Investment and Interest Rate Policy: A Discrete Time Analysis

Transcription

1 w o r k i n g p p e r 3 Invesmen nd Ineres Re Poliy: A Disree Time Anlysis by Chrles T. Crlsrom nd Timohy S. Fuers FEDERAL RESERVE BAN OF CLEVELAND

2 Working ppers of he Federl Reserve Bnk of Clevelnd re preliminry merils iruled o simule disuion nd riil ommen on reserh in progre. They my no hve been subje o he forml edioril review orded offiil Federl Reserve Bnk of Clevelnd publiions. The views sed herein re hose of he uhors nd re no neerily hose of he Federl Reserve Bnk of Clevelnd or of he Bord of Governors of he Federl Reserve Sysem. Working ppers re now vilble eleronilly hrough he Clevelnd Fed s sie on he World Wide Web:

3 Working Pper 3- Deember 3 Invesmen nd Ineres Re Poliy: A Disree Time Anlysis by Chrles T. Crlsrom nd Timohy S. Fuers This pper nlyzes he resriions neery o ensure h he ineres re poliy rule used by he enrl bnk does no inrodue lol rel indeerminy ino he eonomy. I ondus he nlysis in Clvo-syle siky prie model. A key innovion is o dd invesmen spending o he nlysis. In his environmen, lol rel indeerminy is muh more likely. In priulr, ll forwrd-looking ineres re rules re subje o rel indeerminy. JEL Clifiion: E4, E5 ey Words: ineres res, monery poliy, enrl bnking Chrles T. Crlsrom is he Federl ReserveBnk of Clevelnd nd my be rehed Chrles..rlsrom@lev.frb.org or (6) , Fx: (6) Timohy S. Fuers is Bowling Green Se Universiy nd my be rehed.fuers@b.bgsu.edu Or (49) ; Fx: (49)

4 I. Inroduion. The elebred Tylor (993) rule posis h he enrl bnk uses firly simple rule when onduing monery poliy. This rule is reion funion linking movemens in he nominl ineres re o movemens in inflion nd poibly oher endogenous vribles. A rule is lled ive if he elsiiy wih respe o inflion (denoed by τ) is greer hn one; he rule is lled pive if τ is le hn one. Reenly here hs been onsiderble moun of ineres in ensuring h suh rules do no hrm. The problem is h by following rule in whih he enrl bnk responds o endogenous vribles, he enrl bnk my inrodue rel indeerminy nd sunspo equilibri ino n oherwise deermine eonomy. These sunspo fluuions re welfre-reduing nd n poenilly be quie lrge. A sndrd resul is h o void rel indeerminy he enrl bnk should respond ggreively (τ > ) o eiher expeed inflion (see Bernnke nd Woodford (997) nd Clrid, Gli, Gerler ()) or urren inflion (see err nd ing (996)). We will refer o he former rule s forwrd-looking nd he ler s urren-looking. These nlyses re ll redued-form siky prie models, where he underlying sruurl model is lbor-only eonomy nd money is inrodued vi money-in-he-uiliy funion (MIUF) model wih zero ro-pril beween onsumpion nd rel blnes (ie., U m =). Our nlysis differs from hose of Bernnke nd Woodford (997), Clrid, Gli, Gerler ()), nd err nd ing (996) on one imporn dimension, he ddiion of Benhbib, Shmi-Grohe nd Uribe () nlyze Tylor rules in oninuous ime MIUF environmen wih n rbirry ro-pril U m nd demonsre h he ondiions for deerminy

