Managing Beliefs about Monetary Policy under Discretion

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Managing Beliefs about Monetary Policy under Discretion Elmar Mertens 2010-11 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Managing Beliefs about Monetary Policy under Discretion∗ Elmar Mertens† Federal Reserve Board August 2009 ∗Iwouldliketothankthemembersofmydissertationcommittee,inparticularthedirectorJean-PierreDanthine, aswellasPhilippeBacchetta,RobertG.King,andPeterKuglerfortheiradviceandstimulatinginputs. Furthermore I would like to thank seminar participants at the University of Basel, Boston University, the Federal Reserve Bank of Richmond, the Federal Reserve Board as well as the University of Lausanne for their comments. All remaining errorsareofcoursemine. ThisresearchhasbeencarriedoutwithintheNationalCenterofCompetenceinResearch “FinancialValuationandRiskManagement”(NCCRFINRISK).NCCRFINRISKisaresearchprogramsupportedby theSwissNationalScienceFoundation. †Forcorrespondence:ElmarMertens,BoardofGovernorsoftheFederalReserveSystem,WashingtonD.C.20551. emailelmar.mertens@frb.gov. Tel.: +(202)4522916. Theviewsinthispaperdonotnecessarilyrepresentthe viewsoftheFederalReserveBoard,oranyotherpersonintheFederalReserveSystemortheFederalOpenMarket Committee. Anyerrorsoromissionsshouldberegardedasthosesolelyoftheauthor.

Managing Beliefs about Monetary Policy under Discretion Abstract Inmodelsofmonetarypolicy,discretionarypolicymakingoftenlackstheabilitytomanage publicbeliefs,whichexplainsthetheoreticalappealofpolicyrulesandcommitmentstrategies. Butasshowninthispaper,whenapolicymakerpossessesprivateinformation,beliefmanagementbecomesanintegralpartofoptimaldiscretionpoliciesandimprovestheirperformance. Solving for optimal policy in a simple New Keynesian model, this paper shows how discretionarylossesarereducedwhenthepolicymakerhasprivateinformation. Furthermore,disinflations are pursued more vigorously, when the hidden information problem is larger, even wheninflationispartlybackward-looking. JELClassification: E31,E37,E47,E52,E58 Keywords: OptimalMonetaryPolicy,Discretion,Time-ConsistentPolicy,Markov-PerfectEquilibrium,IncompleteInformation,KalmanFilter

1 Introduction Starting with Kydland and Prescott (1977) and Barro and Gordon (1983b), the theoretical literatureonrulesversusdiscretionhasdocumentedclearbenefitsfromcommitmentinmonetarypolicy. Manyeconomicdecisionsintheprivatesectorareforward-lookinganddependonpolicyexpectations. Insuchanenvironment,policyrulesandcommitmentstrategiesbenefitfromtheirabilityto managepublicbeliefsaboutfuturepolicies. However,mostoftherules-versus-discretionliterature is based on models of perfect information, symmetrically shared between the central bank and the public. Monetary policy is often conducted under imperfect information of various sorts. On the one hand, policymakers may face uncertainty about the state of the business cycle, the nature of structural relationships in the economy or a lack of access to timely data. The problems arising from this perspective have for example been analyzed by Sargent (1999) or Svensson and Woodford (2004) to name but a few. On the other hand, policymakers may also be privy to confidential information, for example arising from staff efforts in gathering and analyzing economic data or supervisory activities. Extending the work of Cukierman and Meltzer (1986), Faust and Svensson (2001, 2002), this paper focuses on the design of optimal monetary policy, when the policymaker hasprivateinformation.1 This paper shows how belief management becomes an integral part of discretionary policies, when the central bank has private information. In this case, the public will make inferences about the hidden information based on observed policy actions such that current policies directly affect inflation expectations.2 The trade-offs faced by a discretionary policymaker resemble then those knownfromcommitmentproblems. Investigating the design of optimal policy when the central bank has private information also contributes to the literature on optimal transparency. Morris and Shin (2002) caution against pro- 1AcompleteliteraturereviewcanbefoundinSection5. 2Of course, in equilibrium inflation always depends on monetary policy. But in perfect information models of discretionary policy, this dependence occurs mostly indirectly and in ways beyond the control of a current period policymaker.

viding the public with too much information. But Woodford (2005) doubts whether their conclusions will be relevant in a forward-looking model, where the economy is mostly affected by expectations of future policies. The model analyzed here provides an intriguing counterexample to this conjecture, in that policy losses are lower under hidden information when comparing discretionary policies in a New Keynesian model. Since hidden information gives scope for belief managementunderdiscretion,theresultoccurspreciselyforreasonsstressedbyWoodford(2005). Attention is limited here to Markov-perfect policies. In the spirit of “bygones are bygones”, Markov-perfect state variables equilibrium must be relevant for current payoffs. When the public is imperfectly informed, its prior beliefs matter for public payoffs and they become a distinct, endogenousstatevariableofthepolicyproblem,whichisinfluencedbypolicyactions. Bymanaging this state of (public) beliefs, the policymaker indirectly responds to past policies, even when reputational mechanisms via history-dependent strategies, known from Barro and Gordon (1983b) or ChariandKehoe(1990),areexcludedfromtheanalysis. In Markov-perfect models, a current decision-maker can influence a future decision-maker only via endogenous state variables, such as capital or government debt. In the model presented here, belief management leads to Markov perfect outcomes that share similarities with those from models with commitment respectively reputational mechanisms. Previous research has already recognized how discretionary outcomes can be improved by adding endogenous state variables to the policy problem. Usually, this is done by modifying the central bank’s loss function, for examplebyaddingconcernsforinterestratesmoothing(Woodford2003b)orbyreplacinginflation stabilizationwithpriceleveltargeting(Vestin2006). What is novel about the present paper, is how beliefs naturally emerge as such an endogenous state variable, without the need for modifying the central bank’s loss function or other aspects of theeconomy. Totheextentthathiddeninformationproblemsareanessentialfeatureofinteractions between policymakers and the public, this suggests that the importance of discretionary biases in practicemightbedifferent,andlikelysmallerthanwhatissuggestedbyfullinformationmodels.3 3SeealsoBlinder(1998), whodownplaystherelevanceoftime-inconsistencyasamajordistortioninreal-world decisionsatcentralbanks.

The problem of “public learns about central bank” studied here is distinct from settings of “bank learns about economy” studied for example by Sargent (1999), Aoki (2003) or Svensson and Woodford (2004). In the latter settings, atomistic individuals take policy as given without regard for inference problems faced by the policymaker. Policy constraints like the Phillips Curve are largely preserved. In the linear quadratic case studied by Svensson and Woodford (2004), certainty equivalence holds and optimal policies are identical to the full information case when actualvaluesarereplacedbypolicymakers’expectations. Akeycomplicationformypaperisthat the central bank is a strategic, not an atomistic player, who takes the public’s inference problem intoaccountwhendevisingitspolicy. Thischangesthepolicyconstraintsinnon-trivialways. The framework adopted here exclusively assigns the policymaker, and not the public, with superior information. This is an extreme assumptions. Reality is best described by dispersed information, endowing different bits and pieces of hidden knowledge to the private sector and policymakers. The policy constraints change in dramatic ways when agents are learning about the policymaker,becauseofhisstrategicpositionintheeconomy. Thosestrategiceffectsarethemain concernofthepaper. The effects of hidden information on optimal policy are illustrated with a simple New Keynesian model — a model not chosen for its realism, but in order to document the differences with the symmetric information benchmark most clearly within a widely studied setting. The paper solves for the optimal discretion policy in a New Keynesian model where the output target of the policymaker is not directly observed by the public. The public only observes policy actions, but cannotdisentanglewhethertheunderlyingshocktotheoutputtargetispersistentortransitory. The policymaker faces a direct feedback from higher inflation expectations when choosing more expansionarypoliciescautioninghimtotemporarilyboostaggregateactivityattheexpenseofhigher inflation. Compared to a full information model, a key difference is how optimal policy contracts the economy in response to inflationary beliefs. Moreover it does so more vigorously, the larger the credibility problems from hidden information. This result has important implications for the conductofoptimaldisinflations.

To the best of my knowledge, my paper provides the first analysis of disinflations with an explicitly optimizing monetary policymaker and unknown policy targets.4 The results confirm conjectures by Sargent (1982) and Bordo et al. (2007) about the necessity to disinflate more quickly, when credibility is at stake. Other economists, for example Gordon (1982), have rather argued for prolongedandmodestdisinflationpathswheninflationispersistent. Strikingly,myresultisshown to carry over also to a setting with a hybrid Phillips Curve, where inflation persistence is partly exogenous. Evidently, disinflation costs are higher in such a setting. However, by bringing down inflation expectations early on a more aggressive disinflation policy still minimizes these costs, sinceitavoidsinflationtopersistbasedonill-foundedbeliefs. The information structure used here is similar to the models of Faust and Svensson (2001, 2002) and Cukierman and Meltzer (1986) who cast their models within similar linear-quadratic settings, but without providing a general framework capable of handling various models with endogenous state variables. Faust and Svensson focus on the welfare effects of credibility with a Lucas-supply curve. Using a forward-looking Phillips Curve, their results can be confirmed and extended here: Policy losses are reduced when output targets are unobservable, such that there is an explicit role for public beliefs. This disciplines the pursuit of persistent output targets, even whentime-consistencyisimposedonpolicy. So far, problems of this kind have mostly been analyzed in highly stylized and often static settings.5 But the models used for policy analysis are typically dynamic and of larger scale. The technical appendix to this paper presents a flexible, yet tractable way to analyze optimal policy underhiddeninformation,whichisapplicabletothekindofDSGEmodelsusedinpolicyanalysis. The procedure remains tractable and transparent by relying ona linear-quadratic representation of the policy problem driven by Gaussian shocks. A key complication for models with imperfect 4TheclosestcounterparttomyanalysisshouldbetheworkofIreland(1995)whoimposesasluggishresponseof publicbeliefstopolicyannouncements. 5SeeforexampletheclassiccontributionsbyBackusandDriffill(1985a), Canzoneri(1985)andCukiermanand Liviatan (1991), where my definition of static includes also repeated play of one-period games. More recent work includes the papers by Ball (1995) and Walsh (2000). Fully dynamic, but limited in size, are the models of Gaspar, Smets, and Vestin (2006), Faust and Svensson (2001, 2002), Cukierman and Meltzer (1986). A more detailed discussionoftheliteraturecanbefoundinSection5.

