working Learning and the Central Bank by Charles T. Carlstrom and Timothy S. Fuerst

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1 working p a p e r 0 7 Learning and he Cenral Bank by Charles T. Carlsrom and Timohy S. Fuers FEDERAL RESERVE BANK OF CLEVELAND

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 been subjec o he formal ediorial review accorded official Federal Reserve Bank of Cleveland publicaions. 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 available elecronically hrough he Cleveland Fed s sie on he World Wide Web:

3 Working Paper 0-7 Ocober 00 Learning and he Cenral Bank by Charles T. Carlsrom and Timohy S. Fuers I is well known ha sunspo equilibria may arise under an ineres rae operaing procedure in which he cenral bank varies he nominal rae wih movemens in fuure inflaion (a forwardlooking Taylor rule). This paper demonsraes ha hese sunspo equilibria may be learnable in he sense of E-sabiliy. JEL Codes: D5, E4, E5 Key Words: general equilibrium, money and ineres raes, moneary policy Charles T. Carlsrom is a he Federal Reserve Bank of Cleveland. Timohy S. Fuers is a Bowling Green Sae Universiy. The auhors hank John Carlson for ineresing hem in his opic. They also hank Jim Bullard and Seppo Honkapohja for helpful commens. Charles T. Carlsrom may be reached a ccarlsrom@frb.clev.org or (6)

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5 I. Inroducion. The celebraed Taylor (993) posis ha cenral bank behavior can be described by a fairly simple rule linking nominal rae movemens o movemens in inflaion and oupu. This seminal paper has spawned a large lieraure concerned wih issues of sabiliy: under wha siuaions can a Taylor-rule formulaion of moneary policy creae real indeerminacy and hus sunspo flucuaions in he model economy? See for example, Benhabib, Schmi-Grohe and Uribe (999), Bernanke and Woodford (997), Carlsrom and Fuers (00a,00b,000a), Clarida, Gali and Gerler (000), and Kerr and King (996). As forcefully argued by Evans and Honkapohja (00), sunspo equilibria are compelling only if hey are no fragile o reasonable assumpions abou learning. We follow Evans and Honkapohja (00), and inerpre learning as E- sabiliy, so ha an equilibrium is fragile if i is no E-sable. The issue raised in his paper is wheher he sunspo equilibria induced by some Taylor-rules are E-sable.,3 A robus resul of he papers on indeerminacy is ha sunspos are mos likely in cases in which he cenral bank responds o forecased inflaion. We will hus focus on Taylor rules in which he cenral bank responds o forecased inflaion. Honkapohja and Mira (00) analyze he basic moneary model considered here, and conclude ha he sunspo equilibria arising from a forward-looking moneary policy are no E-sable. 4 The erm sunspo is in one sense misleading since hese shocks are accommodaed by moneary policy. Bu we use he erm since he cenral bank inroduces real indeerminacy by responding o forecass which can be driven by sunspos. E-sabiliy ypically implies ha leas-squares learning converges o he raional expecaions equilibrium, alhough here are some echnical issues in he case of a coninuum of equilibria (as is he case wih he sunspo equilibria examined below). See Evans and Honkapohja (00) for a deailed discussion. 3 Woodford (990) was he firs o demonsrae he learnabiliy of saionary sunspo equilibria in an overlapping generaions model. 4 However, Honkapohja and Mira (00) demonsrae ha resonan frequency sunspo equilibria may be