5 pil nd invesmen spending. This dded mrgin mkes deerminy muh hrder o hieve. In onrs o hese ppers we demonsre h eenilly ll forwrd-looking rules re subje o lol indeerminy, nd h suffiien ondiion for lol deerminy is for he monery uhoriy o re ggreively o urren movemens in inflion. These findings re of more hn demi ineres sine severl enrl bnks urrenly use inflion fore s n imporn pr of heir deision-mking on poliy iues. A reen pper by Dupor () nlyzes similr siky prie environmen wih invesmen bu omes o subsnilly differen poliy presripions hn hose presened here. He demonsres h pive rule (τ < ) is neery nd suffiien for lol equilibrium deerminy. The eenil differene beween he wo ppers is h Dupor uilizes oninuous-ime model. A key differene beween disree-ime nd oninuous-ime model is he no-rbirge relionship beween bonds nd pil. In disree ime he fuure mrginl produiviy of pil equls he rel ineres re; in oninuous ime ody s mrginl produiviy of pil equls he rel ineres re. This onemporneous ondiion provides n exr resriion in oninuous ime. In model in whih he enrl bnk ondus poliy wih n ineres re insrumen, his exr resriion lers he deerminy ondiions ro he wo models. This pper exends Dupor s () nlysis in hree wys. Firs, s noed, we exmine disree ime model nd demonsre h his iming umpion hs n imporn impliion on equilibrium deerminy. Seond, we onsider more generl uiliy speifiion in h we mke no umpion on he sign of U m. Finlly, he depend on he sign of U m. Their nlysis bsrs from invesmen spending.

6 disree-ime model in he urren pper llows us o exmine boh urren- nd forwrdlooking Tylor rules. The ouline of he pper is s follows. The nex seion develops he bsi model. Seion III provides he deerminy nlysis. Seion IV ompres hese resuls o he oninuous ime nlysis of Dupor (). Finlly seion V onludes. An ppendix proves he min proposiions. II. A Siky Prie Model The eonomy onsi of numerous households nd firms eh of whih we will disu in urn. We re onerned wih iues of lol deerminy. Hene, wihou lo of generliy we limi he disuion o deerminisi model. As is well known, if he deerminisi dynmis re no unique, hen i is poible o onsru sunspo equilibri in he model eonomy. Below we will use he erms rel indeerminy, lol indeerminy, nd sunspo equilibri inerhngebly. Households re idenil nd infiniely-lived wih preferenes over onsumpion, rel money blnes nd leisure given by = β U(,M /P,-L ), where β is he personl disoun re, is onsumpion, nd -L is leisure. We uilize MIUF environmen beuse of is generliy (see Feensr (986)). In onrs o Crlsrom nd Fuers (), we dop he onvenion of end-of-period money blnes in he uiliy funion. We do his o be onsisen wih Dupor s oninuous ime Benhbib e l. (b) ondu globl nlysis of equilibrium deerminy in oninuous ime model 3

7 nlysis. 3 For he purposes of his pper his iming iue is of limied imporne. The uiliy funion is given by: U(,m,-L) V(,m) L. Mos sudies invesiging he ondiions for deerminy hve umed zero ro-pril beween onsumpion nd rel blnes (ie., V m =). In onrs we mke no umpion on he nure of V(,m). The ondiions for deerminy re ompleely independen of V. Of ourse his generliy omes prie. Mos nobly we ume n infinie lbor supply elsiiy. Crlsrom nd Fuers (, b) provide omplemenry deerminy nlysis in flexible prie environmen in whih he deerminy ondiions re independen of lbor supply elsiiy. We hus resri our nlysis o n infinie lbor supply elsiiy. The household begins he period wih M sh blnes nd B - holdings of nominl bonds. Before proeeding o he goods mrke, he household visis he finnil mrke where i rries ou bond rding nd reeives sh rnsfer of s M ( G ) s from he monery uhoriy where M denoes he per pi money supply nd G is he gro money growh re. Afer engging in goods rding, he household ends he period wih sh blnes given by he ineremporl budge onsrin: M s = M M G ) B R B P { w L [ r ( )] } P P ( δ Π. wihou pil. Lol indeerminy is suffiien ondiion for globl indeerminy. Similrly, lol deerminy is neery ondiion for globl deerminy. 3 Crlsrom nd Fuers () refer o his s sh-when-i m-done iming (CWID) s i is umed h he money blnes h id in rnsions re he money blnes h he household hs fer leving he sore. The nurl lernive is sh-in-dvne (CIA) iming. Th is, he money blnes h id in rnsions re he money blnes h he household hs upon enering goods mrke rding so s h A. As noed by Benhbib e l. (), he disree ime nlog o = M M ( G ) B R B oninuous ime MIUF model is CWID iming. 4