information is to track the distribution of public beliefs. In a linear, homoscedastic setting, that collapsestotrackingtheevolutionofmeansviatheKalmanfilter. The remainder of this paper is structured as follows. Section 2 introduces hidden information inatextbookversionoftheNewKeynesianmodelandshowshowhiddeninformationchangesthe policy problem. An extension incorporating belief shocks is shown in Section 3. Implications for disinflation strategies are analyzed in Section 4. The related literature is discussed in Section 5. Section 6 concludes the paper. A technical appendix extends the methods used here to a general classoflinearquadraticpolicyproblems. 2 A Simple Model of Hidden Information Thissectionillustratestheissuesarisingfromhiddeninformationwithasimpletextbookversionof the New Keynesian model. The model model is purely forward-looking and the signal extraction problem is univariate. The next section extends this model to a setting where a hybrid Phillips Curveinteractswithshocksfromaricherinformationstructure. 2.1 New Keynesian Economy The model is largely identical to the textbook model of optimal policy in a New Keynesian model known from Clarida, Gali, and Gertler (1999), Walsh (2003) or Woodford (2003a). The only difference is a stochastic preference shock to the policymaker’s objective function, which is unobservable to the public. Otherwise my model and its notation follow closely Gali (2003) where further details can be found. A key feature of the model is that inflation is determined purely by publicexpectationsofcurrentandfuturepolicies. Thisputscenterstagetheconcernsofthepublic aboutthepolicymaker’sintentions.

PrivateSector As in the textbook model, aggregate decisions of the private sector are represented by the New Keynesian Phillips and IS curves. In this simple model, IS curve and the short term interest rate areevenredundantandtheoutputgapcanbeusedaspolicycontrol. The private sector is populated by a continuum of identical firms and households, which trade goods and labor services. There is no capital accumulation and output equals consumption. Firms are monopolistically competitive and use staggered price-setting as in Calvo (1983). Optimal pricingdecisionsleadtotheNewKeynesianPhillipsCurveasinYun(1996)andKingandWolman (1996). Thelog-linearizedPhillipsCurveis π = βπ +κx (1) t t+1|t t where π is inflation and x is the output gap6. The parameter β is the representative agent’s t t discount factor and κ is a reduced form parameter influenced amongst others by the frequency of price-setting.7 For any variable z , z denotes its private sector forecast. The underlying t+1 t+1|t informationsetwillbeexplainedlater. The output gap measures the difference between actual output and its natural rate. The latter would be the output of the economy if there were no nominal frictions.8 My discussion will exclusively focus on monetary shocks that leave the natural rate unaffected. Conditional on those shocks,variationsintheoutputgaparethusidenticaltovariationsinoutputandconsumption. 6Throughoutthepaper,allvariablesareinlog-deviationsfromsteadystate,whichimplicitlyassumestheexistence anduniquenessofasteadystateunderdiscretionarypolicy. 7DetailsaregivenbyGali(2003,p. 159)fromwhomnotationisadopted. 8KingandGoodfriend(1997)explainhowtheNewKeynesianmodelcanbeseparatedintoacorerealbusinesscycle model (RBC), which evolves as if there were no nominal frictions, and a set of “gap” variables that track the differencebetweentheRBCcoreandtheactualeconomy. Thisseparationhasbeenwidelyadoptedforexampleinthe textbooksofWalsh(2003),Woodford(2003a)andGali(2008).

PolicyObjectives Thepolicymakerseekstominimizeapresentvalueofexpectedlosses ∞ X (cid:8) (cid:9) E βk π2 +α (x −x¯ )2 (2) t t+k x t+k t+k k=0 with α ≥ 0. The expectations operator E reflects the policymaker’s information set, to be dex t scribedlater. Thenon-standardfeatureofthelossfunctionisthetime-varyingtargetfortheoutput gap,x¯ ,whichwillbespecifiedasanexogenousstochasticprocess. t In principle, one could think of various ways to motivate the presence of x¯ in the loss funct tion9. However, the information structure used below will require that x¯ is not observed by the t private sector. To keep the model close to the NK benchmark, I maintain the assumption of a homogeneously informed private sector and follow Cukierman and Meltzer (1986) who interpret the output target as arising from time-varying preferences of the policymaker. Under this view, x¯ t representstheoutcomeofpoliticalinfluencesonmonetarypolicytostimulatetheeconomy. These preferences are assumed to vary exogenously with political representation in the government and the makeup of central banker’s preferences.10 Such hidden pressures could arise even when the independenceofthecentralbankisformallyenshrinedinlaw,sinceactualindependenceisamore fragileconcept. ForexampleAbrams(2006)givesastrikingaccountofhiddenbutforcefulpolicy influences. His study documents how U.S. President Nixon covertly pressured the then Chairman oftheFederalReserve,ArthurBurnstoeasepolicyintherun-uptotheGreatInflation. Under either interpretation, the output target is capturing a form of heterogeneity otherwise not present in the model. In particular (2) does not necessarily represent a social welfare function. Faust and Svensson (2001) use a similar loss function for the policymaker. Their notion of 9For starters, time-variation in the output target could arise from variations in wedges between the frictionless and the efficient level of output. Time-varying markups would for example shrink distortions from monopolistic competition. Therearenon-monetarytoolstofightsuchdistortions,forexamplethekindoffiscaltoolsdiscussedby Gali(2003). x¯ couldthencapturechangesinthegovernment’spolicyofhandlingthesedistortions. t 10Intherealworld,pressuresmountedoncentralbankersappeartobearecurring,thoughnotnecessarilypermanent feature.Forexample,intheshorthistoryoftheECBthereweretheearlyattemptsbyGermanFinanceMinister“Red” OskarLafontaineandlateroverturesfromtheFrenchPresidentNicolasSarkozy.

representative welfare would then be to evaluate (2) at the average output target (here: zero)11, LR = π2 + α x2. But without specifying the underlying heterogeneity and associated welfare t t x t weightsthisisatbestanaggregationwithunknowndistributionalconsequences. In reality, short-term interest rates are the typical instruments of monetary policy. But in this simple model, the short term interest rate can be perfectly substituted by the output gap as policy control. TheIScurveisthenredundantfordeterminingequilibrium. DiscretionaryPolicyunderSymmetricInformation Before turning to the informational structure of the model, it is helpful to study optimal policy when there is symmetric information. For the time being, let the output target follow a univariate AR(1)process x¯ = ρ x¯ +e where e ∼ N(0,σ2) and |ρ| < 1 t+1 t t+1 t+1 e which is mutually observed by the policymaker and the public.12 Under symmetric information, theirexpectationscoincidesuchthatz = E z foranyvariablez . t+1|t t t+1 t Lacking a commitment technology, the policymaker can always reoptimize his policies and for each optimization he takes his future choices as given. Sine there are only exogenous state variables,hetakesthepublic’sinflationexpectationsasgiven,too.13 OnlyMarkov-perfect,discretionary equilibria are considered. This excludes for example trigger strategies to support commitmentoutcomes. Thesolutiontothisproblemiswellknown. Thefirstorderconditionbalancestheinflationcost 11Intheresultsdiscussedbelowtherewillnotbeaconflictinrankingoutcomesunderthismeasureasopposedto thepolicymaker’sobjective. 12Theprocessismeanzeroandallowsalsofornegativetargets. Butallvariablesareindeviationfromsteadystate. Byallowingfora(known)averagetarget,thiswouldleadtotheclassicinflationbiasinsteadystate. (Inthecontextof thepresentmodel,detailscanbefoundinWoodford(2003a).) Tobeconsistentwithnon-zeroinflationinsteadystate, thePhillipsCurveisthenviewedasallowingforindexationtothesteadystaterateofinflationasinYun(1996). 13Ingeneral,thepolicymakercouldnottakeinflationexpectationsasgivennumbersbutasagivenmappingfrom expected future state values, where the latter may be partly under his control. This will be the case under hidden information.

