Foreign Money Shocks and the Welfare Performance of Alternative Monetary Policy Regimes

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1 oregn Money Shocks and the Welfare Performance of Alternatve Monetary Polcy Regmes Ozge Senay and Alan Sutherland July 7, 005 Abstract A two-country stcky-prce general equlbrum model s used to examne the welfare propertes of alternatve monetary polcy regmes for a country whch s subject to foregn money shocks. The expendture swtchng effect of exchange rate changes s found to be mportant for the comparson between money targetng and a fxed rate. Money targetng s welfare superor to a fxed rate when the expendture swtchng effect s relatvely weak, but a fxed rate s superor when the expendture swtchng effect s strong. However, prce targetng s found to be superor to both a fxed rate and money targetng for all values of the expendture swtchng effect. nally, a welfare maxmsng monetary feedback rule s found to yeld lower output and exchange rate volatlty than prce targetng for a wde range of parameter values. Keywords: monetary regmes, expendture swtchng, welfare, foregn monetary shocks. JL: 5, 41, 4 Department of conomcs, Mddle ast Techncal Unversty, Ankara 06531, Turkey. -mal: osenay@metu.edu.tr. Web: Department of conomcs, Unversty of St Andrews, St Andrews, fe, KY16 9AL, UK. mal: ajs10@st-and.ac.uk. Web:

2 1 Introducton redman (1953) and other early proponents of floatng exchange rates argued that floatng exchange rates are desrable because they provde a degree of nsulaton aganst foregn monetary shocks. A floatng rate regme allows a country to set monetary polcy ndependently from monetary polcy n other countres. Ths lmts the transmsson of foregn monetary polcy shocks to the domestc economy. Thus, t was argued, floatng exchange rates act as a shock absorber whch helps to stablse the domestc economy n the face of foregn monetary shocks. Recent lterature on the choce of exchange rate regmes makes use of general equlbrum models wth stcky-prces and explct mcroeconomc foundatons. Ths new modellng approach allows a re-examnaton of the shock-absorber role of the exchange rate. In partcular, t allows a formal and systematc analyss of the mpact of regme choce on the welfare of agents (whch can be measured n terms of aggregate utlty). In ths paper we therefore use a two-country general equlbrum model to compare the welfare performance of fxed and a range of floatng exchange rate regmes n the presence of foregn monetary shocks. In ths way, we am to assess whether redman s arguments n favour of the shock absorbng propertes of floatng rate regmes hold true n a stochastc general equlbrum model of the type used n the recent nternatonal macroeconomcs lterature. Intally we defne a floatng rate to be a regme of money targetng. Our results show that the welfare performance of money targetng relatve to a fxed rate depends on the responsveness of demand to changes n the exchange rate (.e. the strength of the expendture swtchng effect). It s found that money targetng yelds hgher welfare than a fxed rate as long as the expendture swtchng effect s relatvely weak. But when the expendture swtchng effect s strong t s found that a fxed rate can yeld hgher welfare than money targetng. We show that there are two underlyng factors whch help explan ths result. The frst s that the welfare of the home populaton s strongly nfluenced by the volatlty of home output (.e. hgher output volatlty tends to reduce welfare). The second s that foregn monetary shocks create a volatlty trade-off for home monetary polcy. The volatlty of home output depends on the volatlty of world aggregate expendture and the volatlty of the nomnal exchange rate. A monetary polcy whch stablses the exchange rate tends to cause hgh volatlty of world aggregate expendture. 1 But a monetary polcy whch reduces the volatlty of world aggregate expendture tends to cause volatlty of the nomnal exchange rate. The relatve mpact of exchange rate volatlty and world expendture volatlty on the volatlty of output depends on the strength of the expendture swtchng effect. The strength of the expendture swtchng effect therefore has mportant mplcatons for 1 We focus on a two-country framework where the home country s large enough for ts monetary polcy to have an mpact on world aggregate expendture. In the case of a small open economy, wth some degree of home bas n consumpton expendture, home monetary polcy would have a comparable effect on the measure of aggregate expendture relevant for the home economy. Thus a smlar volatlty trade-off would exst n a small open economy settng. 1

3 therelatvewelfareperformanceofdfferent exchange rate regmes. When the expendture swtchng effect s relatvely weak, exchange rate fluctuatons have less mpact on home output. Thus, n ths case, money targetng yelds hgher welfare than a peg. But when the expendture swtchng effect s strong, exchange rate fluctuatons cause hgh volatlty n home output so a peg yelds hgher welfare than money targetng. Money targetng, whle correspondng to the polcy recommended by redman, s only one form of floatng exchange rate. A partcularly relevant alternatve form of floatng rate s a regme of prce targetng. Ths (approxmately) corresponds to nflaton targetng - whch s the regme adopted by many monetary authortes n recent years. Another mportant case to consder s the optmal polcy regme, where the monetary nstrument s adjusted n order to maxmse welfare. We extend our analyss to consder the welfare performance of ths wder set of polcy regmes. We fnd that prce targetng yelds hgher welfare than both money targetng and a fxed rate for all values of the expendture swtchng coeffcent. Ths s because (for a wde range of parameter values) prce targetng acheves a compromse between stablsaton of world aggregate demand and stablsaton of the exchange rate. It therefore les at a superor pont on the volatlty trade-off compared to the money targetng and fxed rate regmes. The volatlty of output thus tends to be lower, and welfare hgher, than n the other two regmes. But t s also found that prce targetng s not (n general) fully optmal. It s shown that optmal polcy (for a wde range of parameter values) mples some stablsaton of the nomnal exchange rate relatve to prce targetng. So optmal monetary polcy, even f t does not completely fx the exchange rate, should am to stablse t to some extent,.e. optmal monetary polcy s not entrely nward lookng. Ths paper s part of an extensve new lterature whch uses stochastc general equlbrum models to analyse the welfare effects of exchange rate and monetary regmes. Recent papers have focused on a range of dfferent aspects of ths ssue. Devereux and ngel (003), Corsett and Pesent (001), Sutherland (005a) and Bacchetta and van Wncoop (000) analyse the role of mperfect pass-through. Bengno and Bengno (003) and Sutherland (005b) examne the mpact of cost-push shocks on regme choce. And Devereux (004) and Bengno (001) consder the effects of dfferent fnancal market structures. Much of ths recent lterature has focused on models where the elastcty of substtuton between home and foregn goods Gven the mportance of the expendture swtchng effect for the analyss whch follows t s useful to note that the emprcal lterature does not provde any clear gudance on an approprate value for the nternatonal elastcty of substtuton between goods. Obstfeld and Rogoff (000) brefly survey some of the relevant lterature and quote estmates for the elastcty rangng between 1. and 1.4 for ndvdual goods (see Trefler and La (1999)). stmates for the average elastcty across all traded goods le n the range 5 to 6 (see for nstance Hummels (001)). Anderson and van Wncoop (003) also survey the emprcal lterature on trade elastctes and conclude that a value between 5 and 10 s reasonable. On the other hand, the real busness cycle lterature typcally uses a much smaller value for ths parameter. or nstance Char, Kehoe and McGrattan (00) use a value of 1.5 n ther analyss.