8 denoes he households umuled pil sok h erns renl re r nd depreies re δ. The rel wge is given by w while Π denoes he profi flow from firms. The firs order ondiions o he household s problem inlude he following: U L ( ) U ( ) = w () { U ( )[ r ( )]} U ( ) β δ () = U ( ) U ( ) = β R (3) P P U m ( ) R U ( ) R = (4) Equion () is he fmilir lbor supply equion, while () is he e umulion mrgin. Equion (3) is he Fisherin ineres re deerminion in whih he nominl re vries wih expeed inflion nd he rel re of ineres on bonds. Equion (4) is he model s money demnd funion. As for firm behvior, we follow Yun (996) nd uilize model of imperfe ompeiion in he inermedie goods mrke. Finl goods produion in his eonomy is rried ou in perfely ompeiive indusry h uilizes inermedie goods in produion. The CES produion funion is given by Y = { [ y ( i) ] di} ( η )/ η η/( η) where Y denoes he finl good, nd y (i) denoes he oninuum of inermedie goods, eh indexed by i [,]. The implied demnd for he inermedie good is hus 5

9 given by y ( i) = Y P ( i P ) η where P (i) is he dollr prie of good i, nd P is he finl goods prie. Perfe ompeiion in he finl goods mrke implies h he finl goods prie is given by ( ) /( ) [ ( ) η η P i ] di}. (5) P= { Inermedie goods firm i is monopolis produer of inermedie good i. Eh inermedie firm rens pil nd hires lbor from households uilizing CRS Cobb- Dougl produion funion denoed by f(,l) α L -α. Imperfe ompeiion implies h for pymens re disored. Wih z s mrginl os, we hen hve r = z f (,L ) nd w = z f L (,L ). Sine for mrkes re ompeiive, he inermedie goods firms ke z s given. As for inermedie goods priing, we follow Yun (996) nd uilize he umpion of sggered priing in Clvo (983). Eh period frion (-ν) of firms ge o se new prie, while he remining frion ν mus hrge he previous period s prie imes sedy-se inflion (denoed by π). This probbiliy of prie hnge is onsn ro ime nd is independen of how long i hs been sine ny one firm hs ls djused is prie. Suppose h firm i wins he Clvo loery nd n se new prie in ime. I s opimizion problem is given by: mx P ( i) j= ( νβ ) j Λ j Λ P ( i) P η P ( i) Y z P 6

10 where Λ ( j) / P j U j ondiion is given by denoes he mrginl uiliy of dollr. The opimizion η j= P ( i) = ( η ) ( νβ ) ( νβπ ) j= j Λ j j P η j Λ Y j j P z η j j Y j (6) If ν = so h ll pries re flexible eh period, z = (η-)/η <. This ler erm z (η-)/η is mesure of he sedy-se disorion rising from monopolisi ompeiion. In he se of siky pries (ν > ), z will no ypilly equl z nd will refle he ime vrying monopoly disorion. A reursive ompeiive equilibrium is given by sionry deision rules h sisfy (3), (4), (5), (6), nd he following: U L ( ) U ( ) = z f ( ) (7) L U ) β{ U ( )[ z f (, L ) ( δ )] (8) ( = } ) ( δ ) = f (, L Y (9) Noe h equions (7)-(9) re eenilly he rel busine yle (RBC) ondiions disored by mrginl os nd he effe of rel money blnes on he mrginl uiliy of onsumpion. This ler disorion is proxied by he nominl re of ineres. If hese disorions were held fixed, we would hve he RBC model, nd would hus be ured of unique equilibrium. Indeerminy rises beuse of endogenous fluuions in he nominl ineres re nd mrginl os. 7