againstthedesiretoattaintheoutputtarget: α (x −x¯ )+κπ = 0 (3) x t t t (Section 2.3 below, will compare this optimality condition against its counterpart under hidden information.) Substitution of (3) into the Phillips Curve yields the following Markov-perfect policies: α (1−βρ) κ x ¯ ¯ x = x¯ ≡ f x¯ and π = f x¯ (4) t κ2 +α (1−βρ) t t t 1−βρ t x Inflationandoutputgapinheritthedynamicpropertiesofthetargetprocess. Awellknownproperty ofoptimalpoliciesinalinearquadraticframeworkistheircertaintyequivalence,whichholdshere, ¯ too, since f does not depend on the volatility σ of the target shocks. Under hidden information, e thiswillbedifferent. ¯ Sensibly,f isboundedbetweenzeroandone. Inprinciple,thepolicymakercouldalwaysattain ¯ the output target by choosing f = 1, but for α < ∞ this has to be weighed against the inflation x ¯ resulting from this policy. At the other extreme, there would be no inflation if f = 0, but only at thecostofmissingthetarget,whichmattersifα > 0. Valuesoutsidethezerotoonerangewould x lead to further target deviations and be associated with unnecessary inflation. This will be useful tobearinmindwhenanalyzingpoliciesunderhiddeninformation. ¯ Policies with f close to unity will be called “bold” and it is instructive to see how policy depends on the preference weight α and the persistence of the target process. Inspection of (4) x revealstheintuitivepropertythatpoliciesgetbolderthehigherthepreferenceweightonoutput,in ¯ factf variesbetweenzeroandonewhenα isvariedbetweenzeroandinfinity. x Policies are less bold, when the target is persistent. Higher persistence of the target causes higher persistence in policy and thus higher inflation. This is a dynamic version of the inflation bias known from Kydland and Prescott (1977) and Barro and Gordon (1983a) and similar to the stabilizationbiasknownfromSvensson(1997). Under hidden information there will be persistent and transitory shocks to the output target,

neither of them being directly observable to the public. As in the full information case, what matters for the inflation response to a policy shock is its perceived persistence. The policymaker will then seek policies that are as bold as possible, while trying to keep perceived persistence as lowaspossible. 2.2 Hidden Information Hidden information is introduced by assuming that the public can observe only policy, x , but t not shocks to the policy target. To make the public’s signal extraction interesting, the target is henceforthdrivenbytwocomponents,onepersistent,onetransitory: x¯ = τ +ε ε ∼ N(0,σ2) (5) t t t t ε τ = ρ τ +η η ∼ N(0,σ2) and 0 < |ρ| < 1 (6) t+1 t t+1 t η The private sector has no structural uncertainty about the economy. All parameters are known, includingthespecificationofthetargetprocess. Thepublicmusthoweverinfertherealizations of τ andε basedontheobservedhistoryofpolicies,denotedxt.14 t t The policymaker observes the complete history of the target components and his expectations are typically different from those of the public. As before, for any variable z , the policymaker’s t expectations are denoted E z = E(z |τt,εt) with the obvious property z = E z . Public t t+1 t+1 t t t expectations are z = E(z |xt). By construction, x = x and π = π (since inflation is a t+1|t t+1 t|t t t|t t choicevariableoftheprivatesector)buttypicallyτ 6= τ andε 6= ε . t|t t t|t t Surprises in z relative to the public’s past information will be called “innovations”. Formally, t theyaredefinedas z˜ ≡ z −z t t t|t−1 14In principle, this includes also the history of inflation rates πt. But as a choice variable of the private sector, inflationmerelyreflectstheprivatesectorsinformationset,withoutprovidingadditionalinformationbeyondxt.

Innovations provide an orthogonal decomposition of the public information set since z˜ = 0. t|t−1 Even though they are unpredictable from the public’s perspective, they may well be predictable basedonthecompleteinformationset,andtypicallyE z˜ willnotbeidenticaltozero. t−1 t Since the model is linear with Gaussian disturbances, rational expectations of the public can be computed recursively from the Kalman filter. Given prior beliefs z and x , the public t|t−1 t|t−1 observesarealizationofpolicyx andupdatesitsbeliefsaccordingto t Cov(z ,x˜ ) t t z = z +K x˜ withKalmangain K ≡ (7) t|t t|t−1 z t z Varx˜ t A convenient property of the Kalman update is that it preserves the linearity of the model. The difference with adaptive expectations is that the gain coefficient is an endogenous parameter, identical to the least squares slope of projecting z on x˜ . The present model is particularly simple t t since there is only one observable, x , such that K is a scalar. (A multivariate setting will be t z illustratedinSection3.) SignalExtractionforGivenPolicy Aswillbeverifiedbelow,theoptimalpolicyislinearandhastheform x = f τ +f ε +f τ (8) t τ t ε t b t|t−1 for some scalars f , f and f . Compared to the symmetric information case, the dependence on τ ε b τ isnovel. Itcapturespolicyresponsestopublicbeliefs. Aswillbeshownshortly,itinfluences t|t−1 thepersistenceofpolicyshocks,whichisacrucialfactorindetermininginflation. The public belief system is a straightforward application of the Kalman filter with (6) as state equation and (8) as measurement equation. In the parlance of time-series econometrics, policy posesanunobservedcomponentsmodeltothepublic. KeyfortheKalmanfilteristheratioofpolicy loadings on the realized components of the output target, f /f . Only these loadings, and not ε τ f ,arerelevantfortheKalmanfilter. (DetailsaregiveninAppendixB.)This“mixingratio”f /f b ε τ

determines how much a policy innovation reveals about τ instead of ε . It allows the policymaker t t tochangethesignal-to-noiseratiointhepublic’ssignalextractionproblem. From the perspective of the public, policy is driven by the iid innovations x˜ and it has an t innovationsrepresentationintheformofanARMA(1,1)process: x = ρx +x˜ +ρψx˜ (9) t t−1 t t−1 with ψ = (f +f )K −1 τ b τ For the public, the above innovations representation is observationally equivalent to the hidden componentsrepresentationofpolicy(8). Bothgeneratethesamevariancesandautocovariancesof policy, whilst implying different impulse responses as will be illustrated below. Via ψ, the persistence of this ARMA depends on the policy coefficients f , f and f . For plausible assumptions τ ε b of the policy coefficients, ψ is bounded between zero and minus one. For ψ = 0, persistence is largest as policy follows an AR(1) with auto-correlation equal to ρ. For ψ = −1 both roots of the ARMA(1,1)cancelandpolicyisiid. (DetailscanbefoundinAppendixA.) Together with the Phillips curve (1), the innovations representation of policy is sufficient to determine inflation in a way which crucially depends on the “average persistence” of policy as capturedbytheARMArootsρandψ. ∞ X κ (cid:0) (cid:1) π = κ βjx = (1+βρψ)x˜ +x (10) t t+j|t t t|t−1 1−βρ j=0 The policy function has two levers to affect the persistence of x : First, there is the mixing ratio, t whichhasbeendiscussedabove. Ifpolicylargelyignoresthepersistenttarget,i.e. iff /f islarge ε τ suchthatK isclosetozero,theMArootgetsclosetocanceltheARrootandpolicyis(correctly) τ perceivedtobealmostiid. Inthiscase,inflationalsoapproachesthesolution(4)undersymmetric informationwithρ = 0. But due to the second lever, f , things need not collapse to the AR(1) case, when the mixing b

ratiotendstozero. Inthiscase,ψ convergestoρ·f ,whichisnotnecessarilyzero.15 f represents b b themarginalreactiontopeople’spriorbeliefsandaffectsthepersistenceofpolicy,too. Anegative f counteracts policy persistence induced by τ . The marginal reaction to beliefs is likely negative b t sincebeliefsτ willbeinflationary;thisconjecturewillbeverifiedinSection2.4. t|t−1 To keep inflation low, it is tempting to conclude that the policymaker should better ignore the persistent output target. Alternatively, a high mixing ratio could be chosen, with a higher responsivenesstotransitorythanpersistentshocks,forexamplef = 1andf = 100. Butneitherchoice τ ε would likely be a sensible policy, since output plays not only an informational role. Attaining the output targets matters, too; calling for f = f = 1 and f = 0. For example, ignoring the τ ε b persistent target by setting f = f = 0 alleviates inflationary cost, but it also leads to persistent τ b shortfalls from the τ-target. Neither would it appear sensible to overshoot the output target, for example by setting f = 100. The optimal trade-off is the subject of the next sections. But an im- ε portant restriction imposed by rational expectations has already become clear: at least on average actualpoliciesmustmatchpublicperceptions. 2.3 The Discretionary Policy Problem This section sets up the discretionary policy problem for the simple, purely forward-looking New Keynesianmodelwhenthereistheabovestructureofhiddeninformation. Extensionsofthemodel, including a hybrid Phillips Curve, will be analyzed in subsequent sections of this paper. The conceptsandmethodspresentedherearegeneralizedtoawiderclassoflinearquadraticmodelsin thetechnicalappendixofthispaper. MarkovPerfectEquilibria AttentionislimitedheretoMarkov-perfectequilibria,whichexcludereputationalmechanismsvia the kind of history-dependent strategies considered by Barro and Gordon (1983b) or Chari and 15Sincex =x inthissimplemodel,itfollowsthatf K +f K =1. Whenthemixingratiogoestozero,this t t|t τ τ ε ε collapsestof K =1. τ τ

Kehoe(1990)andavoidstheassociatedmultiplicityofequilibria. Inthespiritof“bygonesarebygones”, state variables in a Markov-perfect equilibrium must be relevant for current payoffs.16 In the symmetric information setting shown above, these were the contemporaneous values τ and ε t t (but not any elements of their history). Both of these state variables evolve in a purely exogenous fashionwhichaccountsforthemyopicbehaviorofdiscretionarypolicyundersymmetricinformation: In Markov-perfect models, a current decision-maker can influence a future decision-maker only via endogenous state variables, like capital or government debt. This channel is however absentinthesymmetricinformationversionoftheNewKeynesianmodel. Once hidden information is introduced, an additional state variable becomes relevant: Since thepublicobservesonlyx butneitherτ notε ,itisthepublicbeliefsaboutthetargetcomponents t t t whicharerelevantforpublicpayoffs. Precisely,itisthepriorbeliefs(τ andε )andnotthe t|t−1 t|t−1 posteriors (τ and ε ) which qualify as state variables for the time t decision problem, since the t|t t|t latter are already influenced by time t policies. In the present setting, ε is iid and ε = 0 so t t|t−1 onlyτ needstobetracked. ThevectorofMarkov-perfectstatevariablesisthen t|t−1 (cid:20) (cid:21)0 S = τ ε τ t t t t|t−1 The transition equation for the new state variable is given by the Kalman Filter. The response ofbeliefstopolicydependsontheKalmangainK ,whichreflectshowmuchpolicyreactstoτ . τ t τ = ρτ and τ = τ +K x˜ (11) t+1|t t|t t|t t|t−1 τ t The discretionary policymaker retains the freedom to reoptimize his policies at each point in time. On the one hand, this allows a recursive representation of the policy problem as a dynamic program. On the other hand, he does not commit to future policies so these have to be taken as givenin thedecisionproblem. Tobeprecise, whatis takenasgiven ishow thepolicymakerreacts 16Persson and Tabellini (2000, Chapter 11) review applications of Markov-perfect equilibria to macroeconomic policyproblems.