4 s restrcted to unty. The model presented below, however, allows ths parameter to dffer from unty, and thus allows an analyss of the role of the expendture swtchng effect. Ths s one mportant respect n whch the current paper departs from the exstng lterature. Much of the exstng lterature focuses on the welfare effects at the world level of the symmetrc choce of exchange rate regme. Ths paper, on the other hand, analyses the choce of regme from the pont of vew of a sngle (large) country. A thrd mportant departure from the exstng lterature s that ths paper analyses the choce of monetary regme n the face of foregn monetary shocks. Other authors have focused on models where productvty or labour supply shocks predomnate. 3 The paper proceeds as follows: Secton presents the model; Secton 3 descrbes the soluton method and the dervaton of the welfare measure; Secton 4 dscusses the lnks between monetary polcy and welfare. Secton 5 compares money targetng and a fxed nomnal exchange rare. Sectons 6 and 7 analyse prce targetng and optmal polcy respectvely. Secton 8 concludes the paper. The Model The model s a varaton of the stcky-prce general equlbrum structure whch has been wdely used n the recent open economy macroeconomcs lterature (followng the approach developed by Obstfeld and Rogoff (1995, 1998)). 4 Whle the general structure s standard, an mportant dfference between the model presented below and many others used n the recent lterature s that the elastcty of substtuton between home and foregn goods can dffer from unty. Ths allows an analyss of the mplcatons of the expendture swtchng effect for the relatve performance of dfferent regmes. The only source of stochastc shocks n the model s the foregn money supply. Home monetary polcy s modelled as a general feedback rule and the four possble regmes for the home monetary authorty consdered correspond to dfferent settngs for the coeffcent n ths feedback rule..1 Market Structure The world exsts for a sngle perod and conssts of two equal-szed countres, whch wll be referred to as the home country and the foregn country. There s a contnuum of agents of unt mass n each country wth home agents ndexed h [0, 1] and foregn agents ndexed f [0, 1]. Agents consume a basket of goods contanng all home and foregn produced goods. ach agent s a monopoly producer of a sngle dfferentated product. All agents set prces n advance of the realsaton of shocks 3 Sutherland (004), whle analysng the mpact of the expendture swtchng effect on monetary polcy n a small open economy, focuses exclusvely on supply shocks and does not consder the role of foregn monetary shocks. 4 See Lane (001) for a survey of ths lterature. 3

5 and are contracted to meet demand at the pre-fxed prces. Prces are set n the currency of the producer. The detaled structure of the home country s descrbed below. The foregn country has an dentcal structure. Where approprate, foregn real varables and foregn currency prces are ndcated wth an astersk.. Preferences All agents n the home economy have utlty functons of the same form. The utlty of agent h s gven by U (h) = log C (h)+χlog M (h) K P y (h) (1) where χ and K are postve constants, C s a consumpton ndex defned across all home and foregn goods, M denotes end-of-perod nomnal money holdngs, P s the consumer prce ndex, y (h) s the output of good h and s the expectatons operator. The consumpton ndex C for home agents s defned as C = " µ1 1 µ 1 θ θ 1 C θ 1 H + θ C θ 1 θ # θ θ 1 where C H and C are ndces of home and foregn produced goods defned as follows Z 1 C H = c H () φ 1 φ d φ Z φ 1 1, C = c (j) φ 1 φ dj φ φ 1 (3) 0 0 where φ>1, c H () s consumpton of home good and c (j) s consumpton of foregn good j. The parameter θ s the elastcty of substtuton between home and foregn goods. Ths s the key parameter whch determnes the strength of the expendture swtchng effect. The budget constrant of agent h s gven by M(h) =M 0 + p H (h) y(h) PC(h) T + PR(h) (4) where M 0 and M(h) are ntal and fnal money holdngs, T s a lump-sum government transfer, p H (h) s the prce of home good h, P s the aggregate consumer prce ndex and R(h) s the ncome from a portfolo of state contngent assets (to be descrbed n more detal below). The government s budget constrant s () M M 0 + T =0 (5) Changes n the money supply are assumed to enter and leave the economy va changes n lump-sum transfers. 4

6 .3 Prce Indces The aggregate consumer prce ndex for home agents s P = 1 P 1 θ H P 1 θ 1 θ (6) where P H and P are the prce ndces for home and foregn goods respectvely defned as Z 1 P H = 0 1 Z p H () 1 φ 1 φ 1 1 d, P = p (j) 1 φ 1 φ dj 0 (7) The law of one prce s assumed to hold. Ths mples p H () = p H () S and p (j) =p (j) S for all and j where an astersk ndcates a prce measured n foregn currency and S stheexchangerate(defned as the domestc prce of foregn currency). Purchasng power party holds n terms of aggregate consumer prce ndces, P = SP..4 Consumpton Choces Indvdual home demand for representatve home good, h, and foregn good, f, are gven by µ φ µ φ ph (h) p (f) c H (h) =C H, c (f) =C (8) P H P where C H = 1 µ θ C PH, C = 1 µ θ P C P (9) P oregn demands for home and foregn goods have an dentcal structure to the home demands. Indvdual foregn demand for representatve home good, h, andforegn good, f, aregvenby c H (h) =C H µ p H (h) P H φ, c (f) =C µ p φ (f) (10) where CH = 1 µ P θ H C, C P = 1 µ P θ C (11) P ach country has a populaton of unt mass so the total demands for goods are equvalent to ndvdual demands. The total demand for home goods s therefore Y = C H + CH and the total demand for foregn goods s Y = C + C. P 5

