Follow links for Class Use and other Permissions. For more information send to:

Transcription

1 COPYRIGHT NOTICE: Jordi Galí: Moeary Policy, Iflaio, ad he Busiess Cycle is published by Priceo Uiversiy Press ad copyrighed, 008, by Priceo Uiversiy Press. All righs reserved. No par of his book may be reproduced i ay form by ay elecroic or mechaical meas (icludig phoocopyig, recordig, or iformaio sorage ad rerieval) wihou permissio i wriig from he publisher, excep for readig ad browsig via he World Wide Web. Users are o permied o mou his file o ay ework servers. Follow liks for Class Use ad oher Permissios. For more iformaio sed o: permissios@pupress.priceo.edu

2 4 Moeary Policy Desig i he Basic New Keyesia Model This chaper addresses he quesio of how moeary policy should be coduced, usig as a referece framework he basic New Keyesia model developed i chaper 3. To sar, ha model s efficie allocaio is characerized ad show o correspod o he equilibrium allocaio of he deceralized ecoomy uder moopolisic compeiio ad flexible prices oce a appropriaely chose subsidy is i place. As i will be demosraed, whe prices are sicky, ha allocaio ca be aaied by meas of a policy ha fully sabilizes he price level. The objecives of he opimal moeary policy are firs deermied, ad he he issues peraiig o is implemeaio are addressed. Examples of ieres rae rules ha impleme he opimal policy, i.e., opimal ieres rae rules, are provided. Bu a argume is give ha oe of hose rules seems a likely cadidae o guide moeary policy i pracice, for hey all require ha he ceral bak respod coemporaeously o chages i a variable he aural rae of ieres ha is o observable i acual ecoomies. Tha observaio moivaes he iroducio of rules ha a ceral bak could arguably follow i pracice (labeled as simple rules), ad he developme of a crierio o evaluae he relaive desirabiliy of hose rules, based o heir implied welfare losses. A illusraio of ha approach o policy evaluaio is provided by aalyzig he properies of wo such simple rules: a Taylor rule ad a cosa moey growh rule. 4. The Efficie Allocaio The efficie allocaio associaed wih he model ecoomy described i chaper 3 ca be deermied by solvig he problem facig a beevole social plaer seekig o maximize he represeaive household s welfare, give echology ad prefereces. Thus, for each period he opimal allocaio mus maximize he household s uiliy U (C,N )

3 7 4. Moeary Policy Desig i he Basic New Keyesia Model where C ( 0 C (i) ε di) ε ε, subjec o he resource cosrais for all i [0, ] ad C (i) = A N (i) α N = The associaed opimaliy codiios are 0 N (i) di. C (i) = C, all i [0, ] () N (i) = N, all i [0, ] () U, = MPN (3) U c, where MPN ( α) A N α deoes he ecoomy s average margial produc of labor (which i he case of he symmeric allocaio cosidered above also happes o coicide wih he margial produc for each idividual firm). Thus, i is opimal o produce ad cosume he same quaiy of all goods ad o allocae he same amou of labor o all firms. Tha resul is a cosequece of all goods eerig he uiliy fucio symmerically, combied wih cocaviy of uiliy ad ideical echologies o produce all goods. Oce ha symmeric allocaio is imposed, he remaiig codiio defiig he efficie allocaio, equaio (3), equaes he margial rae of subsiuio bewee cosumpio ad work hours o he correspodig margial rae of rasformaio (which i ur correspods o he margial produc of labor). Noe also ha he laer codiio coicides wih he oe deermiig he equilibrium allocaio of he classical moeary model (wih perfec compeiio ad fully flexible prices) aalyzed i chaper. Nex, he facors ha make he equilibrium allocaio i he baselie model subopimal are discussed. 4. Sources of Subopimaliy i he Basic New Keyesia Model The basic New Keyesia model developed i chaper 3 is characerized by wo disorios, whose implicaios are worh cosiderig separaely. The firs disorio is he presece of marke power i goods markes, exercised by moopolisically compeiive firms. Tha disorio is urelaed o he presece of sicky prices, i.e., i would be effecive eve uder he assumpio of flexible prices. The secod disorio resuls from he assumpio of ifreque adjusme of prices by firms. Nex, boh ypes of disorios ad heir implicaios for he efficiecy of equilibrium allocaios are discussed.

4 4.. Sources of Subopimaliy i he Basic New Keyesia Model Disorios Urelaed o Sicky Prices: Moopolisic Compeiio The fac ha each firm perceives he demad for is differeiaed produc o be imperfecly elasic edows i wih some marke power ad leads o pricigabove-margial cos policies. To isolae he role of moopolisic compeiio le us suppose for he ime beig ha prices are fully flexible, i.e., each firm ca adjus freely he price of is good each period. I ha case, ad uder hese assumpios, he profi maximizig price is ideical across firms. I paricular, uder a isoelasic demad fucio (wih price-elasiciy ε), he opimal priceseig rule is give by W P = M MPN where M ε > is he (gross) opimal markup chose by firms ad ε he margial cos. Accordigly, U, W MPN = = <MPN U c, P M W MPN is where he firs equaliy follows from he opimaliy codiios of he household. Hece, i is see ha he presece of a orivial price markup implies ha codiio (3) characerizig he efficie allocaio is violaed. Because, i equilibrium, he margial rae of subsiuio U, /U c, ad he margial produc of labor are, respecively, icreasig ad decreasig (or oicreasig) i hours, he presece of a markup disorio leads o a iefficiely low level of employme ad oupu. The above iefficiecy resulig from he presece of marke power ca be elimiaed hrough he suiable choice of a employme subsidy. Le τ deoe he rae a which he cos of employme is subsidized, ad assume ha he oulays associaed wih he subsidy are fiaced by meas of lump-sum axes. The, uder flexible prices, P = M ( τ)w. Accordigly, MPN U, W MPN = =. U c, P M( τ) Hece, he opimal allocaio ca be aaied if M( τ) = or, equivalely, by seig τ =. I much of he aalysis below i is assumed ha such a opimal subsidy is i place. By cosrucio, he equilibrium uder flexible prices is ε efficie i ha case. 4.. Disorios Associaed wih he Presece of Saggered Price Seig The assumed cosrais o he frequecy of price adjusme cosiue a source of iefficiecy o wo differe grouds. Firs, he fac ha firms do o adjus