11 To lose he model we need o speify he enrl bnk reion funion. In wh follows we ume reion funion where he urren nominl ineres re is funion of inflion. 4 We will onsider wo vriions of his simple rule: τ π i π R = R, where τ, R =, π β where i = is forwrd-looking rule, nd i = is urren-looking rule. Under ny suh ineres re poliy he money supply responds endogenously o be onsisen wih he ineres re rule. I is his endogeneiy of he money supply h leds o he poibiliy of indeerminy. By rel indeerminy, we men siuion in whih he behvior of one or more rel vribles is no pinned down by he model. This poibiliy is of gre imporne s i immediely implies he exisene of sunspo equilibri whih in he presen environmen re neerily welfre reduing. III. Equilibrium Deerminy. We will now disu eh Tylor iming onvenion in urn. Beuse we re ineresed in highlighing he effes of pil on he deerminy ondiions, in eh subseion we will firs presen resuls for lbor-only eonomy nd hen noe how he deerminy rnge is ffeed by he inlusion of he invesmen mrgin. Forwrd-Looking Tylor Rules: Consider lbor-only eonomy in whih produion is liner in lbor (α = ). Combining (3), (7) nd he forwrd-looking poliy rule gives 4 Inluding oupu in he Tylor rule would hve only minor effes on he lol deerminy ondiions. 8

12 z = z π π τ. () The ildes denoe log deviions from sedy-se vlues, exep for he inflion re in whih se i is simply he deviion from he sedy-se vlue. As for mrginl os, Yun (996) demonsres h (5)-(6) n be ombined o yield he following loglinerized Phillips urve π z π, () = λ β where π = (P /P - ), denoes he inflion re, nd λ = (-ν)(-νβ)/ν. Equions ()-() represen sysem in z nd π. This is eenilly he model nlyzed by Clrid, Gli, nd Gerler (). Their nlysis suggesed h o void indeerminy he enrl bnk should re ggreively (bu no oo ggreively) o fuure inflion. In priulr, he neery nd suffiien ondiion for deerminy is ( β ) λ < τ < λ. For plusible prmeer vlues (eg., β is lose o one, λ nd α eh bou /3) indie h he deerminy rnge is quie lrge wih n upper bound of bou 3. Hene, n ggreive (bu no oo ggreive) forwrd-looking rule is deermine in his environmen. Adding pil bsilly elimines ll hese deermine equilibri. One imporn reson is beuse in he model wih pil here is lwys zero eigenvlue. To see his subsiue (8) nd (9) ino (7) o obin R π = z k f ( ) ( δ ). We hus bsr from i for simpliiy. 9

13 This is n rbirge relionship linking he rel reurn on bonds o he rel reurn on pil umulion. Noe h boh elemens re forwrd-looking. In priulr, for forwrd-looking poliy rule his rbirge relionship does no depend on ime vribles. This immediely sugge zero eigenvlue. 5 Sine pil is he only se vrible in he sysem zero eigenvlue implies h if here is deerminy pil mus immediely jump o he sedy se. This sugge h even if here is deerminy he welfre properies of he rule would be dissrous. Conversely, if here re sunspos, one n lwys onsru hem in model in whih pil is se o sedy-se for ll periods. Hene, he ondiions for deerminy in model wih pil re les s igh s in he model wih lbor. Bu in f, he ondiions re suffiienly igher: (NEED MORE DISCUSSION HERE.) Proposiion : Suppose h monery poliy is given by forwrd-looking Tylor rule. In he Clvo siky prie model wih invesmen neery ondiion for deerminy is h τ be in one of he following wo regions: ( 3 β ) ( β ) < λ( τ ) < min,, α α < λ( τ ) < ( β ), α where = β ( δ )( α) nd = β ( ). δ 5 The ppendix formlly shows he presene of zero eigenvlue. The lbor equion hs ime elemens bu n be subsiued ou given he umpion of liner leisure. However, zero eigenvlue rises even in models wihou n infinie lbor supply elsiiy seprble preferenes re suffiien ondiion.

