliy o he resul bu o emphasize ha he loss in welfare arises no specifically from price sickiness bu from prices ha do no deliver he efficien allocaions Of course i is our specific assumpions of nominal price and wage seing ha give rise o he inernal and exernal price misalignmens in his model and indeed moneary policy would be ineffecive if here were no nominal price or wage sickiness Bu one could imagine especially a number of mechanisms ha give rise o deviaions from he law of one price because he lieraure has produced a number of models based boh on nominal sickiness and real facors In he nex secion we modify he CGG model in he simples way allowing localcurrency pricing insead of producer-currency pricing o examine furher he implicaions of currency misalignmens 4 Price and Wage Seing We now inroduce our models of price and wage seing We follow CGG in ha wages are se flexibly by monopolisic suppliers of labor bu goods prices are sicky Wages adus coninuously bu households exploi heir monopoly power by seing a wage ha incorporaes a mark-up over heir uiliy cos of work Governmen is assumed o have only limied fiscal insrumens The governmen can se a consan oupu subsidy rae for monopolisic firms which will achieve an efficien allocaion in 6

18 he non-sochasic seady sae Bu unforunaely he mark-up charged by workers is imevarying because he elasiciy of demand for heir labor services is assumed o follow a sochasic process These shocks are someimes labeled cos-push shocks and give rise o he well-known radeoff in CGG s work beween conrolling inflaion and achieving a zero oupu gap Households are monopolisic suppliers of heir unique form of labor services Household h faces demand for is labor services given by: (35 where W ( h N( h = W η N (36 ( η ( (37 = W W h dh η The firs-order condiion for household h s choice of labor supply is given by: W ( h W σ φ W = ( + μ ( C( h ( N( h where μ P η The opimal wage se by he household is a ime-varying mark-up over he marginal disuiliy of work (expressed in consumpion unis Because all households are idenical we have W = W( h and N = N( h Since all households are idenical we have from equaion (37: W σ φ ( v/ (38 W / PH = ( + μ C N S We will consider hree differen scenarios for firm behavior In he firs prices can be adused freely In he second he PCP scenario ha CGG analyze firms se prices in heir own counry s currency and face a Calvo pricing echnology In he hird when firms are allowed o change prices according o he Calvo pricing rule hey se a price in heir own currency for sales in heir own counry and a price in he oher counry s currency for expors This is he LCP scenario We adop he following noaion For any variable K is he value under flexible prices K is he value of variables under globally efficien allocaions In oher words his is he value for variables if prices were flexible and opimal subsidies o monopolisic suppliers of labor and monopolisic producers of goods were in place This includes a ime-varying subsidy o suppliers of labor o offse he ime-varying mark-up in wages in equaion (37 K : K is he gap: he value of he variable under PCP or LCP relaive o K 7

19 We will rea he PCP and LCP cases separaely so here will be no need o use noaion o disinguish variables under PCP versus LCP Flexible Prices Home firms maximize profis given by equaion (6 subec o he demand curve ( They opimally se prices as a mark-up over marginal cos: (39 (4 P f = E P f = τ + μ W A where ( ( ( ( P H H / When opimal subsidies are in place: P f = E P f = W A H ( H ( / P μ ξ From (37 (39 and (4 i is apparen ha he opimal subsidy saisfies P W (4 ( τ ( + μ ( + μ = Noe from (39 ha all flexible price firms are idenical and se he same price Because he demand funcions of Foreign residens have he same elasiciy of demand for Home goods as Home residens firms se he same price for sale abroad: (4 E P = P and E P = P (43 (44 H H H H From (4 using (38 we have: W P = E P = W /(( +μ A and H H We can conclude: S = S P = E P = W /(( + μ A W F F Because P ( f is idenical for all firms (3 collapses o (45 Y = AN H PCP We assume a sandard Calvo pricing echnology A given firm may rese is prices wih probabiliy θ each period We assume ha when he firm reses is price i will be able o rese is prices for sales in boh markes The PCP firm ses boh prices in is own currency ha is he Home firm ses boh PH ( f and P ( f E P ( f in Home currency (As will become H H apparen he firm opimally chooses he same price for boh markes P f = P f H ( H ( The firm s obecive is o maximize is value Is value is equal o he value of is enire sream of dividends valued a sae-coningen prices Given equaion ( i is apparen ha he firm ha selecs is prices a ime chooses is rese prices P H ( f and P H ( f o maximize 8