to future state variables: Future policies are not given numbers but a given function of future state variables. This distinction is important here, since one of the state variables, τ , is under the t+1|t influence of current policy so that future outcomes can be influenced. The continuation value of hisdynamicprogramisafunctionoffuturestates,denotedV0(S )andthepolicyobjectiveisto t+1 minimize π2 +α (x −τ −ε )2 +E V0(S ) (12) t x t t t t t+1 The linear quadratic nature of the model allows to guess (and verify) that the value function will bequadraticandpolicieslinearinthestatevector,whichsimplifiestheanalysisconsiderably: V0(S ) = S0 V0S +v0 t+1 t+1 t+1 ⇒ E V0(S ) = 2v0 ρτ τ +v0 τ2 +t.i.p. t t+1 13 t t+1|t 33 t+1|t for some positive definite matrix V0 with elements v0 , v0 > 0 and a scalar v0. Throughout this 13 33 paper, a zero superscript “0” indicates coefficients embodying a guess about (future) policy and “t.i.p.”aretermsindependentoftimetpolicy. The time-invariant solution to the discretionary policy problem has the linear form anticipated in(8). Inprinciple,thepolicymakerisfreetodeviatefromthis“rule”atanytime. Hewilljustnot finditoptimaltodoso. An important constraint on the policy problem is the optimality of beliefs and decisions in the private sector. Optimality of beliefs are captured by the Kalman filter (7) and the the policymaker seeshimselffacedwithafixedKalmangainK0 whencontemplatinghispolicyproblem. Optimal τ decisions of the private sector are represented by the Phillips Curve (1) where the policymaker takes as given how inflation expectations are related to future state variables; π = g0τ for t+1|t t+1|t

somescalarg0. Tosumup,thepolicyproblemistominimize V = min π2 +α (x −τ −ε )2 +2v0 ρτ τ +v0 τ2 +t.i.p. (13) t t x t t t 13 t t+1|t 33 t+1|t xt,πt,τ t+1|t s.t. π = βg0τ +κx (14) t t+1|t t τ = ρ(1−K0(f0 +f0))τ +ρK0x (15) t+1|t τ τ b t|t−1 τ t whose solution is indeed of the form anticipated in (8). Whilst beliefs embodied in g0, V0 and K0 are taken as given in the policy problem, in equilibrium they must be consistent with the τ solution to the policy problem. This poses an intricate fixed point problem. Fixed points between current expectations and future policy as in g0 and V0 are common in Markov-perfect models under symmetric information. What is new is the fixed point between current policy and beliefs about the systematic relationship between current policy and states contained in the Kalman gain K0. τ ChangedPolicyTrade-OffswithBeliefManagement Thefirst-orderconditionsof(13)requireoptimalpolicytosatisfy α (x −x¯ )+κπ +ρK0µ = 0 (16) x t t t τ t where µ is the multiplier on the belief constraint (15). It is the term involving µ which distint t guishes the optimality condition (16) from its counterpart under symmetric information (3) discussedabove. To shed some light on the fixed point considerations behind the solution to (13), suppose that the output target is positive and the policymaker must balance an increase in output against its inflationary costs. The marginal value of relaxing the belief constraint is likely positive, owing to the positive autocorrelation in the persistent component of the target. Likewise, the Kalman gain K will be positive, since policy will co-move positively with the target. The new “belief term” τ

ρK0µ in(16)willthencautionthepolicymakeragainstpursuingtheoutputtargettooaggressively. τ t As will be seen in the numerical analysis below, optimal policy will be less bold under hidden information—exceptwhenshockstoτ aresorarethattheKalmangainK isverysmall. t τ The change in policy trade-offs under hidden information can be nicely illustrated with a picture similar to Kydland and Prescott (1977). Under symmetric information, the policymaker’s indifference curves over output and inflation are concentric around π = 0 and x = x¯ = τ +ε . t t t t t Theoptimalitycondition(3)seeksthetangencypointbetweentheindifferencecurvesandthepolicy constraint. The latter being the Phillips Curve with intercept βπ = βg¯0ρτ for some g¯0. t+1|t t ThisisdepictedbythedashedlinesinFigure1. Inequilibrium,g¯0 mustbeidenticaltotheoptimal policy coefficient computed in (4), which is a positive number. That is, the larger policy responds toagivenlevelofthepersistenttarget,thehighertheinterceptitfacesinequilibrium. [Figure1abouthere.] Under hidden information belief management comes into play and changes the picture. To reach some substantive conclusions, I am willing to make the following assumptions about the policycoefficients. Apartfrombeingplausible,theywillbeverifiedtobetrueinthecomputations belowforawiderangeofcalibrations. First,policyshouldreactpositivelytotargetshocks,f0 > 0. τ Second, policy seeks to counteract belief f0 < 0. But third, it still seeks to accommodate a target, b even when its realization coincides with public beliefs: f0 +f0 > 0.17 These imply that K and τ b τ g0 arepositive. AkeyresultisthathiddeninformationsteepenstheslopeofthePhillipsCurvewhencompared against the symmetric information case. Substituting the belief dynamics (11), the Phillips Curve becomes (cid:0) (cid:1) (cid:0) (cid:1) π = βg0ρ 1−K0(f0 +f0) τ + κ+βg0ρK0 x (17) t τ τ b t|t−1 τ t The steepening of the Phillips Curve worsens the policy trade-off and makes policies less bold 17Thisassumptionimpliesthatthepublicexpectspoliciestobeexpansionary,x >0,whenτ >0. t|t−1 t|t−1

withrespecttobothtargetcomponents. Underliningtheimportanceofbeliefs,theinterceptofthe Phillips Curve depends now on the public’s prior beliefs, τ , instead of the actual value of τ . t|t−1 t Coming out of steady state with τ = 0, this alone makes policies bolder than otherwise. An t|t−1 important aspect for the fixed point computations is that, via K , the slope of the Phillips Curve τ becomes ever steeper the bolder policies are with respect to τ , which again tames the boldness of t equilibriumpolicies. Belief management changes the indifference curves as well. Most importantly, output acts as a signal about the persistence of policy targets which again influences the evaluation of future lossesinthepolicyproblem. Thisshiftsoutputpreferences,suchthattheyarenotcenteredaround x¯ = τ + ε anymore. Substituting again the belief dynamics, the indifference curves can be t t t computedfrom π2 +α (x −τ −ε )2 +γ x2 +γ τ x +γ τ x (18) t x t t t 0 t 1 t t 2 t|t−1 t where the scalars γ > 0, γ ≶ 0 and γ > 0 depend on the coefficients of (18). 18 These 0 1 2 indifferencecurvesarecenteredaroundπ = 0and t α 1 1 x∗ = (τ +ε )− γ τ − γ τ t α+γ t t 2 1 t 2 2 t|t−1 0 | {z } x¯t RegardlessofslopeandinterceptofthePhillipsCurve,x∗ isthe“maximallydesirable”levelof t output. Apart from its dependence on the original target term x¯ , it shifts both with the actual and t perceivedlevelofthepersistenttargetcomponentτ . Butforstartersconsideratransitoryshockto t the output target, say ε = 1 whilst τ = τ = 0: Any policy response will partly be attributed t t t|t−1 to a persistent shock and thus increase τ causing future inflation. The associated losses to the t+1|t policymaker are captured by the γ term of the indifference curves. Independently of the Phillips 0 Curve, the policymaker does then not even desire to attain that transitory target but only a fraction 18Itisstraightforwardtoshowthatγ > 0followsfromthepositivedefinitenessofthevaluefunction,andγ > 0 0 2 fromtheaforementionedassumptionsonthepolicycoefficients. Analytically,γ cannotbesigned,butforthevariety 1 ofcalibrationsconsideredinthenumericalsimulationsbelowitturnsouttobepositive.

α/(α+γ )thereof. 0 Since γ > 0, public beliefs τ shift the indifference curves towards lower output levels. 2 t|t−1 Whileγ cannotbesignedanalytically,ithappenstobepositiveovertherangeofcalibrationscon- 1 sideredbelowandthiscontributestomakingpolicylessbold. Allinall,priorbeliefsofthepublic τ > 0 caution policy in two ways: First they increase inflation immediately (the intercept of t|t−1 thePhillipsCurve)and—ifnotcounteractedbycurrentpolicy—theyheraldfutureinflationand shrinkthe“maximallydesirable”levelofoutput,x∗,towardszero. t 2.4 Optimal Policy in the Simple Model This section presents results for the optimal policy. Calibration values are taken from Gali (2003) withequallyweightedpolicypreferences(α = 1)andequal-probableshockstothetargetcompox nents (σ = σ = 1), see Table 1.19 The solution algorithm for the underlying fixed point problem η ε isdiscussedinthetechnicalappendix. [Table1abouthere.] OptimalMixingRatioandBeliefResponses Key statistics of the policy function are the mixing ratio f /f , which governs the Kalman gains, ε τ and f via which policy responds to prior beliefs. As anticipated, f is negative. The policy b b response to τ is synonymous with counteracting inflation expectations of the public formed in t|t−1 the past. There is a one-to-one correspondence between the public’s prior beliefs of the hidden stateandthepublic’sinflationexpectationsinthissimplemodel: κ π = (f +f )τ t|t−1 τ b t|t−1 1−βρ Howoptimalpolicyseekstoquellpastbeliefscanbeseenfromtheimpulseresponseshownin Figure2. Thefirsttwocolumnsshowresponsestoshocksinτ andε . Thethirdcolumndocuments t t 19Given the limited range of shocks considered, the calibration is not designed to match the level of variations observedinthedata.