14 These regions for deerminy re remrkbly nrrow. Suppose h α = λ = /3, δ =., nd β =.99, so h.35 nd.3. In his se, he firs region is n empy se, nd he seond region is < τ <.7. In omprison o he lbor-only eonomy, he presene of pil mkes deerminy eenilly impoible. Curren-Looking Tylor Rules: As before, onsider he lbor-only eonomy. The sysem is given by () nd he ounerpr o () z z = π π τ. I is srighforwrd o show h we hve rel deerminy if nd only if τ >. To undersnd why, ume o he onrry h here is indeerminy when τ >. Consider sunspo inrese in π. Indeerminy implies h z < z nd π < π. If τ > hen mrginl os (z ) nd inflion (π ) mus be inversely reled. Bu he Phillips urve implies h hey move ogeher whih gives us our onrdiion. A similr rgumen n be used o show h when τ < indeerminy is poible. Remrkbly his onlusion is no ffeed by he ddiion of invesmen o he model: Proposiion : Suppose h monery poliy is given by urren-looking Tylor rule. A neery ondiion for deerminy is h τ >. Furhermore if λ > ( β ) where = β ( δ )( α) nd = β ( ) hen in he Clvo siky prie δ model wih invesmen τ > is boh neery nd suffiien ondiion for deerminy.

15 Noe: Given he bove librion α = /3, δ =., nd β =.99, so h.35 nd.3, here n only be indeerminy for some vlues of τ > if pries re exremely siky, λ <.83. Given h λ = (-ν)(-νβ)/ν his implies h le hn 5% of firms djus heir pries every qurer (where ν denoes he probbiliy h firm n djus pries in he urren period). The bove, however, is only suffiien ondiion for deerminy. Even if λ <.5 he rnge for indeerminy is very smll. < τ <.69. IV. A Comprison wih Coninuous Time Anlysis. In reen pper Dupor () ondus similr deerminy nlysis in oninuous ime MIUF model. The environmens pper o be he sme: pries re siky s in Clvo (983), oupu is produed using boh lbor nd pil, nd preferenes re liner over lbor. However, Dupor rehes quie differen ouome: he repors h τ < is neery nd suffiien for lol deerminy. In his seion we will explore he resons for he differen resuls. Dupor umes uiliy funion of he form U(,m,-L) ln() V(m) L. For simpliiy we minin his resriion. Following Proposiion, however, we need no mke ny umpion regrding V(,m). We will firs presen disree ime nlysis of he model using his funionl form, nd hen urn o Dupor s oninuous ime version. To ompre he models we mus speify wheher in disree ime we hve urren- or forwrd-looking Tylor rule. In oninuous ime limi, here is no disinion beween urren-looking nd forwrd-looking rule. We iniilly ume h he disree ime model is given by forwrd-looking rule sine he omprison beween disree nd oninuous ime is espeilly srigh forwrd nd drmi.

16 Using he Fisher equion (3) he pil umulion equion is R π β{ z f, L ) ( δ )} () = ( Equion () is key in wh follows. As noed bove, his is no-rbirge relionship: he rel reurn on bonds mus be equl o he rel reurn on pil umulion. Wih forwrd-looking rule his expreion is enirely in erms of ime vribles, nd is he use of he zero eigenvlue bove. Noe h sine sh blnes re seprble, he money demnd urve (4) is irrelevn for deerminy iues. Money is he residul h n be bked ou he end. Le x L/. The log-linerized equilibrium ondiions re given by z = α x (3) τ ) ) (4) ( π = z ( α x = ( τ ) π (5) ( α) Y x = (6) π z π, (7) = λ β where β ( ) δ, β /(ρ), nd denoes sedy-se onsumpion, e. The oninuous ime ounerpr o his disree-ime sysem is given by z = α x (8) ) x (9) ( τ π = z ( α ) = ( τ ) π () 3