5 74 4. Moeary Policy Desig i he Basic New Keyesia Model heir prices coiuously implies ha he ecoomy s average markup will vary over ime i respose o shocks, ad will geerally differ from he cosa fricioless markup M. Formally, ad deoig he ecoomy s average markup as M (defied as he raio of average price o average margial cos), M = ( τ)(w /MP N ) = W /MP N P P M where he secod equaliy follows from he assumpio ha he subsidy i place exacly offses he moopolisic compeiio disorio, which allows he isolaio of he role of sicky prices. I ha case, U, = W = MPN M U c, P M which violaes efficiecy codiio (3) o he exe ha M = M. The efficiecy of he equilibrium allocaio ca oly be resored if policy maages o sabilize he ecoomy s average markup a is fricioless level. I addiio o he above iefficiecy, which implies eiher oo low or oo high a level of aggregae employme ad oupu, he presece of saggered price seig is a source of a secod ype of iefficiecy. The laer has o do wih he fac ha he relaive prices of differe goods will vary i a way uwarraed by chages i prefereces or echologies, as a resul of he lack of sychroizaio i price adjusmes. Thus, geerally P (i) = P (j) for ay pair of goods (i, j) whose prices do o happe o have bee adjused i he same period. Such relaive price disorios will lead, i ur, o differe quaiies of he differe goods beig produced ad cosumed, i.e., C (i) = C (j), ad, as a resul, N (i) = N (j) for some (i, j). Tha oucome violaes efficiecy codiios () ad (). Aaiig he efficiecy allocaio requires ha he quaiies produced ad cosumed of all goods are equalized (ad, hece, so are heir prices ad margial coss). Accordigly, markups should be ideical across firms ad goods a all imes, i addiio o beig cosa (ad equal o he fricioless markup) o average. Nex, he policy ha will aai hose objecives is characerized. 4.3 Opimal Moeary Policy i he Basic New Keyesia Model I addiio o assumig a opimal subsidy i place ha exacly offses he marke power disorio, ad i order o keep he aalysis simple, he aalysis is resriced o he case where here are o iheried relaive price disorios, i.e., P (i) = P for all i [0, ]. Uder hose assumpios, he efficie allocaio ca be aaied by a policy ha sabilizes margial coss a a level cosise wih firms The case of a odegeerae iiial disribuio of prices is aalyzed iyu (005). I he laer case, he opimal moeary policy coverges o he oe described here afer a rasiio period.

6 4.3. Opimal Moeary Policy i he Basic New Keyesia Model 75 desired markup, give he prices i place. If ha policy is expeced o be i place idefiiely, o firm has a iceive o adjus is price, because i is currely chargig is opimal markup ad expecs o keep doig so i he fuure wihou havig o chage is price. As a resul, P = P ad, hece, P = P for = 0,,,...I oher words, he aggregae price level is fully sabilized ad o relaive price disorios emerge. I addiio, M = M for all, ad oupu ad employme mach heir couerpars i he flexible price equilibrium allocaio (which, i ur, correspods o he efficie allocaio, give he subsidy i place). Usig he oaio for he log-liearized model iroduced i chaper 3, he opimal policy requires ha for all, ỹ = 0 π = 0 i.e., he oupu gap is closed a all imes, which (as implied by he New Keyesia Phillips curve) leads o zero iflaio. The dyamic IS equaio he implies i = r for all, i.e., he equilibrium omial ieres rae (which equals he real rae, give zero iflaio) mus be equal o he aural ieres rae. Two feaures of he opimal policy are worh emphasizig. Firs, sabilizig oupu is o desirable i ad of iself. Isead, oupu should vary oe for oe wih he aural level of oupu, i.e., y = y for all. There is o reaso, i priciple, why he aural level of oupu should be cosa or follow a smooh red, because all kids of real shocks will be a source of variaios i is level. I ha coex, policies ha sress oupu sabiliy (possibly abou a smooh red) may geerae poeially large deviaios of oupu from is aural level ad, hus, be subopimal. This poi is illusraed i secio 4.3., i he coex of a quaiaive aalysis of a simple policy rule. Secod, price sabiliy emerges as a feaure of he opimal policy eve hough, a priori, he policymaker does o aach ay weigh o such a objecive. Isead, price sabiliy is closely associaed wih he aaime of he efficie allocaio (which is a more immediae policy objecive). Bu he oly way o replicae he (efficie) flexible price allocaio whe prices are sicky is by makig all firms coe wih heir exisig prices, so ha he assumed cosrais o he adjusme of hose prices are effecively obidig. Aggregae price sabiliy he follows as a cosequece of o firm willig o adjus is price Implemeaio: Opimal Ieres Rae Rules Nex, some cadidae rules for implemeig he opimal policy are cosidered. All of hem are cosise wih he desired equilibrium oucome. Some, however,

