Comments on Backward-Looking Interest-Rate Rules, Interest-Rate Smoothing, and Macroeconomic Instability

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1 w o r k i n g p a p e r 3 9 Commens on Backward-Looking Ineres-Rae Rules, Ineres-Rae Smoohing, and Macroeconomic Insailiy y Charles T. Carlsrom and Timohy S. Fuers FEDERL RESERVE BNK OF CLEVELND

2 Working papers of he Federal Reserve Bank of Cleveland are preliminary maerials circulaed o simulae discussion and criical commen on research in progress. They may no have een sujec o he formal ediorial review accorded official Federal Reserve Bank of Cleveland pulicaions. The views saed herein are hose of he auhors and are no necessarily hose of he Federal Reserve Bank of Cleveland or of he Board of Governors of he Federal Reserve Sysem. Working papers are now availale elecronically hrough he Cleveland Fed s sie on he World Wide We:

3 Working Paper 3-9 Decemer 3 Commens on Backward-Looking Ineres-Rae Rules, Ineres-Rae Smoohing, and Macroeconomic Insailiy y Charles T. Carlsrom and Timohy S. Fuers Benhai, Schmi-Grohe, and Urie 3 argue ha if you relied solely on local analysis you would e led o elieve ha aggressive, ackward-looking ineres rae rules are sufficien for deerminacy. Bu from he perspecive of gloal analysis, ackward-looking rules do no guaranee uniueness of euilirium and indeed may lead o cyclic and even chaoic euiliria. This commen argues ha his resul is premaure. We uilize a discree ime model and make wo oservaions. Firs, compared o heir coninuous ime model, he cyclic euiliria under a ackward-looking rule are much less likely o arise in a discree ime model. Second, pure ackward-looking rules are less likely o suffer from hese gloal indeerminacy prolems han rules ha also include curren or fuure inflaion. JEL Classificaion: E4, E5 Key Words: ineres raes, moneary policy, cenral anking Charles T. Carlsrom is a he Federal Reserve Bank of Cleveland and may e reached a Charles..carlsrom@clev.fr.org or , Fax: Timohy S. Fuers is a Bowling Green Sae Universiy and may e reached a fuers@ca.gsu.edu or , Fax: The auhors hank Jess Benhai, Sephanie Schmi-Grohe, and Marin Urie for many helpful commens.

4 I. Inroducion Benhai, Schmi-Grohe, and Urie are o e commended for wriing a clear and houghful paper. Mos researchers simply look a local analysis when analyzing differen moneary policy rules. The presen auhors, however, have consisenly worried aou he gloal properies of differen ineres rae rules. In his paper hey argue ha if you relied solely on local analysis you would e led o elieve ha aggressive, ackward-looking ineres rae rules are sufficien for deerminacy. Bu from he perspecive of gloal analysis, ackward-looking rules do no guaranee uniueness of euilirium and indeed may lead o cyclic and even chaoic euiliria. This commen argues ha his resul is premaure. We uilize a discree ime model and make wo oservaions. Firs, compared o he corresponding coninuous ime model, he cyclic euiliria under a ackward-looking rule are much less likely o arise in a discree ime model. Second, pure ackward-looking rules are less likely o suffer from hese gloal indeerminacy prolems han rules ha also include curren or fuure inflaion. By pure we mean rules where only lagged inflaion raes are in he ineresrae rule. This disincion eween lagged and curren or fuure inflaion does no arise in he coninuous ime model of Benhai e al. n ineresing oservaion is ha he discree ime ineres rae rule ha mos closely mirrors he resuls of Benhai e al is he rule ha includes expeced inflaion. We also show ha hese cycles arise ecause Benhai e al. adop a money in he producion funcion MIPF framework. s shown y Feensra 986 in a flexile price environmen, his is euivalen o a money in he uiliy funcion MIUF framework wih a negaive cross parial eween consumpion and real money alances U mc. The

5 exisence of cycles arises ecause he cross parial is significanly negaive U mc. Wih discree ime a rule ha does no include expeced inflaion has o e even more negaive han i does for coninuous ime. I would e ineresing o see wheher he MIPF framework adoped y Benhai e al. can generae cycles in discree ime for eiher he curren or ackward-looking rules. Similarly since oher moives for holding money may lead o a posiive cross parial eween consumpion and real money alances U mc >, cycles may no even emerge in coninuous ime if oher moives for holding money were included in he model. II. Discree Time Model ssume ha preferences are separale in he consumpion-money composie and in laor L: γ L U c, m, L V c, m B, γ and ha producion is linear in laor y L. We follow Yun 996 and uilize he assumpion of saggered pricing in Calvo 983. Specifically marginal cos z evolves according o he following log-linearized Phillips curve, Kz where P /P -, denoes he inflaion rae, he ildes denoe log deviaions, and Κ is a measure of price sickiness K represens perfecly flexile prices. long wih his Phillips curve, he euilirium condiions are given y γ BL U c z