20 (46 θ + H H+ H H+ τ + + = Ε Q P ( f C ( f + P C ( f ( W N ( f subec o he sequence of demand curves given by equaion ( and he corresponding Foreign demand equaion for Home goods In his equaion we define (47 β ( σ Q + C + / C ( P / P+ The soluion for he opimal price for he Home firm for sale in he Home counry is given by: (48 P H ( z = Ε ξ θ Q + ( τ W+ PH+ CH+ / A+ = ξ Ε θ Q + PH+ CH+ = For sale in he foreign marke we have: (49 P H ( z = Ε ξ θ Q + ( τ W+ ( E+ PH+ CH+ / A+ = ξ Ε θ Q + ( E+ PH+ CH+ = Under he Calvo price seing mechanism a fracion θ of prices remain unchanged from he previous period From equaion (7 we can wrie: (5 ξ ξ /( ξ H = θ( H + ( θ( H P P P (5 PCP Tha is S = S ξ ξ /( ξ H = θ( H + ( θ( H P P P Taking equaions (48 (49 (5 and (5 we see ha he law of one price holds under P f = E P f for all f hence H ( H ( P = E P Hence under PCP we have H H LCP The same environmen as he PCP case holds wih he sole excepion ha he firm ses is price for expor in he imporer s currency raher han is own currency when i is allowed o rese prices The Home firm for example ses rese is price a ime chooses is rese prices P H ( f in Foreign currency The firm ha can P H ( f and P ( f o maximize H (5 θ + H H+ H H+ τ + + = Ε Q P ( z C ( f + E P C ( f ( W N ( f The soluion for P ( z is idenical o (48 We find for expor prices H 9

21 (53 P H ( z = Ε ξ θ Q + ( τ W+ ( PH+ CH+ / A+ = ξ Ε θ Q + E+ ( PH+ CH+ = does no hold Subsidies Equaions (5 and (5 hold in he LCP case as well However he law of one price As in CGG we will assume ha subsidies o monopoliss are no se a heir opimal level excep in seady-sae Tha is insead of he efficien subsidy given in equaion (4 we have: (54 ( ( P W τ + μ ( + μ = Here W W μ is he seady-sae level of μ We have dropped he ime subscrip on he subsidy rae τ because i is no ime-varying 5 Log-linearized Phillips Curves Under PCP we can derive a New Keynesian Phillips curve for an open economy: π H = δ( w p H + βεπ H + or: (55 where (56 (57 u σ( + D σ( D πh = δ + φ y + y + βε π H + + u D D = δμ W Similarly for foreign producer-price inflaion we have: σ( + D σ( D πf = δ + φ y + y + βε π F + + u D D In he LCP model he law of one price deviaion is no zero We have: σ( + D σ( D D ( ν πh = δ + φ y + y + Δ + βε π H + + u D D D σ( + D σ( D ν D (58 πf = δ + φ y + y + Δ + βε π F + + u D D D There are also price adusmen equaions for he local prices of impored goods: (59 σ( + D σ( D D+ ν πh = δ + φ y + y Δ + βε π H + + u D D D

22 (6 σ( + D σ( D D+ ν πf = δ + φ y + y + Δ + βε π F + + u D D D From (57-(58 and (59-(6 we see π π = π π Assuming a symmeric iniial condiion so ha F H F H F H F H p p = p p we conclude s = s as we noed above Tha is he relaive price of Foreign o Home goods is he same in boh counries We emphasize ha his is rue in general for a firs-order approximaion The efficien allocaions canno be obained wih moneary policy alone because of he sicky-price exernaliy and because we assume he policymaker does no have access o fiscal insrumens aside from seing a consan subsidy rae o firms The policymaker has home and foreign nominal ineres raes as insrumens As is sandard in he lieraure we can model he policymaker as direcly choosing oupu gaps inflaion levels and (in he LCP case deviaions from he law of one price subec o consrains From he firs-order condiions we can back ou he opimal choice of nominal ineres raes using a log-linearized version of equaion ( and is foreign counerpar given by: (6 r Eπ + = σ ( Εc+ c (6 r Eπ = σ( Ε c c + + In hese equaions π and π refer o Home and Foreign consumer price inflaion respecively: (63 (64 ν ν π = πh + πf ν ν π = πf + πh 6 Opimal Policy under PCP As is familiar in he New Keynesian models wih Calvo price adusmen he loss funcion can be rewrien in he form: where: β + β + = = Ε Χ = Ε Ψ ν ( ν σ ( σ σ + ( φ (( ( ξ ( ( 4D δ (65 Ψ y y y + y ( π + π H F This loss funcion exends he one derived in CGG o he case of home bias in preferences or nonraded goods (ie ν raher han ν =