6 They show ha he only equilibria ha are E-sable are he minimum sae vecor (msv) soluions where inflaion depends only on fundamenal shocks. McCallum (00) concludes from his ha only he msv soluion is empirically relevan. In his paper we consider wo varians of he analysis in Honkapohja and Mira (00) and demonsrae he exisence of E-sable sunspo equilibria. Firs, we consider a differen iming scenario. A microfoundaion of he model analyzed by Honkapohja and Mira is ha money balances a he end of goods marke rading is wha aids in ransacions. Carlsrom and Fuers (00) refer o his as cash-when-i m-done (CWID) iming. In a model wih CWID iming Honkapohja and Mira demonsrae ha sunspo equilibria are no E-sable. Bu CWID is a peculiar iming convenion. In conras, suppose ha cash balances held in advance of goods rading are he balances ha aid in ransacions, wha Carlsrom and Fuers (00) call cash-in-advance (CIA) iming. One conribuion of his paper is o demonsrae ha in a model wih CIA iming here exis E- sable sunspo equilibria. Our second modeling variaion is a differen assumpion on he naure of learning. Honkapohja and Mira (00) examine a model in which here is symmeric learning by boh he public and he cenral bank. Tha is, boh he cenral bank and privae secor have common expecaions. This can be inerpreed as he privae secor learning, and he cenral bank operaing off of privae secor forecass. In conras, his paper examines a case in which he forecass of he cenral bank and privae secor differ, and coincide only in he long run. There are many possible differenial learning scenarios. Here we ake one exreme: We assume ha only he cenral bank is subjec o a learning process, while learnable under cerain policy rules.

7 privae secor expecaions are always raional. This assumpion is analogous o he assumpion in Sargen s (999) analysis of The Conques of American Inflaion. A second conribuion of his paper is o demonsrae ha in he case of cenral bank learning he sunspo equilibria are ypically E-sable. In essence, cenral bank policy can lead he public o believe in sunspos. The ouline of he paper is as follows. In he nex secion we presen he basic CWID moneary model and he resuls of Honkapohja and Mira (00). We hen consider he CIA varian of his model. Here sunspos can be learnable. In secion 3, we demonsrae ha sunspos are ypically learnable when here is asymmeric learning and i is he cenral bank doing he learning. We also briefly discuss he limplicaions of differen ypes of asymmeric informaion. Secion 4 concludes. I. Symmeric Learning in a Sicky Price Model. A. Sunspos and Learnabiliy in he CWID Model. The analysis is conduced using he now-sandard sicky price model ha is given by he following wo equaions: = λ + βe + z + u () z () [ ] = R E + + E z+ where u = ρ + ε, u ln( P ) ln( P ) denoes he inflaion rae from ime - o ime, z denoes marginal cos, R is he nominal ineres rae beween and +, and u denoes a shock o he 3

8 pricing equaion. 5 All variables are expressed as log deviaions from he non-sochasic seady-sae. Below we find i convenien o make he weak assumpion ha β+λ>. To close he model we need o specify he cenral bank reacion funcion. In wha follows we assume a reacion funcion where he curren nominal ineres rae responds o expeced inflaion: R = τ, + E (3) where τ > 0 is he response of he nominal ineres rae o movemens in expeced inflaion. Under any such ineres rae policy he money supply (no modeled) responds endogenously o saisfy he ineres rae rule. I is his endogeneiy of he money supply ha leads o he possibiliy of real indeerminacy and sunspo equilibria. Tha is, here is real indeerminacy if differen money growh rules suppor he ineres rae arge (3). 6 These are hen associaed wih differen real oucomes because of he sicky price assumpion (). To proceed, use () o eliminae z from he sysem: E β + u = λ R E + ] + E + E β + [ ρ u. (4) Using (3) o eliminae he nominal rae, we have a second-order difference equaion in. For deerminacy, we need boh roos of he corresponding characerisic equaion o be ouside he uni circle. Sraighforward calculaions imply ha here is real deerminacy if 5 See Clarida, Gali, and Gerler (000), and he references herein. Following Yun (996), Carlsrom and Fuers (000b) demonsrae ha wih a linear producion echnology, he sysem can be wrien in he marginal cos form used above. In his case, λ represens he link beween marginal cos and prices, while in he Clarida, Gali, and Gerler (000) framework λ represens he link beween oupu and prices. One can ransform he curren model by replacing our λ wih Clarida e al. s λσ, where σ is he elasiciy of ineremporal subsiuion. 6 See Carlsrom and Fuers (000b) for a discussion. 4