7 .5 Prce Settng Goods prces are chosen before shocks are realsed and are fxed at these pre-chosen levels. Prces are set n the currency of the producer. 5 The frst-order condton for prce settng for home agents mples the followng P H = φ K [Y ] (1) φ 1 [Y/(PC)] A smlar expresson can be derved for foregn agents, as follows P = φ K [Y ] (13) φ 1 [Y /(P C )] Notce that the expectatons terms n these expressons mply that a form of rsk premum s bult nto goods prces. Ths rsk premum arses because agents, who are rsk averse, face uncertanty over the level of work effort. The rsk premum depends on the varances and covarances of work effort, the margnal utlty of consumpton and consumer prces. In ths framework, where all goods prces are fxed, a prce-targetng regme s defned n terms of targetng the prce level that producers would set f they were able to respond to shocks. Ths prce level s smply gven by the expresson n (1) after removng the expectatons operators from the rght hand sde,.e. PH V = φ KY PC (14) φ 1 where the superscrpt V ndcates that ths s a vrtual or shadow prce level. Ths structure can be thought of as a lmtng case of a more general model where the populaton s splt nto a set of flexble-prce agents and a set of fxed-prce agents. The shadow prce PH V corresponds to the prce of flexble-prce producers as the proporton of flexble-prce producers tends to zero. 6.6 nancal Markets and Rsk Sharng The asymmetrc structure of shocks and monetary polcy, coupled wth a non-unt elastcty of substtuton between home and foregn goods, makes t necessary to adopt a more explct structure for nternatonal asset markets than s usual n the recent lterature. 7 It s assumed that suffcent contngent fnancal nstruments exst 5 Other contrbutons to the recent lterature have shown that the currency of prcng s potentally mportant for the relatve welfare performance of exchange rate regmes. Ths ssue s not the man focus of the current paper, so, for smplcty, we concentrate on the case of producer currency prcng. 6 See Sutherland (004) for an example of a model where the proporton of flexble prce producers s strctly greater than zero. 7 When θ s equal to unty, the trade balance between the two countres automatcally balances n all states of the world, n whch case fnancal markets are rrelevant. When θ 6= 1t becomes necessary to consder the structure of fnancal markets. Addtonally, when shocks are asymmetrc and when the focus of nterest s the polcy choce and welfare of a sngle country, t becomes necessary explctly to consder how polcy choces affect asset prces and portfolo decsons. 6

8 to allow effcent sharng of consumpton rsks. All consumpton s fnanced out of real ncome so the only source of consumpton rsk s varablty n real ncome. ffcent sharng of consumpton rsk can therefore be acheved by allowng trade n two state-contngent assets, one whch has a pay-off correlated wth home aggregate real ncome and one wth a pay-off correlated wth foregn real ncome. or smplcty, t s assumed that each asset pays a return equal to the relevant country s real ncome,.e. a unt of the home asset pays y = YP H /P and a unt of the foregn asset pays y = Y P /P. 8 The portfolo pay-offs for home and foregn agents are gven by the followng R (h) =ζ H (h)(y q H )+ζ (h)(y q ) (15) R (f) =ζ H (f)(y q H )+ζ (f)(y q ) (16) where ζ H (h) and ζ (h) are holdngs of home agent h of the home and foregn assets, ζ H (f) and ζ (f) are the holdngs of foregn agent f of home and foregn assets and q H and q are the unt prces of the home and foregn assets. It s mportant to specfy the tmng of asset trade. It s assumed that asset trade takes place after the choce of exchange rate regme. Ths mples that agents can nsure themselves aganst the rsk mpled by a partcular exchange rate regme, but they can not nsure themselves aganst the choce of regme. 9 The Appendx shows that rsk sharng mples the followng relatonshp between consumpton, asset prces and expected output levels n the two countres C C = q H q =.7 Money Demand and Supply h y y+y The frst-order condton for the choce of money holdngs s h y y+y (17) M = χc (18) P The money supply n each country s assumed to be determned by the relevant natonal monetary authorty. The foregn money supply s subject to stochastc 8 Note that asset pay-offs are correlated wth aggregate ncome. Indvdual agents therefore treat pay-offs as exogenous. Ths mples that the exstence of contngent assets has no drect mpact on optmal prce settng. 9 If, alternatvely, asset trade takes place before the exchange rate regme s chosen, t would be possble for agents to nsure themselves aganst the choce of regme. Ths could have very sgnfcant mplcatons for the optmal choce of regme. The home monetary authorty would be tempted to choose a regme whch mples hgh volatlty of demand for home goods. The hgh volatlty of demand would dscourage home labour supply and reduce home work effort but the level of home consumpton would be protected by the rsk-sharng arrangement. Ths alternatve rsk-sharng structure rases some nterestng ssues but t also nvolves some techncal problems whch go beyond the scope of ths paper. 7

9 shocks such that log M s symmetrcally dstrbuted over the nterval [, ] wth [log M ]=0and Var[log M ]=σ. Home monetary polcy s defned n terms of a general feedback rule for the home money supply of the followng form µ M = M M δ (19) M In what follows the value of δ wll be determned by the monetary regme under consderaton. We consder four dfferent regmes: a fxed nomnal exchange rate; money targetng; prce targetng; and welfare maxmsng monetary rule. In the case of a fxed exchange rate, the home monetary authorty chooses δ so that the exchange rate s mantaned at a target level, S. In the case of money targetng, the home monetary authorty sets δ =0so that the home money supply s constant at M. In the case of prce targetng, δ s chosen to mantan the vrtual producer-prce level, PH V,atatargetlevel, P H V.Andfnally, n the case of optmal polcy, δ s chosen to maxmse home welfare. The approprate values of δ for each regme are derved below. Note that the values of M, S and V P H only serve to provde an anchor for the equlbrum level of nomnal varables. The equlbrum level of nomnal varables has no effect on real varables or on welfare and s thus rrelevant to the analyss presented below. 3 Welfare and Model Soluton Our man objectve s to analyse the welfare performance of dfferent monetary polcy regmes. The dfferent regmes are represented by dfferent choces for the polcyfeedbackparameterδ, sothefrst task s to derve an expresson for welfare ntermsofthsparameter. Aggregate welfare of home agents s measured usng the followng 10 K Ω = log C Y (0) It s not possble to derve an exact soluton to the model descrbed above. 11 A second-order approxmaton of the welfare expresson s therefore derved. Before proceedng t s necessary to defne and explan some notaton. The nonstochastc equlbrum of the model s defned as the soluton whch results when M = 1 wth σ =0. or any varable X defne ˆX = log X/ X where X s the value of varable X n the non-stochastc equlbrum. ˆX s therefore the logdevaton of X from ts value n the non-stochastc equlbrum. 10 ollowng Obstfeld and Rogoff (1998, 00) t s assumed that the utlty of real balances s small enough to be neglected. 11 The complcaton arsng n ths model s contaned n equaton (6). When θ s dfferent from unty ths equaton s not lnear n logs. 8