responses to initial conditions τ = 0,ε = 0,τ = 1. This corresponds to a situation where t t t|t−1 the policymaker is faced with erroneous beliefs about his inflationary output preferences. The optimalresponseisaprolongedcontractionuntilbeliefsandoutcomeshavesettledbackinsteady state after about four periods. Given that the New Keynesian model generally lacks endogenous persistence,thelengthofthislearningprocessisaremarkableoutcomeechoingtheresultsofErceg and Levin (2003). Moreover, the effect of fighting past beliefs is also present in the other impulse responses. When the true target shock is iid, this leads to a contractionary policy one period after the shock. This pattern is similar (though not fully identical) to commitment policies under full information. In both cases, a credible promise to undo expansionary shocks in the future lowers inflation expectations; similar to the disciplinary channel emphasized by Walsh (2000), Faust and Svensson(2001)andGaspar,Smets,andVestin(2006). [Figure2abouthere.] The other lever of policy is the mixing ratio, which is higher compared to the full information case.20 Underhiddeninformation,policyislessboldinitspursuitofpersistentoutputtargets. This lowersthesignal-to-noiseratiointhepublic’ssignalextractionproblemandthepublic(correctly) placesalowerprobabilityonapolicyinnovationx˜ beingcausedbyapersistenttargetshock. t InnovationResponses The model with hidden policy components is observationally equivalent to a symmetric information model where the policy target follows a univariate ARMA(1,1). Both yield the same second moments and have identical likelihoods. But there is an important difference: The hidden components model distinguishes different sets of impulses responses, which can be associated with differentepisodesinmonetarypolicy. [Figure3abouthere.] 20For the baseline calibration, the mixing ratio is 1.2340 under symmetric information and 1.2866 under hidden information.

The differences between true impulse responses and public beliefs are illustrated in Figure 3. For output and inflation, the figure shows two sets of impulse response: First, the expected responsescomputedbythepublic,afterobservingaunitinnovationinpolicy,x˜ ,attimezero. After t itsinitialupwardsjump,outputremainsexpandedatabouthalfitsimpactvalueanddecayspersistentlythereafter. Theinflationpathisequallyequallyelevatedandpersistent. Secondly,thefigureshowsthetrueresponsestothestructuralshocksτ andε ,computedunder t t thefullinformationmeasurespannedby(τt,εt). Theyarescaledsuchastoyieldaunitinnovation in output as well. After a shock to the persistent target, τ , policy is persistently more expansive t thanoriginally expectedbythe public. Thedifferencebetween thesetwosets ofimpulseresponse represents the errors of public forecasts made in the initial period. As the structural responses unfold, the public learns about the true nature of the shock. The figure also shows how public beliefs are updated in subsequent periods, leading to persistent upwards, respectively downwards revisions of beliefs. The innovations responses are rational and on average correct. Persistently positive forecast errors to a shock in τ are offset by persistently negative forecast errors when a t shocktoε occurs. t Whenparticularperiodsaresupposedtohavebeendominatedbyonesetofshocksratherthan another, patterns of persistent forecast errors in public beliefs should be reflected in survey data. For example, Erceg and Levin (2003) use survey data to characterize the Volcker disinflation as a period of persistently excessive inflation forecasts. Their model uses a Gaussion information structure similar to mine, but for a fixed policy rule. The methods presented here can be used to derivetheparametersofsucharulewithinanexplicitlyoptimizingframeworkofmonetarypolicy underhiddeninformation. SensitivityAnalysisofPolicyCoefficients Policytrade-offsareparticularlyaffectedbytwoparameters: Therelativevarianceoftransitoryto persistenttargetshocksandthepreferenceweightα ,whereasincreasesintheslopeofthePhillips x

Curve, κ, affect policy trade-offs similarly to decreases in α .21 When considering changes in the x importanceofthetargetcomponentsε andτ ,theoverallvarianceoftheoutputtargetwillbefixed t t atsomelevelσ2. Denotingtheweightonτ byω ∈ [0;1]thistranslatesinto x¯ t σ2 = (1−ω)σ2 and σ2 = ω(1−ρ2)σ2 ε x¯ η x¯ Figure 4 documents changes in the policy coefficients f , f , f as well as the mixing ratio τ ε b f /f due to variations in ω and α . The upper panels also show the corresponding values of ε τ x f and f under symmetric information. Because of certainty equivalence, their surfaces are flat τ ε alongtheω-axis. Whenthereishiddeninformation,f isuniformlysmallerthanundersymmetric ε information. This is caused by the public’s inability to distinguish between realizations in the two target components. Any innovation in x will be partly attributed to have been caused by the t persistent component τ . This has two adverse effects in the first-order condition (16): First, if the t true shock was to the iid component ε , inflation will be higher compared to the full information t case. Second, since the public expects the target change to have some persistence, there will also be inflationary costs in the future, tracked by µ . Both effects caution the policymaker and lower t f comparedtothefullinformationcase. ε [Figure4abouthere.] Because of the second effect, f is mostly smaller under hidden information as well. As the τ publicunderestimatesthepersistenceofpolicyafterashocktoτ (seeFigure3),inflationislower t than under full information. This would give the policymaker some slack in pursuing the output target, if it were not mostly outweighed by the marginal effect of policy on beliefs, which are represented by the term ρK µ in the first-order condition (16). However, when the probability of τ t a persistent shock is very small (ω close to zero) or when the policymaker is known not to care muchaboutattainingit(α small),theKalmangainK willbesmallandpublicbeliefsτ willbe x τ t|t very insensitive to policy and their importance vanishes in (16). In those cases, persistent shocks 21ThelossfunctioncanbewrittenasL =κ2π¯2+α (x −x¯ )2,whereπ¯ = P∞ βjx . t t x t t t j=0 t+j|t

areveryhardtodetectforthepublic. Butwhentheyoccur,thepolicyresponsecanbebolderthan undersymmetricinformationasisshowninPanel(a)ofFigure4. It is worth recalling that a higher mixing ratio and a more negative reaction to prior beliefs increasethepersistenceofthepolicyprocess,causinghigherinflation. Indeed,ascanbeseenfrom Panel (c) of the figure, f is negative everywhere. The belief reaction is strongly negative, when b there is more weight on inflation in the loss function (smaller values of α ) and when persistent x shocks are more prevalent (ω close to one). Both cases make it more important, respectively more likely,thatinflationarybeliefsarekeptincheck. Under discretion, the policymaker takes the public’s belief system as given, without actively seeking influence it, whereas commitment policymaker would have to consider the systematic effectsofhisactionsontheKalmangainforexample. Stillitisinstructivetoseehowpolicyaffects the public’s signal-to-noise ratio via the mixing ratio. As shown in Panel (d), this ratio increases when policy preferences place more weight on output than inflation. As inflation becomes more and more costly for the policymaker, f is decreasing faster than f , which makes it ever harder τ ε for the public to detect persistent policy changes. When changing ω, the mixing ratio is largest for intermediary values, typically above ω ≥ 0.5. In this range, hidden information problem is most prevalent and a high mixing ratio helps to lessen the sensitivity of beliefs to policy. When ω approaches unity, the public can expect any policy to be caused by a persistent shock with near certainty and the mixing ratio drops to its full information level. When the target is almost exclusively driven by iid shocks (ω → 0) expected future inflation drops towards zero and the mixingratiodropstooneasf approachesf . τ ε PolicyLosses Comparing policies under hidden information against outcomes under full information begs the question what would be the preferred setting. Considering the loss function of the policymaker, it turns out that the ex-ante expectation of the policy loss, E(V ) is improved under hidden informat tion over the wide range of calibrations discussed above. Figure 5 reports how the improvement

in policy losses under hidden information are large enough that an average inflation rate corresponding to about one-standard deviation unit would have to be added to inflation under hidden information for policy losses to be equal; the compensating inflation is somewhat smaller when the volatility weight on persistent shocks, ω, is very small; and it can be considerably larger when persistentshocksareveryprevalentandtheweightoninflationstabilizationislarge.22 [Figure5abouthere.] Moreover, the same holds when considering the notion of “representative” loss discussed in Section 2.1. The reason is simply that the improvement in outcomes is due to the policymaker’s restraint in pursuing the output targets. By lowering inflation and output gap, this is clearly beneficial for the “representative” loss, which would be minimized by keeping the output gap at zero anyway. The benefits of reduced inflation also outweigh the policy losses from staying away from the targets, at least from an ex-ante perspective considering both persistent and transitory shocks as wellastheirrespectivelikelihoods.23 Conditionalontheoccurenceofaniidshockε ,inflationisof t coursehigher,andthepolicymakermissesthetargetbymorethanhewouldunderfullinformation, see Figure 2. On average, this is however outweighed by the benefits incurred when a persistent shockoccurs. Considering the different levers present in the policy function (8) under hidden information, a quantitativedecompositionofthereductioninpolicylosslooksasfollows: First,thereistheoptimal policy under symmetric information. Feeding this same policy through the system but under the hidden information, losses drop by an amount, correspond to a compensating rate of average inflation of about one third of the standard deviation of inflation in the new equilibrium. The optimalpolicyunderhiddeninformationthenseekstoimproveuponthisbychangingthemixingratio andbyreactingtopastbeliefs. Usingtheoptimalmixingratio,butneglectingtheresponsetoprior 22AdditionaldetailsaregiveninAppendixC. 23Expectedloss,E(V ),istheunconditionalexpectationofthepolicymaker’svaluefunctionacrossstatesofnature. t (SeetheTechnicalAppendixforcomputationaldetails.) Optimalpolicyisofcoursedefinedonastate-by-statebasis.