9 and only if ( β + ) + λ < τ <. λ For reasonable calibraions (β =.99, λ =.3), he upper bound is quie high, abou 4, so ha he basic conclusion is ha a τ greaer han uniy will achieve deerminacy. If here is deerminacy, he equilibrium can be wrien as = γ msv u where γ msv is unique and denoes he minimum sae vecor (msv) soluion. If τ lies ouside he deerminacy region, hen we sill have he MSV soluion above bu more imporanly for his analysis we also have an AR soluion. There are wo cases o consider. For τ < only one roo of he characerisic equaion given by (4) is explosive, while he oher is in (0,). If τ > ( β + ) + λ, one roo is explosive while he oher is in λ (-,0). In eiher case we have real indeerminacy and muliple equilibria. In paricular here are sunspo equilibria given by + = α + γu + σε + + σ s+ (5) where α (-,) is unique, γ γ msv is unique, σ and σ are arbirary, ε + is he innovaions in he u process, and s + is an arbirary iid, mean-zero sunspo shock. Noe ha alhough he msv soluion uniquely deermines he response of + o ε +, σ is arbirary in he case of sunspo equilibria because boh ε + and s + are whie noise. Are hese sunspo equilibria learnable? Following he mehodology oulined in Evans and Honkapohja (00), posi he following perceived law of moion (PLM): 5

10 s. (PLM) = a + bu + cε + d Noice ha his PLM has he same form as he sunspo equilibria (5). Using his PLM scrolled forward o eliminae he forecass in he equilibrium condiion (4), we can hen solve for he implied acual law of moion (ALM): s. (ALM) = a + bu + cε + d By replacing all expecaions wih his common PLM, we are assuming symmeric learning beween he public and he cenral bank. 7 We now have he mapping T(a,b,c,d ) = (a,b,c,d ). The fixed poins of his T-mapping are he raional expecaions equilibria. An equilibrium is said o be E-sable if his mapping is sable evaluaed a he equilibrium in quesion. Bullard and Mira (000) sudy he E-sabiliy of he msv equilibrium. 8 Our focus is on sunspo equilibria. I is sraighforward o demonsrae ha if agens know when forecasing + and +, hen he coefficien a maps ino zero so ha he sunspo equilibria are no E- sable. Hence, Honkapohja and Mira (00) exend he analysis by assuming ha when forming expecaions agens do no know, so ha ime- forecass are funcions only of - and he exogenous shocks. As noed by Evans and Honkapohja (00), his increases he chances for E-sabiliy. One conribuion of Honkapohja and Mira (00) is o demonsrae ha even in his case he sunspo equilibria are sill no E-sable so ha sunspos are no learnable. 9 7 In he nex secion we will consider a paricular form of asymmeric learning in which only he cenral bank is learning. In his case we replace only he cenral bank s forecas wih he PLM. 8 I is imporan o noe ha our PLM does no include a consan erm, while a consan erm is cenral o he resuls in he Bullard-Mira paper. 9 However, Honkapohja and Mira (00) demonsrae ha a differen ype of equilibria, resonan frequency sunspo equilibria, may be learnable under cerain policy rules. 6

11 B. Sunspos and Learnabiliy in he CIA Model. Before abandoning he possibiliy of E-sable sunspos in he case of symmeric learning, consider he alernaive money-demand iming convenion suggesed by Carlsrom and Fuers (00). The Fisher equaion given by () has as is microfoundaions he assumpion ha money balances a he end of he period (afer leaving he goods marke) aid in ransacions wha Carlsrom and Fuers call cashwhen-i m-done iming (CWID). If we insead assume ha cash available before enering he goods marke aid in ransacions wha Carlsrom and Fuers call cash-in-advance iming (CIA), equaion () becomes z. (6) = [ R+ E + ] + E z+ As before we use () o eliminae z from he sysem. E β + u = λ R + E + ] + E + E β + [ ρu (7) In his case Carlsrom and Fuers (00) demonsrae ha here is real indeerminacy under he forward-looking Taylor rule for all values of τ. Are any of hese sunspo equilibria E-sable? Yes, bu only a few. We firs characerize he indeerminacy, and hen look a E-sabiliy. Proposiion : Under he assumpion of CIA iming here is real indeerminacy for all values of τ. In paricular: a. If τ <, he equilibria are characerized by he AR() process AR() (8) + = α + γu + σε + + σ s+ where 0 < α < is unique, γ is unique, and σ and σ are arbirary. 7