10 The only exogenous forcng varable n the model s the foregn money supply, M, so all log-devatons from the non-stochastc equlbrum are of the same order as the shocks to ˆM, whch (by assumpton) are of maxmum sze. Whenpresentng an equaton whch s approxmated up to order n t s therefore possble to gather all terms of order hgher than n n a sngle term denoted O ( n+1 ). 1 A second-order approxmaton of the welfare measure s gven by o Ω = nĉ K Ȳ hŷ + Ŷ + O 3 (1) where Ω s the devaton of the level of welfare from the non-stochastc equlbrum. 13 Notce that ths expresson ncludes the frst moments of consumpton and output and the second moment of output. Welfare s ncreasng n the expected level of consumpton and decreasng n the expected level and varance of output. Second-order accurate solutons for varances can be obtaned from frst-order accurate solutons for the relatonshps between endogenous varables and the shock varable. The analyss of volatlty therefore nvolves workng wth a log-lnearsed (.e. frst-order approxmated) verson of the model. But a full second-order expresson for welfare requres second-order accurate solutons for both the frst and second moments of varables. So, a full analyss of welfare nvolves workng wth a second-order approxmaton of the model. The detaled dervaton of a second-order accurate soluton of the model s descrbed n the Appendx. The resultng expressons for the expected level of output and consumpton and the varance of output are [Ĉ] =1 6+3θ θ +( +θ + θ )δ ( +5θ + θ )δ σ + O 3 () 8 [Ŷ ]= 1 4 [1 + δ θ + θδ] σ + O 3 (3) [Ŷ ]= 1 4 [1 + δ θ + θδ] σ + O 3 (4) Thus welfare can be wrtten as a functon of δ, the feedback parameter n the polcy rule, as follows Ω = 1 6+3θ θ +( +θ + θ )δ ( +5θ + θ )δ σ + O 3 (5) 8 1 Thus, when the term O appears n an equaton the varables n that equaton should be understood to be accurate up to order one. Whle an equaton whch ncludes the term O 3 should be understood to contan varables whch are accurate up to order two. And an equaton whch does not nclude any term of the form O ( n ) should be understood to hold exactly. 13 In the non-stochastc equlbrum ndvdual budget constrants mply that P C = Ȳ P H. Combnng ths expresson wth equaton (1) shows that Ȳ =[Kφ/(φ 1)] 1/, thus KȲ = (φ 1) /φ. It s apparent from ths expresson that Ȳ depends on the monopoly mark-up, φ/(φ 1). A common practce n the recent lterature s to ntroduce a producton subsdy to offset the monopoly dstorton so that Ȳ s at ts welfare maxmsng level. It wll become apparent below that the man welfare results n our paper are ndependent of the value of Ȳ and are therefore ndependent of the monopoly dstorton (and would therefore also be ndependent of any producton subsdy). 9

11 Note from (3) and (4) that [Ŷ ]+[Ŷ ]=0,sowelfareseffectvely determned by [Ĉ] alone.14 4 Monetary Polcy and Welfare Before analysng the ndvdual monetary polcy rules t s useful to trace and explan the man lnkages between monetary polcy and welfare. As shown above, because [Ŷ ]+[Ŷ ]=0, welfare depends drectly (and only) on the expected level of consumpton, [Ĉ]. The expected level of consumpton depends, va ndvdual country and world resource constrants, on the expected level of output. And the expected level of output depends (amongst other thngs) on monetary polcy va the mpact of monetary polcy on the volatlty of output. As a general rule, hgher output volatlty tends to dscourage work effort (snce agents are rsk averse). Thus, a monetary rule whch leads to hgh output volatlty tends to lead to a low expected level of output. And, conversely, a monetary rule whch leads to low output volatlty tends to lead to hgh expected output. 15 The mpact of the expected level of output on the expected level of consumpton and welfare can be postve or negatve dependng on the value of θ (the elastcty of substtuton between home and foregn goods). There are thus two mportant lnks between monetary polcy and welfare. rst, the lnk between monetary polcy and the volatlty of output. And second, the lnk between the expected level of output and the expected level of consumpton. It s useful to set out some of the mportant equatons whch explan these two lnks before consderng each of the ndvdual monetary polcy rules. 4.1 Monetary Polcy and Output When consderng the mpact of monetary polcy on the volatlty of output t s suffcent to look at frst-order accurate solutons to the model. 16 A frst-order expanson of equaton (17) shows that rsk sharng mples the followng relatonshp between realsed consumpton levels n the two countres Ĉ Ĉ =0+O where, as explaned above, the term O ( ) ndcates that the varables n ths relatonshp should be understood to be accurate up to a frst-order approxmaton. When combned wth the purchasng power party relatonshp (whch mples 14 Thus, as prevously stated, the value of Ȳ s rrelevant. 15 It s mportant to note that the level of expected output does not depend exclusvely on the volatlty of output. Other factors are also mportant, such as the volatlty of consumpton and prces and the covarances between output, consumpton and prces. 16 Terms of order two and hgher n expressons for realsed values become terms of order three and hgher n expressons for varances. Hgher order terms n expressons for realsed values are therefore rrelevant for the second-order accurate analyss of welfare. 10