beliefs (f = 0) makes expected losses drop further; compared to full information the compensatb ing average inflation amounts to about one standard deviation of inflation. In addition to this, the optimalpolicyreactsalsonegativelytopriorbeliefsandtheaverageinflationcompensatingforthe improvementinlossesoverfullinformationequalsalmosttwostandarddeviationsofinflation. For comparison, the difference in full information losses of discretion and commitment corresponds to a compensating average inflation of about two-and-a-half standard deviations of inflation under discretion. 3 Belief Shocks The simple New Keynesian model analyzed so far has only one communication channel between policymaker and public: Policy actions themselves. Since policy is driven by more shocks than there are communication channels, the public cannot perfectly infer the drivers of policy, not even in equilibrium. In reality, there are however other communication channels than the policy instrument itself. If these channels are informative, they will alleviate the public’s inference problems andaffectthescopeofbeliefmanagementforpolicy. Thissectionextendstheinformationstructure ofthesimplemodeltoarichersetting,nestingthecasesoffullandhiddeninformationconsidered before. In addition to observing policy, the public is now assumed to receive a noisy signal about the persistentoutputtarget. Thetargetsignaliscontaminatedbynoiseshocksn ,whichwillbecalled t “beliefshocks”. Theyareiidandthepublic’smeasurementvectoris (cid:20) (cid:21)0 Z = x (τ +n ) where n ∼ N(0,σ2) t t t t t n andthenotationforpublicbeliefsofavariablez isnowadaptedtoz ≡ E(z |Zt). Thepresence t t|t t of two correlated observables in the public’s inference problem requires to extend the univariate filtering methods discussed in the previous section. Also, the state vector needs to be augmented by n . (Notice that n = 0.) A detailed presentation of handling this and larger settings has t t|t−1

beenrelegatedtothetechnicalappendix. The belief shocks are uncorrelated with fundamentals (here: τ and ε ) and play no role under t t symmetric information. But under asymmetric information they matter since they are correlated with an informative signal about fundamentals, giving rise to fluctuations driven by “non fundamental”shocks.24 Inflationisaffectedbybeliefshocksviatheforward-lookingPhillipsCurve(1), making it suboptimal for policy to ignore belief shocks. As with given prior beliefs about the output target (τ ), they will raise inflation and optimal policy should want to fight their effects by t|t−1 contractingoutput. Inthepresentmodel,economicresponsestonoiseshockswillexhibitpatterns similartocost-pushshocks,echoingresultsofAngeletosandLa’O(2008b). By changing the volatility of noise shocks, the extended model also nests the cases of hidden and full information analyzed in the previous section. The scope for hidden information increases with the volatility of belief shocks. For σ = 0, the model is identical to the full information n model, since τ is perfectly observable. The opposite occurs when σ is very large. In this case, t n the signal becomes useless and the model converges to the hidden information setting from the previoussectionwherepolicyistheonlyobservable. [Figure6abouthere.] ImpulseresponsestoanoiseshockareshowninFigures6,againusingthebaselinecalibration fromTable1. Underthisconfiguration,eachofthethreeshocksinthismodeloccurswiththesame probability. When the target signal τ +n goes up because of a noise shock, this leads to ample t t confusion for the public. Current and expected inflation rise, since the public attributes part of thesignaltothepersistenttargetτ ,Tocounteracttheseerroneousbeliefs,policycontractsoutput. t This is sensible in two ways: First it directly lowers inflation via the output term in (1). Second, it signals that the target τ may in fact not have gone up and thus reduces expected inflation. In the t baselinecalibration,ittakesaboutfourperiods(oneyear)tofighttheseerroneousbeliefs. The resulting pattern of contracting output and elevated inflation is similar to the dynamics known from cost-push shocks. Figure 6 also documents that public beliefs of future output and 24Themeaningof“fundamentals”isintendedhereinthesenseofthefull-informationeconomy.

inflation are both elevated during the entire episode, which distinguishes belief shock induced dynamicsfromcost-pushbehavior,sincethelatterwouldtypicallybeaccompaniedbyanexpected recessionaswell. VaryingTransparency How does policy change with the volatility of belief shocks? To answer this question, Figure 7 shows how policy coefficients and expected losses change when σ is varied between zero and n infinity. As discussed above, the limit points in this experiment are the symmetric information model, respectively the previously studied model with no target signal except for policy. The policycoefficientsf ,f andf varysmoothlyandmonotonicallybetweenthecomparativestatics τ ε b of hidden vs full information studied before. They are all smaller and policy losses are reduced as theextentofhiddeninformationincreaseswithσ . n [Figure7abouthere.] The policy response to noise shocks is always negative. The reasons are similar to what has been discussed in the previous section for the negative response to prior beliefs f . A contraction b lowers inflation directly via the Phillips Curve and indirectly via beliefs. For better comparison withtheothercoefficients,themiddleleftpanelofFigure7showshowthepolicyreactiontoonestandarddeviationshock,f ·σ changeswiththeshockvariance. Thenoiseresponsepeaksatan n n intermediary level of the noise variance, where the public places roughly equal weight on policy andthetargetsignalinitsupdatingofbeliefs. Formally,thepublicbeliefsevolveas τ = τ +K x˜ +K (τ˜ +n ) (19) t|t t|t−1 x t s t t ChangesintheKalmangainsK andK forvariousnoiselevelsareshowninthebottomleftpanel x s of Figure7. Inthe extremesthe response ofpolicy tonoise shocksis zero,as either thesize ofthe shockshrinkstozero(andtherearenoerroneousbeliefstofight)orthepublicpaysnoattentionto asignalwithinfinitenoise.

A natural interpretation of variations in noise variance is to view these as changes in transparency about the central bank’s output target. The above results then document clear disadvantagesfromtransparency. Inasomewhatrelatedmodel,FaustandSvensson(2001,Proposition6.3) appear to establish the opposite: Namely that central bank losses were increasing, not decreasing, in transparency. The difference lies here in the definition of “transparency”, and it is instructive to seehowapparentlyinnocuousdifferencesinamodel’ssettingcanleadtodifferentconclusions. In the experiments of Faust and Svensson, transparency means that targets can be perfectly inferred once policy is observed. In the experiments above, transparency (σ = 0) makes the n target component τ directly observable, regardless of policy. Both imply the same information t sets in equilibrium. But the constraints faced by the discretionary policymaker differ in profound ways. Under discretion, the policymaker takes the public beliefs system and its Kalman gains as given. Whenthetargetisperfectlyobservable,asisthecaseabove,thecurrentpolicymakercannot influencebeliefs,sincetheKalmangainK in(19)iszero.25 Incontrast,whentargetsareperfectly x inferablefromobservedpolicies,thislinkisretainedcausingthedifferenceinoutcomes. 4 (In-)Credible Disinflations and Exogenous Persistence A pertinent question in monetary policy is whether to conduct disinflations quickly or gradually. The answer involves a minimization of the economic costs incurred by the necessary output contractionsalongthedisinflationpath. Thesecostshingeonthepersistenceofinflation. Ifpersistence is large, a larger or more protracted contraction might be necessary. A pertinent policy question is then whether to chose the “cold turkey” approach of a quick disinflation, involving a large initial contraction, or whether to chose a more gradual approach, implementing a longer sequence of smallercontractions. Academicresearchhasoffereddifferentadviceontheseissues,seeforexamplethediscussion between Gordon (1982) and Sargent (1982). Arguments for or against either approach differ in 25ThisissimilartowhatFaustandSvensson,p. 374calltheregime“OG:observablegoalandintention”,forwhich theyfindresultscorrespondingtowhathasbeenfoundinthispaper.

whether credibility is assumed to have an effect on inflation persistence or not. A quick disinflation could enhance the credibility of the policymaker’s intention to disinflate and help reducing the inflation rate by itself. Taking this view, Sargent (1982) favors the “cold turkey” approach. Being more concerned with exogenous sources of inflation persistence makes Gordon (1982) lean towardsadvocatingmoregradualdisinflationpaths. The framework presented in this paper allows to address these questions in a fully dynamic frameworkwithanoptimizingpolicymaker. Thelinearquadraticapproachallowstohandlemultiple, endogenous state variables, including those arising from partially backward-looking inflation dynamics. To the best of my knowledge, this is the first explicitly optimizing analysis of disinflationstrategieswhenpolicygoalsareunobserved.26 The following disinflation experiment is considered: How should policy react to a surge in inflation due to unfounded public beliefs? In an admittedly stylized way, this resembles the initial conditions of the Volcker disinflation as discussed by Erceg and Levin (2003) and Goodfriend and King (2005). Such inflation beliefs may be caused by a belief shock, n , or inherited via t π = g0τ . Qualitatively, results are similar in either case and the discussion below will t|t−1 t|t−1 focusonresponsestobeliefshocks. In the model of the previous sections, the cost of disinflation depends largely on the policymaker’s capability to lower policy expectations quickly. In the belief shock model of the previous section, policy induces a stronger contraction of the economy in response to beliefs τ when t|t−1 credibility problems are larger, see the middle right panel of Figure 7.27 While this suggests that disinflationsshouldbemoreaggressive,itdoesnotyetspeaktoconcernsaboutthetrade-offsunder exogenousinflationpersistence. To see how policy changes in the presence of exogenous persistence, the Phillips Curve is 26RelatedistheworkofIreland(1995)whofindssimilarresultswhenimposingasluggishresponseofpublicbeliefs onpolicyannouncements. Asanalternative,Ireland(1997)seekstoreconciletheconjecturesofSargent(1982)and Gordon(1982)bydifferentiatingbetweendisinflationsathighorlowlevelsofinflation. 27Similarly, the negative response to a belief shock is stronger for larger values of σ , up to the point where the n growingnoisevariancebeliefsreactlessandlesstotheseshocksas.