12 b. If ( + β + λ) 4β * < τ < τ, here are wo sable real roos o he characerisic 4λ equaion, so ha here are wo disinc AR() processes of he form (8) where 0 < α < akes on one of hese wo values. There are also AR() equilibria characerized by + β + λ + = + + γu + σε + + σ s+. AR() (9) β + λτ β + λτ c. If τ > τ *, he roos of he characerisic equaion are complex wih norm in (0,) so ha he equilibria are characerized by he AR() process (9). Proof: Since quesions of deerminacy depend only upon deerminisic dynamics, he proof focuses only on he AR coefficiens wihou loss of generaliy. The characerisic equaion of (7) is given by h( e) = ( β + λτ ) e ( + β + λ) e +. We have h(0) > 0, h (0) < 0, and h() = λ(τ-). Hence, if τ < here is one roo in (0,) and one ouside (0,). Since here are no predeermined variables we have real indeerminacy. Now suppose ha τ >. In his case we have h () > 0. Hence, if he roos are real, hey are boh in (0,). These wo roos are boh possible AR() coefficiens. Alernaively, we can wrie his as he AR() in (9). The roos are real if and only if ( + β + λ) > 4( β + λτ) Solving his for τ yields he τ * in he proposiion. If he roos are complex, heir norm is in (0,) and he equilibria are hen characerized by he AR(). QED In conras o he CWID model in which here is indeerminacy only for very small 8

13 or very large values of τ, Proposiion implies ha in he case of CIA iming real indeerminacy arises for all values of τ. Noe ha he naure of he equilibria varies around τ =. For τ <, he sunspo equilibria are of he AR() form given by (8), while for τ > here are sunspo equilibria of he AR() form given by (9). I urns ou ha sunspos may be learnable wih CIA-iming precisely because for τ > here is double indeerminacy so ha sunspo equilibria are of he AR() form. We will now urn o E-sabiliy of hese equilibria. If we assume ha is known when generaing forecass he earlier discussion applies and he sunspo equilibria are no E-sable. Hence, we once again mus resric he informaion se by assuming ha is no known when generaing forecass. Proposiion : Assume CIA iming and ha is no observable for ime- forecasing. For τ < he AR() equilibria given by (8) are no E-sable. However, for < τ < ( + β + λ) λ β he AR() equilibrium given by (9) are E-sable. Proof: Le us firs consider he AR() case. Suppose ha he PLM is given by s. = a + bu + cε + d Under he assumpion ha is no observable for ime- forecasing, we have E + = a + ab u + ac ε + ads + b ρu + b 3 E a + a b u + a c ε + a d s + b a + ρ)( ρu + ε ) + = ( Subsiuing his ino (7) we have ha he PLM maps ino he ALM via: ε T ( a a ) = ( + β + λ) a ( β + λτ ) 3 9