23 The policymaker chooses values for y y π and H π F o minimize he loss subec o he Phillips curves (55 and (56 Under discreion he policymaker akes pas values of y y π H and π F as given and also does no make plans for fuure values of hese variables undersanding ha fuure incarnaions of he policymaker can aler any given plan The π + policymaker a ime canno influence Ε H and Ε because fuure inflaion levels are π F + chosen by fuure policymakers and here are no endogenous sae variables ha can limi he pahs of fuure inflaion levels Hence he policymaker s problem is essenially a saic one o maximize (65 subec o (55 and (56 aking Ε π H + and Ε π F + as given Even hough we have inroduced home bias in consumpion he opimal policy rules are he same as in CGG The firs-order condiions are given by: (66 (67 y + y + ξπ ( + π = H F y y + ξπ ( π = H F These wo arge crieria can be rewrien as: (68 y + ξπ = and H y + ξπ = F The crieria given in (68 are idenical o hose ha arise in he closed-economy version of his model There is a radeoff beween he goals of eliminaing he oupu gap and driving inflaion o zero and he elasiciy of subsiuion among goods produced in he counry deermines he weighs given o oupu gaps and inflaion I is worh emphasizing ha equaion (68 indicaes he opimal policy enails a radeoff beween he oupu gap and he producer price inflaion level In a closed economy wih no inermediae goods here is no disincion beween producer and consumer prices Bu in an open economy here is an imporan disincion The policies described in (68 imply ha policymakers should no give any weigh o inflaion of impored goods In conuncion wih he Phillips curves (55 and (56 equaion (68 allows us o solve for he Home and Foreign oupu gaps and π H and π F as funcions of curren and expeced fuure cos-push shocks u and Wih he oupu gap deermined by opimal policy he erms of rade mus adus o insure goods marke clearing Bu he erms of rade adus freely in he PCP world because nominal exchange rae changes ranslae direcly ino impor price changes In essence he impor secor is like a flexible-price secor so policymakers can ignore inflaion in ha secor as in Aoki ( u 7 Opimal Policy under LCP The ransformed loss funcion Ψ can be wrien as:

24 (69 ν( ν σ( σ σ + φ ν( ν Ψ ( y y (( y + ( y Δ 4D 4D ξ ν ν ν ν ( πh + ( πf + ( πf + ( πh δ The loss funcion is similar o he one under PCP The main poin o highligh is ha squared deviaions from he law of one price maer for welfare as well as oupu gaps and inflaion raes Deviaions from he law of one price are disorionary and are a separae source of loss in he LCP model The policymaker under discreion seeks o minimize he loss subec o he consrains of he Phillips curves (57-(58 and (59-(6 There is an addiional consrain in he LCP model Noe ha π F π H = s s Bu from equaion (3 we have σ ( ν (7 s s = ( y y ( y y ( Δ Δ D D Using (7 in conuncion wih (57 and (6 we derive: σ ( ν ( y y ( y y ( Δ Δ = D D (7 σ ( ν δ + φ ( y y Δ + βε( πh+ πf+ + u u D D This consrain arises in he LCP model bu no in he PCP model precisely because impor prices are sicky and subec o a Calvo price-adusmen mechanism raher han free o respond via nominal exchange-rae changes Anoher conras wih he PCP model is ha here are four sicky prices in he LCP model so non-zero inflaion raes for each of he four maer for welfare Indeed we noe ha we can rewrie he loss funcion as: (7 ν( ν σ( σ σ + φ ν( ν Ψ ( y y (( y + ( y Δ 4D 4D ξ ν( ν ( π + ( π + ( s s δ Under his formulaion he loss funcion is seen o depend on he aggregae CPI inflaion raes π and π and he change in he erms of rade s s raher han he four individual inflaion raes given in equaion (69 This formulaion is paricularly useful when we consider a simplificaion below under which s s is independen of policy The LCP opimizaion problem under discreion becomes very messy and difficul because of he addiional consrain given by equaion (7 Tha is because here now are 3

25 endogenous sae variables he choices of Home oupu gap relaive o he Foreign oupu gap and he deviaion from he law of one price pus consrains on he evoluion of fuure oupu gaps inflaion raes and deviaions from he law of one price In he LCP case he dynamic game beween curren and fuure policymakers is non-rivial Bu inspecion of equaion (7 reveals a special case in which he policy decision under uncerainy can be seled under he same simple condiions as in he PCP model When φ = so uiliy is linear in labor equaion (7 simplifies considerably Indeed we can rewrie i as: (73 s s = δs + βε ( s s + u u + Wih φ = we have expecaional difference equaion: s s s s a a = = ( β (74 s = s + Ε s+ + ϑ + δ + β + δ + β + δ + β where ϑ = δ( a a + u u = δs So we can wrie (73 as a second-order The poin here is ha equaion (74 deermines he evoluion of s independen of policy choices So while s is a sae variable i is no endogenous for he policymaker One nice hing abou considering his special case is ha he parameer φ does no appear in eiher he arge crieria or he opimal ineres rae rule in he CGG model so we can compare he crieria and rules direcly o he LCP model We also noe ha Devereux and Engel (3 make he same assumpion on preferences We can derive he firs-order condiions for he policymaker which do no depend on any assumpion abou he sochasic process for s We replace equaion (7 wih equaion (3 expressed in gap form as a consrain on he choice of opimal values by he policymaker In fac in his case he policy problem can be simplified more by using version (7 of he loss funcion A useful way o rewrie (7 when φ = is: σ σ ν( ν ξ 4D 4 4D 4δ R W R W (75 Ψ ( y ( y Δ (( π + ( π + ν( ν( s s where we are using he R superscrip o represen Home relaive o Foreign Tha is π = π π u = u u ec Likewise he W superscrip refers o he sum of Home and R R Foreign variables: π = π + π W W u = u + u ec Since s s is independen of policy we can express he policymaker s problem as choosing relaive and world oupu gaps R y and W y relaive and world CPI inflaion raes π R and π W and he currency misalignmen Δ o 4