7 76 4. Moeary Policy Desig i he Basic New Keyesia Model are also cosise wih oher subopimal oucomes. I all cases, ad i order o aalyze is equilibrium implicaios, he cadidae rule cosidered is embedded i he wo equaios describig he o-policy block of he basic New Keyesia model iroduced i chaper 3. Those wo key equaios are show here agai for coveiece ỹ = E {ỹ + } (i E {π + } r ) σ (4) π = βe {π + }+κ ỹ. (5) A Exogeous Ieres Rae Rule Cosider he cadidae ieres rae rule i = r (6) for all. This is a rule ha isrucs he ceral bak o adjus he omial rae oe for oe wih variaios i he aural rae (ad oly i respose o variaios i he laer). Such a rule would seem a aural cadidae o impleme he opimal policy sice (6) was show earlier o be always saisfied i a equilibrium ha aais he opimal allocaio. Subsiuig (6) io (4) ad rearragig erms represes he equilibrium codiios uder rule (6) by meas of he sysem [ ỹ π ] [ E { ỹ = A 0 E { π + + ] } } (7) where [ A 0 ] σ κ β+ κ. σ Noe ha ỹ = π = 0 for all he oucome associaed wih he opimal policy is oe soluio o (7). Tha soluio, however, is o uique: I ca be show ha oe of he wo (real) eigevalues of A 0 always lies i he ierval (0, ), while he secod is sricly greaer ha uiy. Give ha boh ỹ ad π are opredeermied, he exisece of a eigevalue ouside he ui circle implies he exisece of a mulipliciy of equilibria i addiio o ỹ = π = 0 for all. I ha case ohig guaraees ha he laer allocaio will be precisely he oe ha will emerge as a equilibrium. Tha shorcomig leads o he cosideraio of aleraive rules o (6). See, e.g., Blachard ad Kah (980).

8 4.3. Opimal Moeary Policy i he Basic New Keyesia Model A Ieres Rae Rule wih a Edogeous Compoe Le us cosider ex he followig ieres rae rule i = r + φ π π + φ y ỹ (8) where φ π ad φ y are o-egaive coefficies deermied by he ceral bak, ha describe he sregh of he ieres rae respose o deviaios of iflaio or he oupu gap from heir arge levels. As above, subsiue he omial rae ou usig he assumed ieres rae rule, ad represe he equilibrium dyamics by meas of a sysem of differece equaios of he form where [ ỹ π ] [ E {ỹ + } = A T E {π + } [ σ A T σκ ad. σ +φ y +κφ π Oce agai, he desired oucome ( y = π = 0 for all ) is always a soluio o he dyamical sysem (9) ad, hece, a equilibrium of he ecoomy uder rule (8). Ye, i order for ha oucome o be he oly (saioary) equilibrium, boh eigevalues of marix A T should lie wihi he ui circle. The size of hose eigevalues ow depeds o he policy coefficies (φ π,φ y ), i addiio o he o-policy parameers. Uder he assumpio of o-egaive values for (φ π,φ y ), a ecessary ad sufficie codiio for A T o have wo eigevalues wihi he ui circle ad, hece, for he equilibrium o be uique, is give by 3 ] βφ π κ + β(σ + φ y ) ] (9) κ(φ π ) + ( β) φ y > 0. (0) Thus, roughly speakig, he moeary auhoriy should respod o deviaios of iflaio ad he oupu gap from heir arge levels by adjusig he omial rae wih sufficie sregh. Figure 4. illusraes graphically he regios of parameer space for (φ π,φ y ) associaed wih deermiae ad ideermiae equilibria, as implied by codiio (0). Ieresigly, ad somewha paradoxically, if codiio (0) is saisfied, boh he oupu gap ad iflaio will be zero ad, hece, i = r for all will hold ex-pos. Thus, ad i coras wih he case cosidered above (i which he equilibrium oucome i = r was also ake o be he policy rule), i is he presece of a hrea of a srog respose by he moeary auhoriy o a eveual deviaio of 3 See Bullard ad Mira (00) for a proof.

9 78 4. Moeary Policy Desig i he Basic New Keyesia Model Deermiacy. φ Ideermiacy Figure 4. Deermiacy ad Ideermiacy Regios for a Coemporaeous Ieres Rae Rule φ y he oupu gap ad iflaio from arge ha suffices o rule ou ay such deviaio i equilibrium. Some ecoomic iuiio for he form of codiio (0) ca be obaied by cosiderig he eveual implicaios of rule (8) for he omial rae, were a permae icrease i iflaio of size dπ o occur (ad assumig o permae chages i he aural rae) di = φ π dπ + φ y dỹ ( ) φ y ( β) = φ π + dπ () κ where he secod equaliy makes use of he log erm relaioship bewee iflaio ad he oupu gap implied by (5). Noe ha codiio (0) is equivale o he erm i brackes i () beig greaer ha oe. Thus, he equilibrium will be uique uder ieres rae rule (8) wheever φ π ad φ y are sufficiely large eough o guaraee ha he real rae eveually rises i he face of a icrease i