12 Ŝ = ˆP ˆP ) and the expressons for home and foregn money demand (whch mply ˆM = ˆP + Ĉ and ˆM = ˆP + Ĉ ), the followng expresson for the exchange rate s obtaned Ŝ = ˆM ˆM + O (6) Thus the nomnal exchange rate depends on relatve monetary polcy,.e. the dfference between the home money supply and the foregn money supply. The assumpton of fxed goods prces mples ˆP H = ˆP =0+O ( ), so aggregate consumer prces are gven by ˆP = 1 Ŝ + O, ˆP = 1 Ŝ + O (7) These expressons, combned wth the money demand relatonshps, mply that consumpton levels are Ĉ = Ĉ = 1 ³ ˆM + ˆM + O (8) Thus consumpton n the two countres responds equally (because of the rsk-sharng structure) to aggregate world monetary polcy,.e. the sum of home and foregn money supples. rst-order approxmatons for home and foregn aggregate output levels yeld Ŷ = 1 ³Ĉ + Ĉ + θ Ŝ + O (9) Ŷ = 1 ³Ĉ + Ĉ θ Ŝ + O (30) These expressons show that the realsed value of output depends on aggregate monetary polcy (va Ĉ + Ĉ ) and relatve monetary polcy (va Ŝ). Thus homecountrymonetarypolcyaffects the volatlty of home output va ts mpact on the volatlty of aggregate monetary polcy ( ˆM + ˆM ) and ts mpact on relatve monetary polcy ( ˆM ˆM ). Notce that, gven the exogenous stochastc shocks affectng M, the home monetary authorty faces a trade-off: a monetary rule whch stablses ( ˆM + ˆM ) wll necessarly destablse ( ˆM ˆM ) and vce versa. The relatve mportance of these two factors for the volatlty of output wll depend on the value of θ. The larger the value of θ (.e. the stronger the expendture swtchng effect) the more mportant s volatlty n ( ˆM ˆM ). The lnk between monetary polcy and output volatlty therefore depends on the value of θ. Ths s one respect n whch the welfare performance of dfferent monetary polcy regmes s affected by the strength of the expendture swtchng effect. 4. xpected Output and Consumpton xpected levels of varables depend on second moments so, when consderng the lnk between the expected level of output and the expected level of consumpton, t s necessary to consder a second-order accurate soluton to the model. 11

13 Second-order expansons of the home and foregn output relatonshps yeld 17 1 hŷ = ³Ĉ + Ĉ θ + O ˆτ 3 (31) hŷ 1 = ³Ĉ + Ĉ + θ + O ˆτ 3 (3) rom whch t follows that hĉ + Ĉ = hŷ + Ŷ + O 3 (33) Thus the expected level of world aggregate consumpton [Ĉ + Ĉ ] must equal the expected level of world aggregate output [Ŷ + Ŷ ]. Ths shows one part of the lnk between home output and home consumpton. An ncrease n home output rases the total of output avalable for consumpton n the world. The second part of the lnk between home output and home consumpton s determned by the consumpton rsk sharng relatonshp, (17). Ths relatonshp determnes how expected consumpton s shared between the two countres. A second-order expanson of equaton (17) shows that rsk sharng mples that the frst moments of consumpton and output n the two countres are related as follows hĉ Ĉ = hŷ Ŷ +ˆτ + O 3 (34) where τ s the terms-of-trade. Thus relatve consumpton, (Ĉ Ĉ ), depends on relatve output, (Ŷ Ŷ ), and the terms of trade. It s smple to show, usng equatons (31) and (3), that relatve output and the terms of trade are related as follows hŷ Ŷ = θ [ˆτ]+O 3 (35) Ths shows that an ncrease n home output relatve to foregn output requres a fall n the prce of home goods relatve to the prce of foregn goods (.e. a fall n the terms of trade). The sze of the requred adjustment n ˆτ clearly depends on θ, the elastcty of substtuton between home and foregn goods. Combnng (34) and (35) mples hĉ Ĉ =(θ 1) hŷ Ŷ + O 3 (36) quatons (33) and (36) show that the mpact of the expected level of home output on the expected level of home consumpton depends on the mpact of home output on world aggregate consumpton (equaton (33)) and the mpact on relatve consumpton (equaton (36)). When θ>1 an ncrease n home output rases both 17 In general, equatons (31) and (3) should nclude terms whch depend on the second moments of home and foregn consumpton. However, the perfect cross-country correlaton of consumpton levels mples that these terms are equal to zero. They are therefore omtted from (31) and (3). 1

14 aggregate consumpton, [Ĉ +Ĉ ], and relatve consumpton, [Ĉ Ĉ ],sotheres an unambguous ncrease n [Ĉ]. Butwhenθ<1an ncrease n home output leads to an ncrease n aggregate consumpton but a reducton n relatve consumpton. In ths case, there s an ambguous effect on [Ĉ]. If θ s suffcently small an ncrease n home output can cause a decrease n [Ĉ]. The lnk between expected output and expected consumpton thus depends on the value of θ. Ths s another respect n whch the relatve welfare performance of dfferent monetary polcy regmes wll depend on the strength of the expendture swtchng effect. 5 Money Targetng versus a xed Rate The essental feature of each of the targetng rules consdered n ths paper s that each rule ensures that the varance of the targeted varable s zero. Thus the money targetng rule ensures that the varance of M s zero. And the fxed exchange rate rule ensures that the varance of S s zero. The relevant value of δ, the polcy feedback parameter, n each case can be derved by settng the varance of the relevant targeted varable to zero. Second-order accurate expressons for varances can be obtaned by consderng frst-order accurate solutons for realsed values. So, the value of δ for each rule can be derved by consderng frst-order accurate expressons for the targeted varables. The money supply rule (19) mples ˆM = δ ˆM. Thus, n the case of money targetng, the varance of ˆM s zero when δ =0. It was shown above (equaton (6)) that the realsed value of the exchange rate s gven by Ŝ = ˆM ˆM + O ( ). Ths mmedately shows that the varance of the exchange rate s zero when the home money supply s set equal to the foregn money supply,.e. ˆM = ˆM.Sothefxedrateregmemplesδ =1. The welfare level yelded by the money targetng and fxed rate regmes can now be derved by substtutng the relevant value of δ nto the welfare expresson (5). The resultng expressons are shown n Table 1. Ths table also shows the mpled varances of output, the exchange rate and consumpton n the fxed rate and money targetng regmes. It follows from the welfare expressons shown n Table 1 that the money targetng regme yelds hgher welfare than the fxedrateregmewhenθ Thus money targetng yelds hgher welfare when the expendture swtchng effect s relatvely weak, but a fxed exchange rate regme s superor when the expendture swtchng effect s strong. These results can be understood by consderng the mpact of exchange rate volatlty and the expendture swtchng effect on output volatlty. It was shown above n equaton (9) that monetary polcy affects the volatlty of home output va ts mpact on aggregate consumpton and va ts mpact on the exchange rate. Money targetng (.e. δ =0) mples a low varance of aggregate consumpton but hgh varance of the exchange rate. Ths contrasts wth a fxed exchange rate regme 13