augmentedwithabackward-lookingterm,representingpriceindexationattherateγ.28 γ β κ π = π + π + x (20) t t−1 t+1|t t 1+βγ 1+βγ 1+βγ ∞ X = γπ +κ βkx t−1 t+k|t k=0 In this hybrid Phillips Curve (20), inflation is not only determined by the expected path of future policies known from (1), but also by lagged inflation. Policy innovations are still the ultimate driverofinflation,buttheycarrylessweightinchangingcurrentinflation. [Figure8abouthere.] Figure 8 compares impulse responses to a belief shock n when varying the indexation rate t γ ∈ {0;0.5;1}; for γ = 0 the model is identical to what has been studied above.29 For better comparison of the disinflation policies, the belief shocks have been scaled such as to yield a unit innovation in inflation on impact. As higher indexation rates increase exogenous inflation persistence,optimalpolicycontractstheeconomyevermoreaggressivelytoabeliefshock—bolstering the case for the “cold turkey” approach. The lower panel of Figure 8 confirms this also over a widerrangeofvaluesforthenoisevarianceσ2. n Even though policy contracts the economy more vigorously when exogenous persistence is larger, disinflations are not necessarily quicker. Due to the higher exogenous persistence in inflation it takes longer for inflation to fall when γ is larger. Policy cannot avoid the higher degree of backward-lookingness in inflation. But this is precisely why an aggressive initial contraction is warranted. Itdoesnotonlyfightbeliefsasintheprevioussection. Byreducingcurrentinflation,it reduces also the amount of future inflation caused by ill-founded beliefs to be carried forward via thebackward-lookingterminthePhillipsCurve. 28As in Woodford (2003a) or Christiano, Eichenbaum, and Evans (2005) this can be derived from the optimizing behaviorofFirmsunderCalvopricing. Firmswhodonotoptimizetheirpricesaresupposedtochangepricesatthe rateΠγ whereΠ islastperiod’slevel(notlog)ofinflation. AsshownbyWoodford(2003a),thischangesalso t−1 t−1 welfare functions such as (2) to be concerned with quasi-differenced inflation π −γπ instead of inflation. The t t−1 pointoftheexperimentishereishowevertoconsiderhowexogenouspersistencechangespolicieswhilstkeepingthe objectivefunctionconstant. Thepolicymaker’slossfunctionisthuskeptunchanged. 29OtherparametersarecalibratedatthevaluesshowninTable1.

5 Related Literature Since asymmetric information is such a pertinent issue in policymaking, it is no wonder, that there is a wide body of related literature. General surveys can be found in Rogoff (1989), Walsh (2003, Chapter 8) and Persson and Tabellini (2000, Chapter 15). The literature can roughly be classified by answering the following questions: Who learns about what and how? How is policy described,asanexplicitoptimizationproblemsorbyabehavioralpolicyrule? Inthispaper,policy isoptimizedwhilethepublicsolvesasignalextractionproblemabouthiddenpolicytargets. The tractability of the solution method presented here stems from the unobservable states following smooth, Gaussian processes as opposed to regime switches. Discrete regime switches are attractiveformodelingcentralbank“types”likeweak/softorcommitment/discretionasinBackus and Driffill (1985b), Cukierman and Liviatan (1991), Ball (1995), Walsh (2000) and King, Lu, and Pasten (2008). Unobserved regime switches lead to important non-linearities in the public’s inference problem, which complicate the constraints in an optimal policy problem considerably. Theaforementionedliteraturehascorrespondinglyfocusedonverysmallstatespacesand/orfinite horizonproblems,sinceunobservedregimesswitchesarehardtoincorporateintothekindofgeneral dynamic settings commonly used for policy analysis. Svensson and Williams (2006) discuss theresultingdifficultiesinmoredetail. Learning about regime-switches does not pose such problem when it is the central bank who learns about economic conditions as in Sargent (1999). This is because of the strategic behavior of the policymaker when facing agents learning about, respectively from, him as opposed to the non-strategicbehaviorofatomisticprivateagents. Svensson and Woodford (2003, 2004), Aoki (2006) study optimal policy with an imperfectly informed central bank in linear quadratic settings similar to mine. A convenient feature of this approachisthatthepublic’s“learning”reducestoatime-invariantsignalextractionproblem. This isdifferentfromthekindofevolutionarybeliefsystemstudiedinthelearningliteraturerepresented for example by Evans and Honkapohja (2001). Adaptive learning leads to interesting dynamics where past data drives changes in regression coefficients, but is so far hard to capture in optimal

policy problem. For fixed policy rules, the issue is analyzed by Orphanides and Williams (2005, 2006). AnexceptionistheworkofGaspar,Smets,andVestin(2006)whoendowthepublicwitha time-varying,adaptivelearningrule. TheyderiveaMarkov-perfectpolicywithhistorydependence inducedsimilarlyashereviathereactiontopeople’sbeliefs. Theirpoliciesgeneratedatawithlow inflation persistence to influence people’s constant gain learning. Thanks to the lower complexity oftheinferenceproblemadoptedhere,theirresultscanbecorroboratedinaverytransparentway. Hidden information is modeled here as a signal extraction problem where the private sector doesnotobservetherealizationofshocks,butwherethestructureoftheeconomyanditsparameter valuesaremutuallyknown. Inarationalexpectationsequilibrium,theprivatesectorthenknowsthe correctpolicyfunctionbutcanonlyimperfectlyinferthenatureofshocks. Thisequilibriumnotion is stronger than the self-confirming equilibria considered by Fudenberg and Levine (1993) and Sargent (1999) or the recursive learning schemes studied for example by Evans and Honkapohja (2001) or Orphanides and Williams (2005). In those cases, the public beliefs about structural relations may be erroneous as long as they are justified by the data generated from the model. In the rational expectations equilibrium pursued here, the public knows the true policy function — but not the states driving it. This serves as a useful, non-trivial benchmark for evaluating the consequencesofasuperiorlyinformedpolicymakerindynamiceconomies. Closest to the simple model studied in Section 2 are the studies by Cukierman and Meltzer (1986) and Faust and Svensson (2001, 2002). This paper shares with them not only the linear framework and the Kalman filtering of the public, but also that it casts the policy problem around unobserved policy goals. New is the general framework capable of handling various models with endogenous state variables. Faust and Svensson focus on the welfare effects of credibility. Within a slightly different economic structure (forward-looking Phillips Curve instead of Lucas-supply curve),30 their results are broadly confirmed here: Outcomes are improved when output targets are unobservable. (See also the discussion on varying transparency in Section 2.4.) The common force at work is that the updating of public beliefs depends directly on observed policy. Similar 30Afurtherdifferenceisthattheiranaloguetotheiidshockε isnotatargetcomponentbutacontrolerrorofpolicy. t

to mechanisms discussed by Walsh (2000), this disciplines policy while retaining Markov-perfect time-consistency. 6 Conclusions This paper has solved for the optimal discretion policy in a New Keynesian model with unknown output targets and finds that the policy seeks to contract the economy in response to inflationary beliefs. Inaddition,thepursuitofoutputtargetsisscaledback,becauseoftheirinflationaryeffects on public beliefs. This policy, in particular its history dependence, shares some similarities with commitment policies. To the extent that hidden information is a realistic feature of actual policymaking,thissuggestsmuchsmallercostsforreal-worldpolicymakersfromretainingsomedegree ofdiscretionaslongastheykeeppublicbeliefsabouttheirintentionsincheck. Thestylizedmodelanalyzedhereimpliesthatintransparencyofpolicytargetsispreferable— at least under discretion. However, an important caveat is that public beliefs matter here only for linking economic activity to pricing decisions in the New Keynesian Phillips Curve. The welfare effects of transparency might be subject trade-offs, leading to more differentiated results, when expectationsoffutureactivityweretoaffectbothinvestmentandpricingdecisions. Themodelgivesalsorisetobeliefshocksasasourceofbusinesscyclefluctuations. Similarto theworkofLorenzoni(2006),suchshocksshiftpublicperceptionsabouteconomicsfundamentals, whilst the actual fundamentals remain unchanged. Under imperfect information, these shifts in public beliefs are rational since the belief shocks are correlated with informative signals about fundamentals. Theoptimaldiscretionpolicyseekstoquelltheerroneousbeliefsarisingfromthese shocks. In the New Keynesian model studied here, belief shocks induce dynamics similar to cost push shocks, which is similar to belief shock dynamics found by Angeletos and La’O (2008b) basedonhigher-orderdynamics. Apartfromillustratingtheoptimalpolicyunderimperfectinformationwithinawidelystudied New Keynesian model, the technical appendix to this paper provides a general solution method

which allows to extend the analysis to larger settings, relevant for practical policy analysis. By relying on linear quadratic approximations and Gaussian uncertainty, the optimal policy problem becomestractablewithoutlosingitseconomicintricacy. Fortheclassofmodelsconsideredhere,publicbeliefsaboutthepolicymaker’shiddeninformationbecomeadistinctstatevariableofthepolicyproblem,whenapolicymakerisbetterinformed than the public. These public beliefs are shaped by observed policy actions, giving a scope for managing beliefs about future policies that is otherwise absent in a discretionary policy problem. Since public beliefs are a natural state variable under imperfect information, managing this state ofbeliefsisMarkov-perfectandtimeconsistent.31 A fruitful area for future research would be to extend the analysis to a policymaker’s private information about economic fundamentals, for example determinants of potential output. Such information would be widely dispersed amongst different members of the public as well as the central bank, suggesting to combine the policy concepts studied here with the dispersed informationsettingsstudiedforexamplebyLorenzoni(2006)orAngeletosandLa’O(2008a). 31ThisexcludestheexplicitreputationalmechanismsbasedonhistorydependentstrategiesknownfrombyBarro andGordon(1983b)orChariandKehoe(1990)areexcludedfromtheanalysis.