10 4.3. Opimal Moeary Policy i he Basic New Keyesia Model 79 iflaio (hus edig o couerac ha icrease ad acig as a sabilizig force). The previous propery is ofe referred o as he Taylor priciple ad, o he exe ha i preves he emergece of muliple equilibria, i is aurally viewed as a desirable feaure of ay ieres rae rule A Forward-Lookig Ieres Rae Rule I order o illusrae he exisece of a mulipliciy of policy rules capable of implemeig he opimal policy, le us cosider he followig forward-lookig rule i = r + φ π E {π + }+φ y E {ỹ + } () which has he moeary auhoriy o adjus he omial rae i respose o variaios i expeced iflaio ad he expeced oupu gap, as opposed o heir curre values, as assumed i (8). Uder () he implied dyamics are described by he sysem where [ ] [ ] y E { = A y + } π F E {π + } [ A F σ φ y σ φ π κ( σ φ y ) β κσ φ π ]. I his case, he codiios for a uique equilibrium (i.e., for boh eigevalues of A F lyig wihi he ui circle) are wofold ad give by 5 κ(φ π ) + ( β) φ y > 0 (3) κ(φ π ) + ( + β) φ y < σ( + β). (4) Figure 4. represes he deermiacy/ideermiacy regios i (φ π,φ y ) space, uder he baselie calibraio for he remaiig parameers. Noe ha i coras wih he coemporaeous rule cosidered i subsecio 4.3.., deermiacy of equilibrium uder he prese forward-lookig rule requires ha he ceral bak reacs eiher oo srogly or oo weakly o deviaios of iflaio ad/or he oupu gap from arge. Ye, figure 4. suggess ha he kid of overreacio ha would be coducive o ideermiacy would require raher exreme values of he iflaio ad/or oupu gap coefficies, well above hose characerizig empirical ieres rae rules. 4 See Woodford (00) for a discussio. 5 Bullard ad Mira (00) lis a hird codiio, give by he iequaliy φ y <σ( + β ), as ecessary for uiqueess. Bu i ca be easily checked ha he laer codiio is implied by he wo codiios (3) ad (4).

11 80 4. Moeary Policy Desig i he Basic New Keyesia Model Ideermiacy φ 5 0 Deermiacy Figure 4. Deermiacy ad Ideermiacy Regios for a Forward-Lookig Ieres Rae Rule φ y 4.3. Pracical Shorcomigs of Opimal Policy Rules Subsecio 4.3. provided wo examples of ieres rae rules ha impleme he opimal policy, hus guaraeeig ha he efficie allocaio is aaied as he uique equilibrium oucome. While such opimal ieres rae rules appear o ake a relaively simple form, here exiss a impora reaso why hey are ulikely o provide useful pracical guidace for he coduc of moeary policy. The reaso is ha hey boh require ha he policy rae be adjused oe-for-oe wih he aural rae of ieres, hus implicily assumig observabiliy of he laer variable. Tha assumpio is plaily urealisic because deermiaio of he aural rae ad is movemes requires a exac kowledge of (i) he ecoomy s rue model, (ii) he values ake by all is parameers, ad (iii) he realized value (observed i real ime) of all he shocks impigig o he ecoomy. Noe ha a similar requireme would have o be me if, as implied by (8) ad (), he ceral bak should also adjus he omial rae i respose o deviaios of oupu from he aural level of oupu, because he laer is also uobservable.

12 4.4. Two Simple Moeary Policy Rules 8 Tha requireme, however, is o early as bidig as he uobservabiliy of he aural rae of ieres, for ohig preves he ceral bak from implemeig he opimal policy by meas of a rule ha does o require a sysemaic respose o chages i he oupu gap. Formally, φ y i (8) or () could be se o zero, wih uiqueess of equilibrium beig sill guaraeed by he choice of a iflaio coefficie greaer ha uiy (ad o greaer ha + σ( + β)κ i he case of he forward-lookig rule). The pracical shorcomigs of opimal ieres rae rules discussed above have led may auhors o propose a variey of simple rules udersood as rules ha a ceral bak could arguably adop i pracice ad o aalyze heir properies. 6 I ha coex, a ieres rae rule is geerally cosidered simple if i makes he policy isrume a fucio of observable variables oly, ad does o require ay precise kowledge of he exac model or he values ake by is parameers. The desirabiliy of ay give simple rule is hus give o a large exe by is robusess, i.e., is abiliy o yield a good performace across differe models ad parameer cofiguraios. I he followig secio, wo such simple rules are aalyzed a simple Taylorype rule ad a cosa moey growh rule ad heir performace is assessed i he coex of he baselie New Keyesia model. 4.4 Two Simple Moeary Policy Rules This secio provides a illusraio of how he basic New Keyesia model developed i chaper 3 ca be used o assess he performace of wo policy rules. A formal evaluaio of he performace of a simple rule (relaive, say, o he opimal rule or o a aleraive simple rule) requires he use of some quaiaive crierio. Followig he semial work of Roemberg ad Woodford (999), much of he lieraure has adoped a welfare-based crierio, relyig o a secod-order approximaio o he uiliy losses experieced by he represeaive cosumer as a cosequece of deviaios from he efficie allocaio. As show i appedix 4., uder he assumpios made i his chaper (which guaraee he opimaliy of he flexible price equilibrium), ha approximaio yields he followig welfare loss fucio ) ] ϕ + α ε W = E 0 β [(σ + ỹ + π α λ =0 where welfare losses are expressed i erms of he equivale permae cosumpio declie, measured as a fracio of seady sae cosumpio. 6 The volume edied by Joh Taylor (999) coais several impora coribuios i ha regard.