14 T ( b ) = ( + β + λ)( a + ρ) b ( β + λτ )[ a + ρ( a + ρ)] b + ( ρ) ρ T3 ( c) = ( + β + λ)( ac + b ) ( β + λτ )[ a c + b ( a + ρ)] + ( ρ) T ( a d 4 d) = ( + β + λ) ad ( β + λτ ) where he variable in parenhesis is wha he funcion maps ino. Since his sysem is diagonal he eigenvalues are T (a ), T (b ), T 3 (c ), and T 4 (d ). I is sraighforward o demonsrae ha a a = α, T 3 (c ) = T 4 (d ) =, ie., here are no learning dynamics for he coefficiens on he innovaions. Tha is, your iniial guess of c and d are immediaely learned. Following Evans and Honkapohja (00), his implies ha for E- sabiliy of he sunspo equilibria we need focus only on he mappings of a and b. The E-sabiliy condiion is ha T (a ) < and T (b ) <, evaluaed a he sunspo equilibria. Consider a firs: T ( a a) = ( + β + λ) a 3( β + λτ ) The AR() soluion is α such ha T (α) = α. Using his fac we have ha E-sabiliy requires α > + β + λ I is sraighforward o show ha only he larger of he wo real roos saisfies his condiion. When τ <, he larger roo is ouside he uni circle so he AR() equilibria are no E-sable. 0 0 If < τ < τ *, he larger roo is inside he uni circle so ha a = α high is E-sable. In his case, we mus examine T (b ): T ( b ) = + ( + β + λ) ρ ( β + λτ) ρ( a + ρ) For sabiliy, we need his wihin he uni circle. Using a = α and T(α) = α we have ( + β + λ) ρ ( β + λτ) ρ( a + ρ) < 0 0

15 We now analyze he case where τ > so ha we have AR() equilibria. Le he PLM be given by s. = a + a + bu + cε + d Using his PLM and he assumpion ha is no par of he informaion se, we have he following T-mapping from PLM o ALM: 3 T ( a) = ( + β + λ)( a + a ) ( β + λτ )( a + aa) T ( a ) = ( + β + λ) a a ( β + λτ )( a a + ). a T3 ( b ) = [ + β + λ a( β + λτ )]( a + ρ) b ( β + λτ )( a + ρ ) b + ρ( ρ) T4 ( c) = [ + β + λ a( β + λτ )]( ac + b ) ( β + λτ )( ac + b ρ) + ( ρ) T ( a d 5 d) = [ + β + λ a( β + λτ )] ad ( β + λτ ) Noe firs ha afer T (a ) and T (b ) his sysem of derivaives is once again block recursive. This implies ha hree of he eigenvalues of he sysem are given by T 3 (b ), T 4 (c ), and T 5 (d ). As before we have T 4 (c )=T 5 (d ) =, ie., here are no learning dynamics in hese coefficiens. Our focus is on he sysem in a, a, and b. Evaluaing he derivaives a he equilibrium values of a a β + λ = + β + λτ = β + λτ we have T 3 (b ) = -(β+λτ)ρ <. Hence, we need only examine he subsysem in a and Noe ha if ρ = 0, we have insabiliy, bu in his case he sunspo equilibria would no depend upon u -. If ρ > 0, hen E-sabiliy requires + β + λ α > ρ. β + λτ

16 a. The characerisic equaion of his sub-marix (evaluaed a he AR() values) is g( e) = e ( + β + λ) + e, where β + λτ 4 Noe ha g() > 0 and g(0) = -. Recall from Proposiion ha he roos of h (he characerisic equaion of (7)) are real when > 0. If > 0, g(0) < 0 so ha he wo roos of g are below uniy and we have E-sabiliy. If < 0, he roos of g are complex, and we need he real par o be less han uniy. Expressing his condiion in erms of τ yields he expression in he proposiion. QED Proposiion implies ha he AR() sunspo equilibria are learnable for an empirically relevan range. For example, wih β =.99, λ =.3, we have E-sabiliy for < τ < This region includes he celebraed Taylor coefficien of.5. III. Asymmeric Learning in a Sicky Price Model. The former secion made an exreme assumpion: boh he public and he cenral bank have common forecass, boh of which are raional only in he limi. In conras, in his secion we assume ha he privae secor s forecass are raional bu ha he cenral bank uses a forecasing rule ha is raional only in he limi. In his case i is much more likely for real indeerminacy o be learnable. If he cenral bank uses curren inflaion o Curiously, his range ges arbirarily large as he economy approaches a flexible price model (λ ). Ye for an economy which is perfecly flexible (so ha equilibrium is given by (6) wih z = 0) here is real indeerminacy bu sunspos are never learnable (his is immediae given ha no longer eners ino he sysem). This suggess here migh be a problem in he above analysis. The problem may lie wih Honkapohja and Mira s (00) assumpion ha when forming expecaions agens do no know. Bu acual inflaion in he pricing equaion () was assumed observable. Following Yun (996) he microfoundaions of his pricing equaion are ha firms who se prices in ime- base heir prices on he curren price level and forecass of fuure prices. If he curren price level is assumed o be no observable, hen presumably we should replace in equaion () wih he expecaion of given curren informaion. In his case we would have an acual law of moion (ALM) solely in - so ha he coefficien a maps ino zero. (A similar argumen holds in he case of he AR() equilibria.) Under his inerpreaion he sunspo