15 x MT PT OPT δ 1 0 θ 1 θ+3 Ω σ 1 8 hŷ hŝ hĉ σ θ +θ θ +5θ ³ ³ 6+3θ θ σ 5 4θ+θ σ 5θ+θ σ (3+θ) +5θ+θ 1 θ σ 0 σ 4 3+θ 1 θ 3+θ σ σ 1 1+θ 4 σ 3+θ σ ³ σ ³ 3θ+θ +5θ+θ σ 4θ +5θ+θ σ ³ +3θ+θ +5θ+θ σ Table 1: Key xpressons for the our Monetary Polcy Regmes (.e. δ =1)whch mples a hgh varance of consumpton and a completely stable exchange rate. The relatve mpact of consumpton varablty and exchange rate varablty on the volatlty of output depends on the strength of the expendture swtchng effect. or low values of θ, the varablty of the exchange rate s relatvely unmportant so money targetng mples relatvely low output volatlty. 18 But for hgh values of θ, exchange rate volatlty becomes relatvely more mportant so a fxed exchange rate tends to delver much lower output volatlty than money targetng. The net result s that welfare n the money targetng regme declnes as the expendture swtchng effect becomes stronger. Ths s because of the negatve mpact of output volatlty on the expected level of output and consumpton. or large values of θ, ths effect can become so strong that t mples that a fxed rate regme s welfare superor to a money targetng regme. 6 Prce Targetng The prevous secton compared money targetng wth a fxed rate. However, money targetng s only one form of floatng rate regme. A partcularly relevant alternatve form of floatng rate regme s one where the monetary authorty uses polcy to stablse prces,.e. a prce targetng regme. Ths secton analyses a regme of ths type and compares t to money targetng and a fxed rate. In the prce targetng case the feedback term n the rule (equaton (19)) s chosen to stablse the producer prce level. Because all prces are assumed to be pre-fxed, 18 In fact, because consumpton and the exchange rate are negatvely correlated n a money targetng regme, the varance of output wll be very low for θ close to unty. 14

16 prce targetng s defned n terms of stablsng vrtual prces (.e. the prces that producers would choose f prces were fully flexble). 19 In order to derve the value of δ t s necessary to derve a frst-order accurate expresson for PH V as a functon of the money supply. A frst-order expanson of (14), combned wth (6), (7), (8) and (9) mples that ˆP V H = 3+θ ˆM + 1 θ ˆM + O Ths expresson shows that the varance of ˆP V H s zero when δ = θ 1 θ +3 (37) When ths value for the δ s substtuted nto (5), an expresson for welfare n the prce targetng regme s obtaned. Ths s shown n Table 1. A comparson of the welfare expressons n Table 1 shows that prce targetng, for all θ 6= 1, yelds hgher welfare than both money targetng and the fxed rate. (In the case where θ =1, money targetng and prce targetng are equvalent.) 0 In order to understand these results, frst consder the case where θ>1. Notce that, n ths case, prce targetng mples that 0 <δ<1. Thus monetary polcy responds postvely to a foregn monetary shock, but less than one-for-one. In ths sense, prce targetng s a compromse between money targetng (where δ =0)and a fxed exchange rate (where δ =1). Ths n turn mples that prce targetng yelds lower exchange rate volatlty than money targetng and lower aggregate consumpton volatlty than a fxed exchange rate. The net effectsthatprcetargetng yelds lower output volatlty than both money targetng and a fxed rate. So, prce targetng leads to lower rsk for producers and therefore a hgher expected level of output and consumpton. And hgher expected consumpton mples a hgher level of welfare than both the money targetng and a fxedexchangerateregmes. Inthecasewhenθ<1, the explanaton s more complex. In ths case δ<0. Ths mples that prce targetng yelds hgher exchange rate and consumpton volatlty, and thus hgher output volatlty, than the other two regmes. Yet, prce targetng stll yelds hgher welfare than the other two regmes. The key to understandng the contrast between the θ>1case and the θ<1case les n the lnk between the expected level of output and the expected level of home consumpton. When θ>1, an ncrease n home output leads to an ncrease n both aggregate expected consumpton and relatve expected consumpton (see equatons (36) and (33)). Ths means that a monetary polcy whch leads to an ncrease n 19 An alternatve prce targetng regme would be one that stablses consumer prces. However, we focus on producer prce targetng because t s not straghtforward, n the context of our fxedprce model, to defne a vrtual consumer prce ndex. 0 Sutherland (004) shows that, n a smlar model, for very hgh values of θ a fxed rate can yeld hgher welfare than prce targetng. Sutherland, however, focuses on a small open economy case where the man sources of dsturbances are home and foregn labour supply shocks. 15

17 the expected level of home output wll also lead to an ncrease n the expected consumpton possbltes for home consumers and thus an ncrease n home welfare. On the other hand, equaton (33) shows that when θ<1an ncrease n home output causes a fall n relatve consumpton for home agents whch can outwegh the rse n aggregate consumpton. Thus, n contrast to the θ>1case, a monetary polcy whch leads to a decrease n the expected level of home output can, n the θ<1 case, nevertheless lead to an ncrease n expected home consumpton. Ths explans why prce targetng, whch generates hgh exchange rate and output volatlty when θ < 1, nevertheless yelds hgher welfare than the other two regmes. The hgh output volatlty reduces expected home output whch n turn, ncreases expected home consumpton and welfare. 7 Optmal Polcy The prevous sectons have compared three smple monetary polcy regmes. ach regme mples a dfferent value for the polcy feedback parameter δ. A natural benchmark aganst whch to compare these smple polcy rules s the optmal regme where the value of δ s chosen to maxmse welfare. The optmal value of δ s obtaned by maxmsng welfare, as gven n (19), wth respect to δ. The frst order condton mples δ OP T = θ + θ θ +5θ (38) It s useful to note that the second-order condton for ths maxmsaton problem mples that 5θ θ < 0, whch n turn mples that a vald maxmum only exsts for θ>( 5+ 33)/ The welfare level yelded by the optmal feedback rule s shown n Table 1. A comparson of the welfare results n Table 1 shows that, n general, prce targetng s not fully optmal. Only n the case where θ =1does prce targetng correspond to the welfare maxmsng polcy. Notce that, n ths case money targetng also corresponds to the welfare maxmsng polcy. A fxed exchange rate, on the other hand, does not correspond to the optmal polcy rule for any value of θ. 1 Other authors (e.g. Goodfrend and Kng (001), Kng and Wolman (1999) and Clarda, Gal and Gertler (001)) have argued that prce targetng s a welfare maxmsng polcy because t reproduces the flexble prce equlbrum and thus perfectly 1 Another way of representng optmal monetary polcy s to derve the optmal targetng rule. The optmal targetng rule expresses polcy n terms of a rule for stablsng a weghted average of varables. In the current model t can be shown that the optmal targetng rule s the rule whch stablses T OP T,whereT OPT θ ˆP H V +(θ 1)Ŝ. It s smple to show that the monetary feedback rule wth δ = δ OP T can equvalently be represented n terms of the optmal feedback rule or n terms of the optmal targetng rule. Notce that the optmal targetng rule shows that prce targetng s not fully optmal n general but t s optmal when θ =1. The optmal targetng rule also shows that a fxed rate wll not be fully optmal for any value of θ. 16