13 8 4. Moeary Policy Desig i he Basic New Keyesia Model The average welfare loss per period is hus give by he followig liear combiaio of he variaces of he oupu gap ad iflaio L = [( σ + ϕ + α ) var(ỹ ) + ε ] var(π ). α λ Noe ha he relaive weigh of oupu gap flucuaios i he loss fucio is icreasig i σ, ϕ, ad α. The reaso is ha larger values of hose curvaure parameers amplify he effec of ay give deviaio of oupu from is aural level o he size of he gap bewee he margial rae of subsiuio ad he margial produc of labor, which is a measure of he ecoomy s aggregae iefficiecy. O he oher had, he weigh of iflaio flucuaios is icreasig i he elasiciy of subsiuio amog goods ε because he laer amplifies he welfare losses caused by ay give price dispersio ad he degree of price sickiess θ (which is iversely relaed o λ), which amplifies he degree of price dispersio resulig from ay give deviaio from zero iflaio. Give a policy rule ad a calibraio of he model s parameers, oe ca deermie he implied variace of iflaio ad he oupu gap ad he correspodig welfare losses associaed wih ha rule (relaive o he opimal allocaio). Tha procedure is illusraed ex hrough he aalysis of wo simple rules A Taylor-ype Ieres Rae Rule Le us firs cosider he followig ieres rule, i he spiri of Taylor (993) i = ρ + φ π π + φ y ŷ (5) where ŷ log(y /Y) deoes he log deviaio of oupu from is seady sae ad where φ π > 0 ad φ y > 0 are assumed o saisfy he deermiacy codiio (0). Agai, he choice of iercep ρ log β is cosise wih a zero iflaio seady sae. Noe ha (5) ca be rewrie i erms of he oupu gap as i = ρ + φ π π + φ y ỹ + v (6) where v φ y ŷ. The resulig equilibrium dyamics are hus ideical o hose of he ieres rae rule aalyzed i chaper 3, wih v ow reierpreed as a drivig force proporioal o he deviaios of aural oupu from seady sae, isead of a exogeous moeary policy shock. Noe ha he variace of he shock v is o loger exogeous, bu icreasig i φ y, he coefficie deermiig he respose of he moeary auhoriy o flucuaios i oupu. Formally, he equilibrium

14 4.4. Two Simple Moeary Policy Rules 83 Table 4. Evaluaio of Simple Moeary Policy Rules Taylor Rule Cosa Moey Growh φ π φ y (σ ζ,ρ ζ ) (0, 0) (0.0063, 0.6) σ(ỹ) σ(π) welfare loss dyamics are described by he sysem [ ] [ ] y E{ y = A + } π T + B E { π } T ( r v ) + where A T ad B T are defied as i chaper 3. Assumig ha variaios i he echology parameer a represe he oly drivig force i he ecoomy, ad are described by a saioary AR() process wih auoregressive coefficie ρ a, he followig equaliy holds: r v = σψ ( ρ )a φ y ψ a ya a ya = ψ [σ( ρ a ) + φ y ] a ya +ϕ where, as i chaper 3, ψya > 0. From he aalysis i chaper 3, he σ +ϕ+α( σ) variace of he oupu gap ad iflaio uder a rule of he form (6) is proporioal o ha of B T ( r v ), which is sricly icreasig i φ y. Hece, a policy seekig o sabilize oupu by respodig aggressively o deviaios i ha variable from seady sae (or red) is boud o lower he represeaive cosumer s uiliy by icreasig he variace of he oupu gap ad iflaio. 7 The lef pael of able 4. displays some saisics for four differe calibraios of rule (5), correspodig o aleraive cofiguraios for φ π ad φ y. The firs colum correspods o he calibraio proposed by Taylor (993) as a good approximaio o he ieres rae policy of he Fed durig he Greespa years. 8 The secod ad hird rules assume o respose o oupu flucuaios wih a very aggressive ai-iflaio sace i he case of he hird rule (φ π =5). Fially, he fourh rule assumes a srog oupu-sabilizaio moive (φ y =). The remaiig parameers are calibraed a heir baselie values, as iroduced i chaper 3. For each versio of he Taylor rule, able 4. shows he implied sadard deviaios of he oupu gap ad (aualized) iflaio, boh expressed i perce 7 Noice ha i his simple example he opimal allocaio ca be aaied by seig φ y = σ( ρ a ). I ha case, he simple rule is equivale o he opimal rule i = r + φ π π. 8 Taylor s proposed coefficie values were.5 for iflaio ad 0.5 for oupu, based o a specificaio wih aualized iflaio ad ieres raes. The choice of φ y = 0.5/4 is cosise wih Taylor s proposed calibraio because boh i ad π i he model are expressed i quarerly raes.

15 84 4. Moeary Policy Desig i he Basic New Keyesia Model erms, as well as he welfare losses resulig from he associaed deviaios from he efficie allocaio, expressed as a fracio of seady sae cosumpio. Several resuls sad ou. Firs, i a way cosise wih he aalysis above, versios of he rule ha ivolve a sysemaic respose o oupu variaios geerae larger flucuaios i he oupu gap ad iflaio ad, hece, larger welfare losses. Those losses are moderae (0.3 perce of seady sae cosumpio) uder Taylor s origial calibraio, bu hey become subsaial (close o perce of seady sae cosumpio) whe he oupu coefficie φ y is se o uiy. Secod, he smalles welfare losses are aaied whe he moeary auhoriy respods o chages i iflaio oly. Furhermore, hose losses (as well as he uderlyig flucuaios i he oupu gap ad iflaio) become smaller as he sregh of ha respose icreases. Hece, ad a leas i he coex of he basic New Keyesia model cosidered here, a simple Taylor-ype rule ha respods aggressively o movemes i iflaio ca approximae arbirarily well he opimal policy A Cosa Moey Growh Rule Nex, a simple rule cosisig of a cosa growh rae for he moey supply is cosidered, which is a rule geerally associaed wih Friedma (960). Wihou loss of geeraliy, a zero rae of growh of he moey supply is assumed, which is cosise wih zero iflaio i he seady sae (give he absece of secular growh). Formally, m = 0 for all. Oce agai, he assumpio of a moeary rule requires ha equilibrium codiios (4) ad (5) be supplemeed wih a moey marke clearig codiio. Take he laer o be of he form l = y ηi ζ where l m p deoes (log) real balaces ad ζ is a exogeous moey demad shock followig he process ζ = ρ ζ ζ + ε ζ where ρ ζ [0, ). I is coveie o rewrie he moey marke equilibrium codiio i erms of deviaios from seady sae as l = ỹ + ŷ η î ζ. Leig l + l + ζ deoe (log) real balaces adjused by he exogeous compoe of moey demad, i = (ỹ + ŷ l + ). η