17 forecas fuure inflaion, and if he public knows ha he cenral bank is doing so, hen he AR() and AR() sunspo equilibria may be learnable. The cenral bank can lead he economy o indeerminacy. A. The CWID Model. Le us begin wih he case of CWID iming. The relevan equilibrium is given by E β + u = λ R E + ] + E + E β + [ ρ u (0) The sunspo equilibria are of he AR() form in (8). Since only he cenral bank is subjec o learning we subsiue he PLM only ino he bank s forecas: R = τe + + cb = τ[ a bu ] () As will soon be eviden, because of he dynamics of asymmeric learning, he sunspo equilibira can be E-sable even if he cenral bank observes when forecasing +. Wihou loss of generaliy, we hus proceed under his assumpion. Noice ha wih asymmeric learning he forward rule wih parameer τ corresponds o (roughly) a curren rule wih parameer τa. Subsiuing () ino (0), we have a second order sysem in. This sysem is indeerminae, wih one roo in he uni circle. This roo is he ALM. Under his mapping, is he AR() coefficien ever E-sable? Yes. Proposiion 3: Assume CWID iming, and cenral bank learning. If τ <, hen here is real indeerminacy and he AR() equilibria of he form (8) are E-sable. If τ > ( β + ) + λ, hen here is real indeerminacy bu he AR() equilibria of he form (8) λ are no E-sable equilibria are no E-sable. This criicism does no apply o he analysis in Secion III as we assume ha is known when making forecass. 3

18 Proof: Subsiuing () ino (0), we have he following sysem: ( + λτ ) = ( + β + λ) E + E β + + ( ρ λτb a ) u In he neighborhood of he AR() equilibria, a = α, his sysem is subjec o indeerminacy so ha we can use he mehod of undeermined coefficiens o solve i for he ALM: = T( a ) + T( b ) u where wihou loss of generaliy we ignore he sunspo coefficiens. The mapping T(a ) is given by he sable roo (he smaller roo) of he sysem: ( + β + λ) T( a ) = ( β + λ) + ( λ β) + 4βλτ a β so ha dt( a ) / da λτ =. ( β + λ) + ( λ β) + 4βλτ a For E-sabiliy we need his o be less han one. Exploiing he fac ha T(α) = α, where α is he AR() soluion, we can eliminae he square roo and obain: dt( a) / da λτ = + β + λ αβ () We can now consider he wo cases: Case : τ > ( β + ) + λ. λ dt( a) / da λτ λ + ( β + ) = > + β + λ αβ + β + λ αβ where he inequaliy follows from he resricion on τ. Since he AR() α < 0 in his case, 4