18 offsets the dstortons created by stcky prces. Why then does prce targetng not correspond to the welfare maxmsng polcy for all values of θ? There are two reasons for ths. rstly, prce targetng n the home economy does not reproduce the flexble prce equlbrum because there are dstortons that arse from stcky prces n the foregn country. Secondly, the flexble prce equlbrum s not welfare maxmsng from the home country s pont of vew because the home country has monopoly power over the supply of home goods. In a stcky-prce world, the home country can use monetary polcy to explot ths monopoly power to rase welfare above the level yelded by the flexble prce equlbrum. Theszeofthewelfaredfference between prce targetng and optmal polcy s relatvely small for all values of θ. Ths s llustrated n gure 1 whch plots the welfare levels of the four polcy regmes aganst values of θ. The welfare dfference wth the other regmes dverges for large values of θ. The level of welfare yelded by money targetng declnes partcularly rapdly as θ s ncreased. It s smple to check usng the expressons n Table 1 that, when θ>1, optmal polcy yelds lower exchange rate and output volatlty than prce targetng. Ths s llustrated n gures and 3, whch plot exchange rate and output volatlty for the four regmes. 3 Thus, whle a fxed exchange rate s not optmal, some degree of exchange rate stablsaton s optmal relatve to pure prce targetng (n the θ>1 case). In other words, optmal polcy s not purely nward-lookng. The degree of exchange rate stablsaton compared to the prce targetng rule s however relatvely small. As explaned n the prevous secton, for θ<1the welfare effects of exchange rate and output volatlty are reversed -.e. output volatlty can be welfare ncreasng. So the optmal polcy rule tends to generate hgh output and exchange rate volatlty when θ<1. gure shows that optmal polcy tends to lead to more exchange rate volatlty than prce targetng. And, for partcularly low values of θ, the optmal polcy tends to generate substantally more exchange rate and output volatlty than all three of the other regmes. 8 Conclusons Ths paper has analysed the welfare propertes of a range of monetary regmes n the presence of foregn monetary shocks. Our objectve was to nvestgate whether redman s (1953) arguments n support of floatng exchange rate regmes contnue or the purposes of ths fgure σ s set equal to 0.1. The values of welfare plotted on the vertcal axs can be nterpreted n terms of steady-state consumpton unts. or nstance, Ω = 1 represents a welfare devaton equvalent to a one percent devaton of consumpton from ts non-stochastc equlbrum value. 3 The explanaton for the lnk between monetary polcy and welfare gven above emphasses the mportance of the volatlty of output. Ths may suggest that a monetary polcy rule whch stablses output wll be optmal. The results reported n Table 1 and llustrated n gure 3 shows, however, that optmal polcy does not, n general, fully stablse output. Ths s because, as noted n footnote 15, the volatlty of output s not the only determnant of the expected level of output. 17

19 to hold true n a stochastc general equlbrum model wth a mcrofounded welfare crteron. In our analyss the degree of expendture swtchng was found to be an mportant factor n the comparson between regmes. An analyss of the fxed rate and money targetng regmes shows that money targetng yelds hgher welfare when the expendture swtchng effect s relatvely weak, but a fxed exchange rate regme s superor when the expendture swtchng effect s strong. A prce targetng regme, however, yelds hgher welfare than both a fxed rate and money targetng for all values of the expendture swtchng parameter. Nevertheless, prce targetng s not fully optmal. An analyss of a welfare maxmsng monetary rule showed that optmal polcy should nvolve some measure of exchange rate stablsaton relatve to the prce targetng regme when the elastcty of substtuton between home and foregn goods s greater than unty (.e. the expendture swtchng effect s relatvely strong). On the other hand, when the elastcty of substtuton s less than unty, optmal polcy leads to hgher exchange rate volatlty than the other regmes. Our results suggest, that whle redman s ntuton s supported n the sense that a fxed rate regme s always domnated by a floatng rate regme of one form or another, money targetng (redman s preferred monetary regme) s not always better than a fxed rate. Ths s manly because money targetng can generate hgh exchange rate volatlty whch has destablsng effects on output. Other floatng rate regmes, such as prce targetng, welfare domnate both money targetng and a fxed rate regme. References [1] Anderson, James and rc van Wncoop (003) Trade Costs unpublshed manuscrpt, Boston College and Unversty of Vrgna. [] Bacchetta, Phlppe and rc van Wncoop (000) Does xchange Rate Stablty Increase Trade and Welfare? Amercan conomc Revew, 90, [3] Bengno, Perpaolo (001) Prce Stablty wth Imperfect nancal Integraton CPR Dscusson Paper No 854. [4] Bengno, Ganluca and Perpaolo Bengno (003) Desgnng Rules for Internatonal Monetary Polcy Cooperaton, uropean Central Bank Workng Paper No 79. [5] Char, V. V., Patrck J. Kehoe and llen R. McGrattan (00) Can Stcky Prce Models Generate Volatle and Persstent Real xchange Rates? Revew of conomc Studes, 69, [6] Clarda, Rchard H, Jord Gal and Mark Gertler (001) Optmal Monetary PolcynOpenversusClosedconomes:AnIntegratedApproach Amercan conomc Revew (Papers and Proceedngs), 91,