16 4.5. Noes o he Lieraure 85 I addiio, usig he defiiio of l + ogeher wih he assumed rule m = 0, l + = l + + π ζ. Combiig he previous wo equaios wih (4) ad (5) o subsiue ou he omial rae, he equilibrium dyamics uder a cosa moey growh rule ca be summarized by he sysem y E { y +} r + B M A M,0 π =A M, E {π + } y + l + l ζ where A M,0, A M,, ad B M are defied as i chaper 3. The righ pael of able 4. repors he sadard deviaio of he oupu gap ad iflaio, as well as he implied welfare losses, uder a cosa moey growh rule. Two cases are cosidered, depedig o wheher moey demad is assumed o be subjec o exogeous disurbaces. I boh cases he aural oupu ad he aural rae of ieres vary i respose o echology shocks (accordig o he baselie calibraio of he laer iroduced i chaper 3). Whe moey demad shocks are allowed for, he correspodig process for ζ is calibraed by esimaig a AR() process for he (firs-differeced) residual of a moey demad fucio for he period 989:I 004:IV a period characerized by subsaial sabiliy i he demad for moey compued usig a ieres rae semi-elasiciy of η = 4 (see discussio i chaper 3). The esimaed sadard deviaio for he residual of he AR() process is σ ζ = while he esimaed AR() coefficie is ρ ζ = 0.6. Noice ha i he absece of moey demad shocks, a cosa moey growh rule delivers a performace comparable, i erms of welfare losses, o a Taylor rule wih coefficies φ π =.5 ad φ y =0.Ye, whe he calibraed moey demad shock is iroduced, he performace of a cosa moey growh rule deerioraes cosiderably, wih he volailiy of boh he oupu gap ad iflaio risig o a level associaed wih welfare losses above hose of he baselie Taylor rule. Thus, ad o surprisigly, he degree of sabiliy of moey demad is a key eleme i deermiig he desirabiliy of a rule ha focuses o he corol of a moeary aggregae. 4.5 Noes o he Lieraure A early deailed discussio of he case for price sabiliy i he basic New Keyesia model ca be foud i Goodfried ad Kig (997). Svesso (997) coais a aalysis of he desirabiliy of iflaio argeig sraegies, usig a o-fullymicrofouded model.

17 86 4. Moeary Policy Desig i he Basic New Keyesia Model Whe derivig he opimal policy o iheried dispersio of prices across firms was assumed. A rigorous aalysis of he opimal moeary policy i he case of a iiial odegeerae price disribuio ca be foud i Yu (005). Taylor (993) iroduced he simple formula commoly kow as he Taylor rule, as providig a good approximaio o Fed policy i he early Greespa years. Judd ad Rudebusch (998) ad Clarida, Galí, ad Gerler (000) esimae aleraive versios of he Taylor rule, ad examie is (i)sabiliy over he poswar period. Taylor (999) uses he rule calibraed for he Greespa years as a bechmark for he evaluaio of moeary policy durig oher episodes over he poswar period. Orphaides (003) argues ha he bulk of he deviaios from he baselie Taylor rule observed i he pre-volcker era may have bee he resul of large biases i real ime measures of he oupu gap. Key coribuios o he lieraure o he properies of aleraive simple rules ca be foud i he papers coaied i he volume edied by Taylor (999). I paricular, he paper by Roemberg ad Woodford (999) derives a secod-order approximaio o he uiliy of he represeaive cosumer. Chaper 6 i Woodford (003) provides a deailed discussio of welfare-based evaluaios of policy rules. Appedix: A Secod-Order Approximaio o a Household s Welfare: The Case of a Udisored Seady Sae This appedix derives a secod-order approximaio o he uiliy of he represeaive cosumer whe he ecoomy remais i a eighborhood of a efficie seady sae, i a way cosise wih he assumpios made i his chaper. The geeralizaio o he case of a disored seady sae is lef for chaper 5. A secod-order approximaio of uiliy is derived aroud a give seady sae allocaio. Freque use is made of he followig secod-order approximaio of relaive deviaios i erms of log deviaios Z Z ẑ + ẑ Z where ẑ z z is he log deviaio from seady sae for a geeric variable z. All alog i is assumed ha uiliy is separable i cosumpio ad hours (i.e., U c = 0). I order o lighe he oaio, defie U U(C,N ), U U(C,N ), ad U U(C,N). The secod-order Taylor expasio of U aroud a seady sae (C, N) yields ( ) ( ) ( ) C N C C N U U U c C + U N + U cc C C C N C ( N N ) + U N N

18 Appedix 87 I erms of log deviaios, ( ) ( ) σ + ϕ U U U c C ŷ + ŷ + U N + where σ U cc C ad ϕ U N, ad where use of he marke clearig codiio U c U ĉ = ŷ has bee made. The ex sep cosiss i rewriig i erms of oupu. Usig he fac ha Y N = ( ) α A ( P (i) ) ε α di, 0 P ( α) = ŷ a + d where d ( α) log ( P (i) ) ε α di. The followig lemma shows ha d is 0 P proporioal o he cross-secioal variace of relaive prices. Lemma : I a eighborhood of a symmeric seady sae, ad up o a secodorder approximaio, d = ε var i {p (i)}. Proof: Le p (i) p (i) p. Noice ha ( ) ε P (i) = exp [( ε) p (i)] P = + ( ε) p (i) + ( ε) p (i). P (i) Noe ha from he defiiio of P, = 0 ( P ) ε di. A secod-order approximaio o his expressio hus implies (ε ) E i {p (i)}= E i {p (i) }. I addiio, a secod-order approximaio o ( P (i) ) ε α yields P ( ) ε ( ) P (i) α ε ε = p (i) + p (i). P α α Combiig he wo previous resuls, i follows ha ( ) ε ( ) P (i) α ε di = + E i {p (i) } 0 P α = + ( ) ε vari {p (i)} α