19 we have ha dt a ) / da, so ha he soluion is no E-sable. ( > Case : τ <. Expression () is increasing in α. Seing α = we have dt( a ) / da λτ < + λ β < where he las inequaliy follows from τ <. Hence, we mus proceed o he T(b ) mapping: λτb ( ρ) T ( b ) =. ( + β + λ) β( α + ρ) For he case τ < we have 0 < α <, so ha T (b ) <. Hence, in he case of τ < we have E-sabiliy. QED Remark: I is curious o noe ha he sunspos fail o be E-sable only when τ is large so ha he equilibria are oscillaory, α < 0. However, Honkapohja and Mira (00) demonsrae ha he resonan frequency sunspo equilibria are learnable when he equilibria are oscillaory. B. The CIA Model. In he case of CIA iming, he relevan equilibrium is given by E β + = λ[ R + E + ] + E + E β + (3) As before, since only he cenral bank is subjec o learning we only replace heir forecass wih he relevan PLM. Recall ha in he CIA model here are wo forms of indeerminacy, depending upon he size of τ. For τ <, we have AR() equilibria of he form in (8), so ha we replace he ineres rae wih 5

20 cb R E = τ [ a b u ] + = τ. (4) In he case of τ >, we have indeerminacy of he AR() form given in (9), and replace he ineres rae wih cb R E = τ [ a + a b u ] + = τ. (5) We now sae: Proposiion 4: Assume CIA iming, and asymmeric learning (cenral bank learning). For τ < he AR() equilibria in (8) are learnable if ρ < β + λ +. For τ > he AR() equilibria in (9) are learnable for all values of ρ. Proof: Case : τ <. Subsiue (4) ino (3). This sysem is indeerminae, wih wo posiive roos, one in (0,). This smaller roo is he ALM and is given by T(a ): x x 4 / β λ( τ a ) + ( + β ) T ( a ) = where x = > 0. β E-sabiliy is given by dt(a )/da <. dt( a) da = dt( a) dx dx da = + x λτ λτα = x 4 / β β λ( τα) + + β αβ where he las equaliy comes from exploiing T(α) = α o eliminae he square roo. This las erm mus be less han uniy for E-sabiliy. This implies here is E-sabiliy if and only if β + λ + α <. (6) ( β + λτ) 6

21 For τ < his is always saisfied as α is he smaller roo of he characerisic equaion. We now mus urn o he b coefficien: λτb ρ ( ρ) T( b ) = ( + β + λ) α( β + λτ) βρ where we are evaluaing his a a = α. For E-sabiliy we need T (b ) <. Imposing his and using he fac ha α is he roo of he characerisic equaion, we have αλτρ T ( b ) = <. αβρ Combining his wih (6), we have ha he equilibria are learnable if and only if ρ < β + λ +. Case : τ >. Subsiue (5) ino (3). This sysem is indeerminae and in he neighborhood of he candidae sunspo equilibria can be expressed as an AR(). This AR() is our ALM: β [( + β + λ λτa ) ( + λτa ) + (/ ρ)( ρ ρλτb u ] + = + ) + We hus have he mapping a ( a + β + λ λτ ) / β a + λτ ) / β. b ( a ( b ρ ρλτ ) / βρ Inspecion reveals ha his is E-sable. QED 7

22 C. Oher Asymmeries. The previous discussion has considered only one of he hree possible asymmeric learning scenarios. In his secion we briefly discuss he oher wo logical possibiliies. Firs, suppose ha he cenral bank has raional expecaions, bu ha he public is subjec o learning. This case is easily deal wih. Since privae secor expecaions are par of bond-pricing, hen he law of ieraed expecaions immediaely implies ha he analysis of his ype of asymmeric learning will exacly parallel Secion II s discussion of symmeric learning. Second, suppose ha boh he cenral bank and he public are subjec o learning, bu ha heir learning is asymmeric. Tha is, he form of heir PLM s are he same, bu he iniial coefficien values in hese PLM s may differ. Maers are a bi more complicaed here, bu he appendix demonsraes ha once again he E-sabiliy condiions for his ype of asymmeric learning are idenical o he E-sabiliy condiions for symmeric learning examined in Secion II. In summary, he only case in which asymmeric learning gives novel resuls (compared o he resuls on symmeric learning in Secion II), in when he public has raional expecaions while he cenral bank is subjec o a learning process. As noed earlier, his is also he assumpion ha Sargen (999) uilizes in his analysis of he grea inflaion. IV. Conclusion. This paper has shown ha he developing consensus ha policy-induced sunspos are no learnable may be premaure. This paper has considered wo modificaions o he ypical model, eiher one of which leads o he learnabiliy of sunspo equilibria. Firs, if 8