20 [7] Corsett, Gancarlo and Paolo Pesent (001) Internatonal Dmensons of Optmal Monetary Polcy NBR Workng Paper No 830. [8] Devereux, Mchael B. (004) Should the xchange Rate be a Shock Absorber? Journal of Internatonal conomcs, 6, [9] Devereux, Mchael B. and Charles ngel (003) Monetary Polcy n an Open conomy Revsted: Prce Settng and xchange Rate lexblty Revew of conomc Studes, 70, [10] redman, Mlton (1953) The Case for lexble xchange Rates n Mlton redman (ed) ssays n Postve conomcs, Unversty of Chcago Press, Chcago. [11] Goodfrend, Marvn and Robert G. Kng (001) The Case for Prce Stablty NBR Workng Paper No 843. [1] Hummels, Davd (001) Towards a Geography of Trade Costs unpublshed manusrpt, Purdue Unversty. [13] Kng, Robert G. and Alexander L. Wolman (1999) What Should the Monetary Authorty Do When Prces are Stcky? n John B Taylor (ed) Monetary Polcy Rules, Unversty of Chcago Press, Chcago. [14] Lane, Phlp (001) The New Open conomy Macroeconomcs: A Survey Journal of Internatonal conomcs, 54, [15] Obstfeld, Maurce and Kenneth Rogoff (1995) xchange Rate Dynamcs Redux Journal of Poltcal conomy, 103, [16] Obstfeld, Maurce and Kenneth Rogoff (1996) oundatons of Internatonal Macroeconomcs, MIT Press, Cambrdge, MA. [17] Obstfeld, Maurce and Kenneth Rogoff (1998) Rsk and xchange Rates NBR Workng Paper [18] Obstfeld, Maurce and Kenneth Rogoff (000) The Sx Major Puzzles n Internatonal Macroeconomcs: Is There a Common Cause? NBR Macroeconomcs Annual, 15, [19] Obstfeld, Maurce and Kenneth Rogoff (00) Global Implcatons of Self- Orented Natonal Monetary Rules Quarterly Journal of conomcs, 117, [0] Sutherland, Alan J. (00) A Smple Second-Order Soluton Method for Dynamc General qulbrum Models CPR Dscusson Paper

21 [1] Sutherland, Alan J. (004) The xpendture Swtchng ffect, Welfare and Monetary Polcy n an Open conomy IIIS Dscusson Paper No, Trnty College Dubln. [] Sutherland, Alan J. (005a) Incomplete Pass-Through and the Welfare ffects of xchange Rate Varablty Journal of Internatonal conomcs, 65, [3] Sutherland, Alan J. (005b) Cost-Push Shocks and Monetary Polcy n Open conomes Oxford conomc Papers, 57, [4] Trefler, Danel and Huwen La (1999) The Gans from Trade: Standard rrors wth the CS Monopolstc Competton Model unpublshed manuscrpt, Unversty of Toronto. 0

22 Appendx Portfolo Allocaton, Asset Prces and Rsk Sharng There are four frst-order condtons for the choce of asset holdngs. After some rearrangement they mply the followng four equatons C 1 y = C 1 q H, C 1 y = C 1 q (39) C 1 y = C 1 q H, C 1 y = C 1 q (40) The combnaton of the prvate and government budget constrants and the portfolo pay-off functons for each country mply that aggregate home and foregn consumpton levels are gven by C = y + ζ H (y q H )+ζ (y q ) (41) C = y + ζ H (y q H )+ζ (y q ) (4) where n a symmetrc equlbrum ζ H (h) =ζ H and ζ (h) =ζ for all h and ζ H(f) = ζ H and ζ (f) =ζ for all f. qulbrum n asset markets mples ζ H + ζ H =0and ζ + ζ =0. These equatons can be used to solve for q H,q,ζ H,ζ,ζ H,ζ,C and C n terms of y and y. Usng the soluton procedure outlned n Obstfeld and Rogoff (1996, pp. 30-3) t s possble to show that the two asset prces are gven by h y q H = h, q = h (43) 1 1 y+y y+y y+y h y y+y and consumpton levels n the two countres are gven by Thus C = q H (y + y ) q H + q, C = q (y + y ) q H + q (44) C C = q H q = whch s equaton (17) n the man text. Model Soluton h y y+y h y y+y (45) A second-order accurate soluton to the model s derved usng a sngle-perod verson of the soluton method descrbed n Sutherland (00). Second-order expansons of (1) and (13) yeld ˆP H = hŷ + ˆP + Ĉ + λ PH + O 3 (46) 1

23 ˆP = hŷ + ˆP + Ĉ + λ P + O 3 (47) where λ PH = 1 4Ŷ ³Ŷ Ĉ ˆP λ P = 1 4Ŷ ³Ŷ Ĉ ˆP (48) (49) Notce that these expressons both nclude terms (denoted λ PH and λ P ) whch depend on the second moments of output, consumpton and consumer prces. These terms represent a form of rsk premum whch s bult nto goods prces by rsk-averse agents who have to set prces before shocks are realsed. The rsk premum depends on the varances and covarances of work effort, the margnal utlty of consumpton and the consumer prces. The money demand and supply relatonshps mply that h h ˆP + Ĉ =0, ˆP + Ĉ =0 (50) Note that the money demand relatonshps are lnear n logs so they do not requre any approxmaton. The expressons for home and foregn goods prces therefore smplfy to ˆP H = hŷ + λ PH + O 3, ˆP = hŷ + λ P + O 3 (51) These expressons can be combned wth second-order expansons of the defntons of consumer prces to yeld h ˆP = 1 λ P H + 1 λ P + 1 hŷ + Ŷ + Ŝ + [λ CPI ]+O 3 (5) h ˆP = 1 λ P H + 1 λ P + 1 hŷ + Ŷ Ŝ + [λ CPI ]+O 3 (53) where λ CPI = 1 8 (1 θ)ŝ (54) Notce that the non-log-lnearty of consumer prces gves rse to another secondorder term (denoted λ CPI ). The expressons for consumer prces can be combned wth the money market equatons to yeld the followng expressons for consumpton h hĉ = ˆP = 1 λ P H 1 λ P 1 hŷ + Ŷ + Ŝ [λ CPI ]+O 3 (55) hĉ h = ˆP = 1 λ P H 1 λ P 1 hŷ + Ŷ Ŝ [λ CPI ]+O 3 (56)