17 ( α) Y = x () π λ () = ρπ z The fundmenl differene in his sysem is h he no-rbirge pil umulion (9) equion is solely in ime- vribles. We n use (8) nd (9) o elimine z nd x from he sysem. Doing so yields he following: π = λ( α) Y ( α) τ ρ λα ( τ ) / ( α) Y ( τ ) π For deerminy, we need exly one negive eigenvlue. By inspeion, we hve one eigenvlue equl o / >. The remining wo re he soluion o he following qudri equion: h q) q [( λα( τ )) ρ ] q λ( α) ( τ ). ( In oninuous ime deerminy requires h one roo be negive nd one be posiive. The sign of h() is given by he sign of (τ-). Sine h s eiher q - or q, τ < is boh neery nd suffiien for deerminy. (This is nlogous o Dupor s Theorem.) If τ > he sysem is eiher overdeermined or underdeermined depending on he sign of h ( ) = [( λα( τ ) / ) ρ]. If his erm is posiive, he remining wo roos re negive nd he sysem is underdeermined. If his erm is negive, he remining wo roos re posiive nd he sysem is overdeermined. In he former se, here re oninuum of equilibri, while in he ler here re no sionry equilibri. This orresponds o Dupor s Theorem. 4

18 How does he nlysis differ in disree ime? The key differene is h he norbirge equion (4) is enirely in ime vribles so h i does no provide ny resriion on ime- behvior. This omprison n be mde ex in he following wy. Sroll he expreions (3), (5), (6) nd (7) forwrd one period. Srolling everyhing bu he pil umulion forwrd implies h he sysem is given by = x z α (3) ) ( ) ( = x z α π τ (4) ) ( = π τ (5) ) ( = x Y α (6) = z π β λ π, (7) plus he following ime resriions x z = α (8) ) ( = π τ (9) x Y ) ( = α (3) = z π β λ π. (3) Using (3) nd (4) o elimine z nd x from he sysem, we n wrie he dynmi pr of he sysem in mrix form s [ ] = ) ( ) ( ) ( ) /( ) ( ) / ( Y Y π τ α α β τ λα β α λ π τ 5

19 Or Aψ Bψ = Noie he similriy in he disree ime versus oninuous ime mries. This sysem hs hree eigenvlues. The ime resriions (7)-(3) provide hree resriions on he equilibri. Bu we hve seven unknowns. Hene, for deerminy, we need ll hree of he eigenvlues o lie ouside he uni irle. By inspeion, one of he eigenvlues is given by /. The remining wo re he soluion o he following qudri equion: g( q) βq { λ( τ )[ α ( α) ] ( β ) } q λα( τ ). Noie he similriy beween his qudri nd he h(q) funion h rises in he oninuous ime model. In oninuous ime we need only one explosive roo; in disree ime we need wo explosive roos. The qudri g(q) is idenil o he qudri equion h rises in he proof of Proposiion (see he Appendix). Hene, here is indeerminy for ll vlues of τ. We hve now duplied he resuls of Dupor nd demonsred how he lol indeerminy h rises wih he disree ime model beomes lol deerminy (wih τ < ) in he se of oninuous ime. The mrix for he oninuous ime se is he oninuous ime nlogue o he disree ime mrix. However, for deerminy we need boh roos o be explosive wih disree ime nd only one explosive roo for oninuos ime. The key differene is h in oninuous ime here is n ddiionl resriion: ime he mrginl produiviy of pil equls he rel ineres re while in disree ime model his relionship is in erms of he fuure relizions of reurns. Therefore he key differene is in he no-rbirge relionship beween bonds nd pil. 6

20 Wh bou he omprison beween oninuous ime nd disree ime model wih urren-looking rule? As noed before wih oninuous ime here is no disinion beween urren nd forwrd-looking rules. The differene beween he disree-ime model wih urren-looking rule nd oninuous ime model is jus s sriking. A neery (nd suffiien for plusible prmeer vlues) ondiion for deerminy wih urren-looking rule is τ >, while wih oninuous ime τ < is neery nd suffiien. One gin he exr resriion presen wih oninuos ime plys n imporn role in he differene beween he wo models. V. Conlusion. The enrl iue of his pper is o idenify he resriions on he Tylor ineres re rule needed o ensure rel deerminy. A li resul in he lierure is h n ggreive response o fuure or foresed inflion is suffiien for deerminy (eg., Bernnke nd Woodford (997) nd Clrid, Gli, Gerler ()). These erlier nlyses ignore he enrl role of invesmen spending. The role of invesmen ro he busine yle hs long rdiion in monery eonomis so h ignoring i seems like bd ide. We hve demonsred bove h in he se of forwrd-looking poliy, inlusion of he invesmen hoie drmilly shrinks he region of deerminy. The end resul is h for nyhing bu he mos exreme prmeer vlues his model wih Clvo sikine implies h monery poliy mus respond ggreively o urren inflion o genere deerminy. In shor, here is ler dnger o ny poliy h is forwrd-looking. Appendix 7