23 we replace CWID money demand iming wih he more inuiive CIA iming, hen sunspos are learnable over a relevan range of he parameer space. Second, sunspos are learnable if he cenral bank is he one doing he learning. There are several naural areas of furher work. Firs, he Taylor rule examined depended only on expeced inflaion. Fuure work will consider he case of including a measure of oupu in he policy rule. Second, he sunspo equilibria arise because of he endogeneiy of he supporing money supply process. Wha feaures of his money supply behavior lead o E-sabiliy? Finally, work by Carlsrom and Fuers (000a, 00b) suggess ha sunspo equilibria are much more likely when invesmen spending is added o he model. Are any of hese sunspo equilibria E-sable? While addressing wheher sunspos are learnable we have lef unanswered he quesion of how a paricular sunspo is coordinaed upon. While far from being a complee answer o his imporan quesion we noe ha if he moneary auhoriy believes in a paricular sunspo, raional expecaions on he par of he public dicaes ha hey oo will believe in ha sunspo. The cenral bank can lead us o real indeerminacy. 9

24 Appendix In his appendix we demonsrae ha he condiions for E-sabiliy when boh he public and he cenral bank are learning bu wih differen coefficiens in heir PLM s, are idenical o he condiions for E-sabiliy when heir PLM s are idenical. The laer is wha we call symmeric learning in Secion II. In eiher he case of CWID or CIA iming, he key difference equaion has he form p cb p p cb = f E + + f E + + f3e + + f 4E E+ + (A) where he f i s are consans, E denoes privae secor expecaions, and E denoes p cb cenral bank expecaions. We have omied he sochasic shocks for simpliciy and wihou loss of generaliy. In he case of CWID iming R eners he Euler equaion so ha f = -λτ and f 4 = 0 (see equaion (4)), while in he case of CIA iming R + is in he Euler equaion so ha f = 0 and f 4 = -λτ (see equaion (7)). Le us consider AR() equilibria firs. Suppose he sunspo equilibria are characerized by = γ -, where γ is in he uni circle. In he case of symmeric learning boh he cenral bank and he public posi he same PLM, = a. The E-sabiliy condiion, evaluaed a a = γ, is 3 4 < γ ( f + f ) + 3γ ( f + f ). (A) In conras, in he case of asymmeric learning suppose ha he privae secor s PLM is given by, = a 0

25 while he cenral bank s PLM is given by. = b Subsiuing his ino (A), we have he PLM o ALM mapping: T ( a, b) + a 3 3 = fa + f b + f 3a f 4. Noe ha we have used an ieraed expecaions assumpion E p E E. (A3) cb p + + = + Since a and b boh map ino he same scalar, he eigenvalues of he E-sabiliy marix are 0 and T (a,b)+t (a,b), evaluaed a a=b=γ. For E-sabiliy we need 4 γ ( f + f ) + 3γ ( f 3 + f ) <. (A4) This condiion is he same as he condiions for symmeric E-sabiliy given in (A). Wha abou he AR() equilibria? Recall ha hese equilibria arise only in he case of CIA iming in which case f = 0. Bu hen he assumpion in (A3) immediaely implies ha symmeric and asymmeric learning are idenical. This assumpion is only imporan for CIA-iming. Noe ha ieraed expecaions is an assumpion. Wihou raional expecaions he law of ieraed expecaions does no necessarily hold. We are basically assuming ha he public s bes guess of he cenral bank s esimae of inflaion is heir own inflaion esimae. We could have made he alernaive heroic assumpion ha he public knew he cenral bank s esimae of inflaion and obained he same resul.

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