6 U c P U c R P U m R U R c Finally o close he model we adop eiher a curren-looking, ackward-looking, or a forward-looking rule: τ R R i, where > and i for curren-looking, i - for ackward-looking, and i for forward-looking ineres rae rules. Leing τ we solve hese rules ackwards o yield he euivalen expression R p i j. j j Noice ha if i or i - and all hree rules converge o he same coninuous ime rule. Boh he curren rule and he ackward rule are in he spiri of Benhai e al. The forward case is included for compleeness sake. s noed aove, he resuls for he forward rule mos closely mirror he resuls of Benhai e al. Wih all hree rules he cenral ank is reacing o a weighed average of all pas inflaion raes. Wih he curren rule his weighed average also includes he curren inflaion rae while wih he forward rule i includes oh he curren and nex period s expeced inflaion rae. The key difference eween he rules is wheher curren or fuure inflaion is par of he rule. In he case of he ackward rule i is no, so he nominal ineres rae is predeermined. In he case of he curren and fuure rules, he 3

7 nominal ineres rae is no predeermined. III. Curren-looking Ineres-Rae Rules We firs analyze he model if he policy rule is curren-looking. Log linearizing he euilirium condiions for he curren looking rule we have p p λ λ where U c λ ss mc DR U K γ ss ss mm ss ss mc DR R U DR R U K γ γ > ss mc mm cc c U U U D The characerisic euaion for he aove sysem is { } { } 3 4 J We are ineresed in siuaions in which he sysem is locally deerminae, u is gloally indeerminae ecause of a supercriical Hopf ifurcaion. In his noe we will no consider he issue of supercriicaliy, u leave his for fuure work. For presen purposes we simply assume ha any ifurcaion is supercriical. We use he erminology 4

8 Locally Deerminae, Gloally Indeerminae LDGI for siuaions in which we have local deerminacy, u a Hopf ifurcaion exiss. Following Benhai e. al. we ask wheher here exiss a LDGI euilirium as we vary. Even when here canno exis a LDGI wih respec o our analysis does no compleely rule ou he possiiliy of cycles ha may emerge from varying oher deep srucural parameers such as and γ. The aove sysem is locally deerminae if wo of he eigenvalues lie ouside he uni circle while wo lie inside he uni circle. By inspecion, one roo of J is zero, so ha we are lef wih he following cuic: J 3 { } { } Wih discree ime a LDGI exiss when one roo of J is inside he uni circle and he oher wo are complex conjugaes wih a norm eual o one. We will henceforh make use of he following assumpions: ssumpions : > and J-. Noe he assumpion ha > is he analogue of Benhai e al. s >. The assumpion ha J- is exremely weak and is euivalen o >. Lemma : Suppose ha assumpions are saisfied. If >, hen one or hree real roos of J lie in he uni circle. Thus, under, a necessary condiion for deerminacy is >. 5

9 Proof: The aove implies J >. Since y assumpion J-, here are an odd numer of real roos in -,. QED We follow Benhai e al. and consider variaions in he coefficien on he weighed average of curren and pas inflaions. The necessary condiion for deerminacy if we varied he coefficien on curren inflaion, τ, in he curren-looking Taylor rule is τ / >. Corollary : If J- > and > hen he sysem is eiher over-deermined or underdeermined so ha here canno exis a LDGI. Lemma : Under he assumpions, J and > / is a necessary and sufficien condiion for a LDGI wih respec o. Proof: Recall ha he produc of he hree roos of J is eual o /. This immediaely esalishes he necessiy of his condiion. s for sufficiency, suppose ha J. The oher wo roos have a norm of uniy. If hey are real, one mus e wihin he uni circle and one mus e ouside he uni circle. Bu from Lemma, here canno e an even numer of roos in he uni circle. Hence, he remaining wo roos mus e complex. QED 6