21 Proposiion : Suppose h monery poliy is given by forwrd-looking Tylor rule. Then in he Clvo siky prie model wih invesmen neery ondiion for deerminy is h τ be in one of he following wo regions: ( 3 β ) ( β ) < λ( τ ) < min,, α α < λ( τ ) < ( β ), α where = β ( δ )( α) nd = β ( ). δ Proof: Given he umpion h uiliy is liner in leisure he firs order ondiions (7) nd (8) n be wrien s x α U ( ) = where x = z ( α), L α ( U ( ) { αz x ( )}) (A) U ( ) = β (A) δ Subsiuing (A) nd equion (A) srolled forwrd one period ino (A) yields x z α x = β z We n expre his s α { αz x ( } α δ ). (A3) x = F(x,z,z ). (A4) The resoure onsrin (9) provides noher equion: α = x ( δ). (A5) 8

22 The money demnd urve (4) implies h rel blnes depend only on nd R. Using (A) we hen hve h depends only on R, z nd x. Using he poliy rule we n hen wrie (A5) s = G(x,π,,z ). (A6) Using (A) nd he poliy rule, we n expre he Fisher equion (3) s z = H(π,x,x,z ). (A7) In summry, we hve hree equions (A4), (A6), nd (A7). The Phillips urve () yields π = P(π, z ). The linerized Euler equions re given by x = F(x,z,z ) = G(x,π,,z ) z = H(x,x, π,z ) π = P(π,z ). Le w denoe he veor [x, z, π, ] so h he linerized sysem n be expreed s Aw = Bw where A nd B re 4x4 mries wih elemens given by he derivives of F, G, H nd P. Afer invering A, we re lef wih he mrix A - B whih hs four eigenvlues. Sine here is only one se vrible in his sysem ( ), we need hree explosive eigenvlues for deerminy. One gin one eigenvlue is zero, while noher is given by > where denoes sedy-se levels. Hene, neery nd suffiien ondiion for deerminy is h he remining wo roos be ouside he uni irle. The relevn qudri equion is given by 9

23 J q Jq J where J = β J = λ τ ) ( β ) ( J = α λ( τ ) = β ( δ )( α) nd = β ( ). δ The bove implies J ( ) = λ ( τ )( α). If τ < hen J() < nd J() > whih mens one roo is in (,). Hene, neery ondiion for deerminy is τ >. Under his resriion, J() > so h ddiionl neery ondiions re J() > nd J(-) >. These pu n upper bound on τ: ( β ) ( τ ) < min α α λ, (A) To mke furher progre nd ighen he bound more losely, we mus onsider he wo ses where he soluions o J re rel or omplex. Suppose firs h he wo roos of J re rel: We firs noe h J is qudri nd onvex. In his se (uming he roos re rel, A, nd τ > ) we hve he following wo poenil regions of deerminy: J J < J nd >, or (A3) J

24 ( 3β ) < λ( τ ) nd ( β ) < λ( τ ) < (A4) Combined wih (A) nd τ > hese wo regions of deerminy (uming he roos re rel) re ( β ) < λ( τ ) < nd (A3) ( 3 β ) ( β ) < λ( τ ) < min,. (A4) α α Suppose insed h he roos re omplex: The norm of hese roos is given by J J αλ( τ ) =. β Sine J >, his yields he following neery nd suffiien ondiion for deerminy (if he roos re omplex): < λ( τ ) < ( β ). (A5) α Combining he rel nd omplex regions nd noing h α < neery ondiion for deerminy is h τ be in one of he following wo regions: ( 3 β ) ( β ) < λ( τ ) < min,, α α < λ( τ ) < ( β ). α