19 88 4. Moeary Policy Desig i he Basic New Keyesia Model α where, ad where he las equaliy follows from he observaio ha, α+ αε up o secod order, (p (i) p ) di (p (i) E i {p (i)}) di 0 0 var i {p (i)}. Fially, usig he defiiio of d ad up o a secod-order approximaio, ( ) P (i) ε α ε d ( α) log di var i {p (i)} 0 P QED. Now, he period uiliy ca be rewrie as ( ) σ U U = U c C ŷ + ŷ + U N ( ŷ + ε var i {p (i)}+ + ϕ (ŷ a ) )+.i.p. α ( α) where.i.p. sads for erms idepede of policy. Efficiecy of he seady sae implies U = MPN. Thus, ad usig he fac U c ha MPN = ( α)(y/n) ad Y = C, U U [ ε vari {p (i)} ( σ) ŷ + + ϕ ] (ŷ a ) +.i.p. U c C α = [ ( ε vari {p (i)}+ σ + ϕ + α ) ( ) ] + ϕ ŷ ŷ a +.i.p. α α = [ ( ε vari {p (i)}+ σ + ϕ + α ) ] (ŷ ŷ ŷ ) +.i.p. α = [ ε vari {p (i)}+ (σ + ϕ + α ) ] ỹ +.i.p. α where ŷ y y, ad where he fac was used ha ŷ = σ( ŷ = ỹ. y + ϕ α) + ϕ+α a ad

20 Refereces 89 Accordigly, a secod-order approximaio o he cosumer s welfare losses ca be wrie ad expressed as a fracio of seady sae cosumpio (ad up o addiive erms idepede of policy) as ( ) W = E 0 β U U U c C =0 ( ( ε ϕ + α ) ) = E 0 β var i {p (i)}+ σ + ỹ. α =0 The fial sep cosiss i rewriig he erms ivolvig he price dispersio variable as a fucio of iflaio. I order o do so, make use of he followig lemma θ Lemma : 0 β var i {p (i)}= 0 β π = ( βθ)( θ) = Proof: Woodford (003, chaper 6) Usig he fac ha λ ( θ)( βθ), he previous lemma ca be combied wih θ he expressio for he welfare losses above o obai [ ( ) ] W = ( E 0 β ε ) π + σ + ϕ + α ỹ. λ α =0 Refereces Blachard, Olivier J., ad Charles Kah (980): The Soluio of Liear Differece Equaios uder Raioal Expecaios, Ecoomerica 48, o. 5, Bullard, James, ad Kaushik Mira (00): LearigAbou Moeary Policy Rules, Joural of Moeary Ecoomics 49, o. 6, Clarida, Richard, Jordi Galí, ad Mark Gerler (000): Moeary Policy Rules ad Macroecoomic Sabiliy: Evidece ad Some Theory, Quarerly Joural of Ecoomics 05, o., Friedma, Milo (960): A Program for Moeary Sabiliy, Fordham Uiversiy Press, New York. Goodfried, Marvi, ad Rober G. Kig (997): The New Neoclassical Syhesis, NBER Macroecoomics Aual 997, 3 8. Judd, Joh P., ad Gle Rudebusch (998): Taylor s Rule ad he Fed: A Tale of Three Chairme, FRBSF [Federal Reserve Bak of Sa Fracisco] Ecoomic Review, o. 3, 3 6. Orphaides, Ahaasios (003): The Ques for Prosperiy Wihou Iflaio, Joural of Moeary Ecoomics 50, o. 3, Roemberg, Julio, ad Michael Woodford (999): Ieres Rae Rules i a Esimaed Sicky Price Model, i J. B. Taylor (ed.), Moeary Policy Rules, Uiversiy of Chicago Press, Chicago, IL.

21 90 4. Moeary Policy Desig i he Basic New Keyesia Model Svesso, Lars E. O. (997) Iflaio Forecas Targeig: Implemeig ad Moiorig Iflaio Targes, Europea Ecoomic Review 4, o. 6, 47. Taylor, Joh B. (993): Discreio versus Policy Rules i Pracice, Caregie-Rocheser Series o Public Policy 39, Taylor, Joh B., ed. (999): A Hisorical Aalysis of Moeary Policy Rules, i J. B. Taylor (ed.), Moeary Policy Rules, Uiversiy of Chicago Press, Chicago, IL. Taylor, Joh B. (999): Moeary Policy Rules, Uiversiy of Chicago Press, Chicago, IL. Woodford, Michael (00): The Taylor Rule ad Opimal Moeary Policy, America Ecoomic Review 9, o., Woodford, Michael (003): Ieres ad Prices: Foudaios of a Theory of Moeary Policy, Priceo Uiversiy Press, Priceo, NJ. Yu, Tack (005): Opimal Moeary Policy wih Relaive Price Disorios, America Ecoomic Review 95, o., Exercises 4. Iflaio Targeig wih Noisy Daa Cosider a model ecoomy whose oupu gap ad iflaio dyamics are described by he sysem π = βe{ π+}+ κ y y = (i E {π + } r ) + E {y + } σ (7) (8) where all variables are defied as i he ex. The aural rae r is assumed o follow he exogeous process r ρ = ρ r (r ρ) + ε where {ε } is a whie-oise process ad ρ r [0, ). o Suppose ha iflaio is measured wih some i.i.d. error ξ, i.e., π = π + ξ where π o deoes measured iflaio. Assume ha he ceral bak follows he rule i = ρ + φ π π o. (9) a) Solve for he equilibrium processes for iflaio ad he oupu gap uder he rule (9). (Hi: you may wa o sar aalyzig he simple case of ρ r = 0.) b) Describe he behavior of iflaio, he oupu gap, ad he omial rae whe φ π approaches ifiiy. c) Deermie he size of he iflaio coefficie ha miimizes he variace of acual iflaio.