10 Proposiion : ssume and >. necessary and sufficien condiion for a LDGI wih respec o is. Proof: From Lemma, we need o analyze J/. 3 3 J / j [ ] ifurcaion exiss if j for some > /. Noe ha j / >. Suppose ha. Then as goes o infiniy, j ecomes negaive. Hence, here exiss a such ha j. Suppose insead ha. > Since j is convex and j / >, here does no exis a > / such ha j. QED. Corollary : necessary condiion for a LDGI wih respec o is ha γ U K U D mc mm. Thus a furher necessary condiion is U U. mc mm IV. Backward-looking Ineres-Rae Rules We nex show ha a LDGI is less likely if he policy rule looks purely ackward. Log linearizing he euilirium condiions for he ackward-looking rule yields 7

11 p p λ λ where and are as defined earlier. Once again one eigenvalue of he sysem is zero so ha we are lef wih he following cuic: { } { } 3 H s efore, we will uilize assumpions : > and H-. Noe ha H- J-. Since H-, H > is necessary for deerminacy as his implies an odd numer of roos in he uni circle. Therefore a necessary condiion for deerminacy is H >. Since H and > we conclude ha > or τ / > is necessary for local deerminacy. Proposiion : If > hen here exiss a LDGI wih respec o if and only if here exiss an a in -, and a >, such ha he following condiions are saisfied:, a a a f, a a g > 8

12 This implies ha a necessary condiion for a LDGI wih respec o is ha. Proof: We can consruc he following polynomial ha has real roo r and complex roos on he uni circle wih a real par eual o a: 3 h a r ra r. Comparing h and H, here exiss a LDGI if and only if here exiss an a in he uni circle and a > such ha he uadraic and linear coefficiens coincide. This provides wo linear resricions: f a, a a a g a,. Since > ga, can only e saisfied if. For i o e an LDGI he real roo of H mus e in he uni circle. s in Proposiion, he real roo mus e he negaive of he consan erm in H:. Given ha S he lower ound is always saisfied. The upper ound is euivalen o he las condiion in he proposiion. QED Under he case of, we can prove a sronger resul: Corollary 3: ssume. There exiss an LDGI wih respec o if and only if 9

13 { S S S} where S. Hence, for any S, a sufficienly large eliminaes he LDGI wih respec o. Proof: Wih, we can sum fa, and ga, and find { a S }. Since >, we can use his o solve for -a and susiue his ack ino ga, which will now e a funcion only of : g { S S } S S. Since g >, here exiss a LDGI if and only if g. This is only possile if S S. QED Corollary 4: ssume. furher necessary condiion for an LDGI wih respec o is 4. Hence a necessary condiion for an LDGI o exis is if >. Proof: From he previous corollary a necessary and sufficien condiion for an LDGI is f S S S. This reaches a minimum a f ' S * *. S

14 Solving for S and susiuing his value ino fs yields he following necessary condiion for an LDGI f 4 * S or 4. QED This condiion is in conras o he condiions for a LDGI when policy includes he curren inflaion rae. Wih he curren rule,, and S, here exiss a LDGI wih respec o for all values of >. In he case of he ackward rule, enough weigh on pas values of inflaion rule ou he exisence of a LDGI. Thus ackward-looking rules make hese cycles less likely.

15 V. Forward-looking Ineres-Rae Rules We now consider he forward-looking rule and demonsrae ha a LDGI exiss under even weaker condiions, and ha hese condiions are essenially he same condiions ha arise in he coninuous ime analysis of Benhai e al. Log linearizing he euilirium condiions for he forward-looking rule yields p p λ λ where and are as defined earlier. The characerisic euaion is given y he following cuic: { } { } 3 H s efore, we will uilize assumpions : > and H-. Once again H- J-. The variales are all seemingly jump variales. Bu since p λ,, p p p, is predeermined excep for is dependence on so ha we need only one explosive eigenvalue o deermine oh and. Hence, deerminacy of he sysem reuires ha wo roos of H lie ouside he uni circle. s efore H-, so ha H is necessary for deerminacy. Since p H

16 and > we conclude ha > or τ / > is necessary for local deerminacy. This assumes ha >. Proposiion 3 shows ha his is a necessary condiion for an LDGI. Proposiion 3: If > hen here exiss a LDGI wih respec o if and only if here exiss an a in -, and a >, such ha he following condiions are saisfied: f a, a a a a g a, > Proof: The proof mirrors ha of Proposiion. QED Noe ha since >, > is necessary for an LDGI. Corollary 5: ssume, and >. necessary and sufficien condiion for a LDGI wih respec o is. Proof: Noe ha ga, implies ha which implies ha > is necessary for a LDGI. s for sufficiency, solving ga, for a and susiuing his ino fa, yields he following necessary and sufficien condiions for a LDGI wih respec o : 3