25 QED Proposiion : Suppose h monery poliy is given by urren-looking Tylor rule. A neery ondiion for deerminy is h τ >. Furhermore if λ > ( β ) where = β ( δ )( α) nd = β ( ) hen in he Clvo siky prie δ model wih invesmen τ > is boh neery nd suffiien ondiion for deerminy. Proof: The linerized Euler equions re given by x = F(x,z,z ) = G(x,π,,z ) z = H(x,x, π,π,z ) π = P(π,z ). Le w denoe he veor [x, z, π, ] so h he linerized sysem n be expreed s Aw = Bw where A nd B re 4x4 mries wih elemens given by he derivives of F, G, H nd P. Afer invering A, we re lef wih he mrix A - B whih hs four eigenvlues. Sine here is one se vrible in his sysem ( ), we need hree explosive eigenvlues for deerminy. One gin one is given by > where denoes sedy-se levels. Hene, neery nd suffiien ondiion for deerminy is h wo of he remining hree roos be ouside he uni irle. Hene, here mus be one rel roo wihin he uni irle. The relevn ubi equion is given by J 3 3q J q Jq J

26 where J = β 3 J J λ ( β = ) = αλ λτ J = α λτ = β ( δ )( α) nd = β ( ). δ The bove implies J() < nd J ( ) = λ ( τ )( α). For deerminy we need exly one rel roo in (,). Hene, τ > is neery for deerminy. The umed ondiion in he proposiion implies h J is onve. Hene, he remining wo roos, eiher rel or omplex, mus be ouside he uni irle. QED 3

27 Referenes Benhbib, J., S. Shmi-Grohe, M. Uribe, Monery Poliy nd Muliple Equilibri, Amerin Eonomi Review 9, Mrh, Benhbib, J., S. Shmi-Grohe, M. Uribe, The Perils of Tylor Rules, Journl of Eonomi Theory 96(), Jnury-Februry b, Bernnke, B. nd M. Woodford, Inflion Fore nd Monery Poliy, Journl of Money, Credi nd Bnking 4 (997), Clvo, G.A., Sggered Pries in Uiliy-Mximizing Frmework, Journl of Monery Eonomis (3), Sepember 983, Crlsrom, C., nd T. Fuers, Forwrd vs. Bkwrd-Looking Tylor Rules, Federl Reserve Bnk of Clevelnd working pper,. Crlsrom, C., nd T. Fuers, Timing nd Rel Indeerminy in Monery Models, Journl of Monery Eonomis, April, Crlsrom, C., nd T. Fuers, Rel Indeerminy in Monery Models wih Nominl Ineres Re Disorions, Review of Eonomi Dynmis 4, b, Clrid, Rihrd, Jordi Gli, nd Mrk Gerler, Monery Poliy Rules nd Mroeonomi Sbiliy: Evidene nd Some Theory, Qurerly Journl of Eonomis 5 (), Dupor, Willim, Invesmen nd Ineres Re Poliy, Journl of Eonomi Theory 98,, err, Willim, nd Rober ing, Limis on Ineres Re Rules in he IS-LM Model, Federl Reserve Bnk of Rihmond Eonomi Qurerly, Spring 996. Tylor, John B. (993), Disreion versus Poliy Rules in Prie, Crnegie- Roheser Series on Publi Poliy 39, Yun, Tk, Nominl Prie Rigidiy, Money Supply Endogeneiy, nd Busine Cyles, Journl of Monery Eonomis 37(), April 996,

28 Federl Reserve Bnk of Clevelnd Reserh Deprmen P.O. Box 6387 Clevelnd, OH 44 PRST STD U.S. Posge Pid Clevelnd, OH Permi No. 385 Addre Correion Requesed: Plese send orreed miling lbel o he Federl Reserve Bnk of Clevelnd Reserh Deprmen P.O. Box 6387 Clevelnd, OH 44