22 Exercises 9 4. Moeary Policy ad he Effecs of Techology Shocks Cosider a New Keyesia ecoomy wih equilibrium codiios y = E {y + } (i E {π + } ρ) σ (0) π = βe {π + }+κ(y y ) () where all variables are defied as i he ex. Moeary policy is described by a simple rule of he form i = ρ + φ π π where φ π >. Labor produciviy is give by y = a where a is a exogeous echology parameer ha evolves accordig o a = ρ a a + ε where ρ a [0, ) ad {ε } is a i.i.d. process. The uderlyig RBC model is assumed o imply a aural level of oupu proporioal o echology y = ψ y a where ψ y >. a) Describe i words where (0) ad () come from. b) Deermie he equilibrium respose of oupu, employme, ad iflaio o a echology shock. (Hi: guess ha each edogeous variable will be proporioal o he coemporaeous value of echology.) c) Describe how hose resposes deped o he value of φ π ad κ. Provide some iuiio. Wha happes whe φ π? Wha happes as he degree of price rigidiies chages? d) Aalyze he joi respose of employme ad oupu o a echology shock ad discuss briefly he implicaios for assessme of he role of echology as a source of busiess cycles. 4.3 Ieres Rae versus Moey Supply Rules Cosider a ecoomy described by he equilibrium codiios ỹ = E {ỹ + } (i E {π + } r ) σ π = βe {π + }+κ ỹ m p = y ηi

23 9 4. Moeary Policy Desig i he Basic New Keyesia Model where all variables are defied as i he ex. Boh y ad r evolve exogeously, idepede of moeary policy. The ceral bak seeks o miimize a loss fucio of he form αvar(y ) + var(π ). a) Show how he opimal policy could be implemeed by meas of a ieres rae rule. b) Show ha a rule requirig a cosa moey supply will geerally be subopimal. Explai. (Hi: derive he pah of moey uder he opimal policy.) c) Derive a moey supply rule ha would impleme he opimal policy. 4.4 Opimal Moeary Policy wih Price Seig i Advace Cosider a ecoomy where he represeaive cosumer maximizes ( ) β M E 0 U C,,N =0 subjec o a sequece of dyamic budge cosrais P C + M + Q B M + B + W N + T ad where all variables are defied as i he ex. Assume ha period uiliy is give by ( ) +ϕ M U C,,N =log C + log M N. + ϕ P P P () Firms are moopolisically compeiive, each producig a differeiaed good whose demad is give by Y (i) = ( P (i) ) ε Y. Each firm has access o he liear P producio fucio Y (i) = A N (i) (3) where produciviy evolves accordig o A A = ( + γ a ) exp{ε } wih {ε } beig a i.i.d. ormally disribued process wih mea zero ad variace σε. The moey supply varies exogeously accordig o he process M M = ( + γ m ) exp{u } (4) where {u } is a i.i.d. ormally disribued process wih mea zero ad variace σ. u

24 Exercises 93 Fially, assume ha all oupu is cosumed, so ha i equilibrium Y = C for all. a) Derive he opimaliy codiios for he problem facig he represeaive cosumer. b) Assume ha firms are moopolisically compeiive, each producig a differeiaed good. For each period, afer observig he shocks, firms se he price of heir good i order o maximize curre profi ( ) W Y (i) P (i) subjec o he demad schedule above. Derive he opimaliy codiio associaed wih he firm s problem. c) Show ha he equilibrium levels of aggregae employme, oupu, ad iflaio are give by ( ) N +ϕ = ε Y = A A π = (γ m γ a ) + u ε. d) Discuss how uiliy depeds o he wo parameers describig moeary policy, γ m ad σ u (recall ha he omial ieres rae is cosraied o be oegaive, i.e., i 0 for all ). Show ha he opimal policy mus saisfy he Friedma rule ad discuss aleraive ways of supporig ha rule i equilibrium. e) Nex, assume ha for each period firms have o se he price i advace, i.e., before he realizaio of he shocks. I ha case hey will choose a price i order o maximize he discoued profi ( )} W E {Q, Y (i) P (i) subjec o he demad schedule Y (i) = ( P (i) ) ε Y, where Q, β C P P C P is he sochasic discou facor. Derive he firs-order codiio of he firm s problem ad solve (exacly) for he equilibrium levels of employme, oupu, ad real balaces. f) Evaluae expeced uiliy a he equilibrium values of oupu, real balaces, ad employme. g) Cosider he class of moey supply rules of he form (4) such ha u = φ ε ε + φ v ν, where {ν } is a ormally disribued i.i.d. process wih zero mea ad ui variace, ad idepede of {ε } a all leads ad lags. Noice ha wihi ha family of rules, moeary policy is fully described by hree parameers: γ m,φ ε, ad φ v. Deermie he values of hose parameers ha maximize expeced A

25 94 4. Moeary Policy Desig i he Basic New Keyesia Model uiliy, subjec o he cosrai of a o-egaive omial ieres rae. Show ha he resulig equilibrium uder he opimal policy replicaes he flexible price equilibrium aalyzed above. 4.5 A Price Level Based Ieres Rae Rule Cosider a ecoomy described by he equilibrium codiios y = E {y + } (i E {π +} r ) σ = βe {π + }+κ y. π Show ha he ieres rae rule i = r + φ p p where p p, where p p is a price level arge, geeraes a uique saioary equilibrium, if ad oly if, φ p > 0.