17 h and >. ssuming implies ha h is concave. Since h >, hen h as so ha a LDGI exiss. QED This condiion is in conras o he condiions for a LDGI when policy includes he curren inflaion rae. Wih he curren rule, and > implied ha a LDGI was possile if and only if S. Wih he forward looking rule, and >, here exiss an LDGI if and only if, a noicealy weaker condiion. Thus forward elemens makes an LDGI more likely. VI. The Coninuous Time Model Our focus has een on he discree ime model, u for compleeness we noe how his MIUF model maches up wih he MIPF model in Benai e al. In he case of coninuous ime, he log-linearized sysem is given y 4

18 λ p r λ p Noe ha his sysem maches up idenically wih Benhai e al, euaions 9-3. Our corresponds o heir 3. Following heir resuls we have ha a necessary condiion for deerminacy is >. s for ifurcaions, in he coninuous ime model here exiss a LDGI as we vary if and only if r. Noe ha his is idenical o he oucome in he discree-ime forward rule, u is in conras o he discree ime curren or ackward rule model. In he case of a curren rule, here exiss a LDGI if and only if. This laer condiion is sronger y he erm >. s prices approach perfec flexiiliy K, an LDGI arises in he coninuous model whenever U cm. Bu for he discreeime-curren rule, his condiion is γ U mc U mm, D a noicealy sronger condiion. We summarize hese resuls in wo oservaions. Firs, he condiion for a LDGI o exis in coninuous ime is asically idenical o he condiion in he discree-ime forward-looking rule. Second, in he discree ime model, an LDGI is mos likely o occur for a forward-looking rule, leas likely for he pure ackward-looking rule, wih he curren rule somewhere in eween. 5

19 VII. Conclusions Benhai e al. sugges ha anoher way o rule ou LDGI is o adop superinerial rules. Taking our curren looking rule aove and leing we can solve forward o generae he euivalen reacion funcion. R f i i i. s he auhors show a superinerial rule is euivalen o an infiniely looking forward rule. One would hink ha such a policy would generae indeerminacy u a weighed average of oday s inflaion plus all fuure inflaion is acually predeermined since R - is given. n imporan resul of Benhai e al is ha such a rule is no e sujec o ifurcaions over. Ye as noed earlier, here sill may e ifurcaions as we vary oher srucural parameers such as and γ. Furhermore, if ifurcaions arise for >, hen as we vary elow uniy hese cycles may come wih us. More deailed simulaions are necessary o deermine wheher or no superinerial rules canno have cycles. noher concern wih hese superinerial rules is wheher hey are learnale in he sense of E-sailiy see, for example, Evans and Honkapohja. relaed concern is wheher cyclic euiliria are learnale. This commen demonsraes ha in a discree ime model LDGI are much less likely han in a coninuous ime model. coninuous ime model where he cenral ank reacs o a weighed average of all pas inflaion raes closely resemles a discree ime model where he cenral ank reacs o a weighed average of omorrow s, oday s, and all pas inflaion raes. I hus appears ha Benhai e al. s policy 6

20 conclusion ha ackward rules are sujec o cycles is premaure. I appears ha rules wih a forward-looking elemen are mos suscepile o LDGI, while pure ackwardlooking rules are leas suscepile o his prolem. 7

21 References Benhai, J., S. Schmi-Grohe, M. Urie, Moneary Policy and Muliple Euiliria, merican Economic Review 9, March a, Evans, George and Seppo Honkapohja,, Learning and Expecaions in Macroeconomics, New Jersey: Princeon Universiy. Feensra, R.C., Funcional Euivalence eween Liuidiy Coss and he Uiliy of Money, Journal of Moneary Economics 7, March 986, 7-9. Yun, Tack, Nominal Price Rigidiy, Money Supply Endogeneiy, and Business Cycles, Journal of Moneary Economics 37, pril 996,

22 Federal Reserve Bank of Cleveland Research Deparmen P.O. Box 6387 Cleveland, OH PRST STD U.S. Posage Paid Cleveland, OH Permi No. 385 ddress Correcion Reuesed: Please send correced mailing lael o he Federal Reserve Bank of Cleveland Research Deparmen P.O. Box 6387 Cleveland, OH