Appendix A Innovation Representation in the Simple Model This section derives the ARMA(1,1) innovations process (9) for policy in the simple model of Section2. First,policycanbeseparatedintoinnovationandpublicexpectations x = f τ˜ +f ε˜ +(f +f )τ (21) t τ t ε t τ b t|t−1 | {z } | {z } x˜t x t|t−1 The ARMA representation follows from using τ = ρ τ and the Kalman filter to express the t+1|t t|t evolutionofpriorbeliefsas x = ρx +ρ ((f +f )K −1) x˜ t+1|t t τ b τ t B Kalman Filter in the Simple Model In the simple model of Section 2, the signal extraction problem of the public uses the policy function (8) as measurement equation and the AR(1) transition of τ (6) as state equation. The t Kalmangainsare 1 Σ 1 σ¯2 τ K = and K = τ f Σ +σ¯2 ε f Σ +σ¯2 τ τ ε τ whereσ¯2 ≡ f2/f2 ·σ2 andΣ solvestheRiccatiequation ε τ ε τ Σ σ2 Σ = σ2 +ρ2σ¯2 τ = η τ η Σ +σ¯2 1−ρ2 σ¯2 τ Στ+σ¯2 As discussed on Section 2.3, it is plausible to assume that 0 ≤ f ≤ 1, 0 ≤ f ≤ 1, f ≤ 0, τ ε b and f +f ≥ 0. It is then straightforward to verify that −1 ≤ ψ ≤ 0, since the Kalman gains are τ b positiveandf K +f K = 1. τ τ ε ε

C Compensating Rate of Average Inflation Section2.4evaluatesex-antepolicylossesbasedonthepolicymaker’sobjectivefunction E(π2 +α (x −x¯ )2) E(V ) = t x t t t 1−β Thedifferenceinlossesunderhiddenversusfullinformation,canbeexpressedasacompensating rate of average inflation, π¯, which would equalize policy losses in both equilibria when added to thedynamicsunderhiddeninformation. (cid:0) (cid:1) E((1−β)V |FullInfo) = E (π +π¯)2 +α (x −x¯ )2|HiddenInfo t t x t t As an alternative measure of policy losses, this “compensating average inflation” abstracts fromthevalidityofthelinearquadraticframeworkfornon-zero(ornon-indexed)inflationratesin steadystate. Since this paper looks only at shocks to output targets, the calibration does not try to match a level of second moments observed in the data and the compensating rate of average inflation is scaledbythestandarddeviationofinflationunderhiddeninformation.

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Figure8: DisinflationwithExogenousPersistence (a)Impulseresponse (b)ImpactCoefficientsofx t Note: Panel(a): Impulseresponsesofoutputandinflationtoabeliefshockwithvaryingdegreesofpriceindexation (γ = {0;0.5;1})inthePhillipsCurve(20). Beliefshocksarenormalizedsuchthattheyproduceaunitresponsein inflationonimpact. Panel(b)reportstheimpactcoefficientsofoutputinresponsetosuchnormalizedbeliefshocks fordifferentvaluesoftheindexationrateγ andnoiselevelσ2. (OtherparameterscalibratedasinTable1.) n

Table1: ModelCalibration PrivateSectorParameters β 0.99 Timepreference σ 1.00 RiskAversion/InverseEIS θ 0.75 CalvoProbabilityofnotrepricing φ 1.00 InverseFrischLaborElasticity κ 0.1717 SlopeofPhillipsCurve: κ=(1−θ)·(1−β·θ)/θ·(σ+φ) PolicyPreferences α 1.00 Policymaker’spreferenceforoutputstabilizationL +π2+α (x +x¯ )2 x t t x t t DrivingProcesses σ 1.00 Volatilityofiidtargetcomponent,ε ∼ N(0,σ2)fromx¯ =τ +ε ε t ε t t t ρ 0.90 Persistenceoftargetcomponentτ =ρτ +η t+1 t t+1 σ 1.00 η ∼ N(0,σ2) η t η σ 1.00 Totalvolatilityofoutputtargetusedforsensitivityanalysis. σ ≡σ2+σ2/(1−ρ2) x¯ x¯ ε τ σ 1.00 Volatilityofbeliefshocksn ∼ N(0,σ2)(Section3) n t n Notes: Private-sectorparameterstakenfromGali(2003)’scalibrationtoquarterlyU.S.data. Innovationvariancesare eachnormalizedtounityandnotintendedtomatchthescaleofanysecondmoments. Thesensitivityofresultstoρ and α is discussed in Section 2.4. As shown there, variations in α are isomorphic to varying κ. As a measure of x x credibility,σ isvariedinSection3. n

MANAGING BELIEFS ABOUT MONETARY POLICY UNDER DISCRETION TECHNICAL APPENDIX April 2009 i

TECHNICALAPPENDIX ii A A Class of Linear Quadratic Models Themechanismsofthehiddeninformationpolicyproblemextendbeyondthesimplemodelofthe previous sections and are applicable to a general class of linear quadratic environments. It is thus beneficial to cast the exposition around this more general class of models. The applications presented in Section 3 (noisy signals, backward looking inflation) have already relied on this general framework. Again, attention is limited to a Markov perfect, discretionary policy problem. In the spirit of “bygones are bygones”, state variables in a Markov-perfect equilibrium must be relevant for current payoffs. The public’s prior beliefs are part of these Markov states since they matter for public payoffs. A current decision-maker can influence future decisions only by manipulating the state of beliefs as well as other endogenous state variables, for example capital, carried forward intofuturedecisionproblems. Thereisnocommitmenttofuturepolicies. In general, the entire distribution of public beliefs needs to be tracked by the policy problem. The framework presented here affords a considerable simplification, which makes the problem welltractable: ThemodeliscastinaGaussianframeworkwithconstantvariances. Trackingentire distributionsthencollapsestotrackingonlytheirmeansandcanbehandledwiththeKalmanfilter. It is the public’s prior, not posterior, beliefs which enter the state vector, since the latter will be formedafterobservingcurrentdatawhichisinfluencedbycurrentpolicy. This section defines a rational expectations equilibrium where the public forms its posterior beliefs consistently with the optimal policy function. The policymaker is free to choose policies which are inconsistent with the public’s belief system, but equilibrium requires that he finds it ex-postoptimalnottodeviatefromthepolicyfunctionassumedinpeople’sKalmanfilter. There are four types of variables: 1) Backward-looking variables, X , corresponding for ext ample to the policy targets τ and ε in the model of the previous sections. 2) Policy controls, U , t t t for example like the output gap above. 3) Publicly observable variables, Z , coinciding with the t outputgapinthesimplemodel. 4)Forward-lookingdecisionvariablesoftheprivatesector,Y ,like t

TECHNICALAPPENDIX iii inflation and the interest rate in the above model.1 They will be treated as vectors of dimensions N ,N ,N ,andN respectively. x u z y The backward looking variables need not only capture exogenous forcing variables like τ and t ε but also endogenous states like capital, habits, or lagged variables, for example inflation in a t modelwithpriceindexation. Theyevolveas X = A X +A Y +B U +Dw (1) t+1 xx t xy t x t t+1 wherew isanexogenousN -dimensionalwhitenoiseprocesswithvarianceEw w0 = I.2 t+1 w t t The policymaker observes the entire history of w , denoted wt and will thus have complete t informationabouttherealizationofallvariablesuntiltimet. Incontrast,theprivatesectorobserves onlyalinearcombinationofpolicycontrolsandbackwardlookingvariables: Z = C X +C U (2) t x t u t Zt = {Z ,Z ,Z ,...} t t−1 t−2 The history Zt spans the public information set.3 A sufficient condition to ensure superior informationofthepolicymakeristhatN < N . Foranyvariablez , z w t z ≡ E(z |Zt) t|t t denotes the expectation of z on the private sector’s information set. Synonymously these expect tations will be called public beliefs. In particular, X are the prior beliefs about X before t|t−1 t observing Z . By construction, Y = Y always holds since public decisions are based on public t t t|t information. In principle, Y could also be added to the measurement vector, but without adding t 1Exceptforsuchsimplemodels,theinterestrateistypicallymodeledasthepolicycontrolandtheoutputgapisa forward-lookingvariableoftheprivatesector. 2Withoutlossofgenerality,X isconstructedsuchthatN ≥N . t x w 3Inaddition,thereisnouncertaintyaboutthestructureoftheeconomyandthepublicwillknowallparametersof themodel,forexamplethematricesA ,A ,B andDofequation(1). xx xy x

TECHNICALAPPENDIX iv newinformation. The optimality conditions of private sector behavior are represented by an expectational linear differenceequationinvolvingonlypubliclyobservablevariablesandpublicsectorexpectations:4 A1 Y = A Y +A X +B U (3) yy t+1|t yy t|t yx t|t y t|t Thepolicymakerseekstominimizetheexpectedpresentvalueofcurrentandfuturelosses ∞ X V = E βkL (4) t t t+k k=0  0   X X t t         L = Y  QY  (5) t  t  t     U U t t where the per period loss function L is quadratic in X , Y and U , Q is assumed to be a positive t t t t definitematrix,andtheexpectationoperatorisconditionalonthehistoryofwt. In principle, one could also allow for public beliefs X and U to enter the loss function. t|t t|t Except for adding algebraic complexity, this would not raise any further methodological issues.5 Inthecurrentform,thelossfunction(5)dependsonpublicbeliefsviaY = Y . t t|t 4Notice that the policy control or parts of X are not precluded from entering directly in this forward looking t constraint. Thiswillbethecasewhen,forexample,thepolicycontrolispubliclyobservablesuchthatU = U . A t|t t moregeneralwaytosetup(3)wouldbetowrite A1 Y =A Y +A2 X +B2U +A3 X +B3U yy t+1|t yy t|t yx t|t y t|t yx t y t withtheunderstandingthatthemeasurementequation(2)impliesA3 X +B3U =A3 X +B3U . Thisreduces yx t y t yx t|t y t|t thento(3)withA =A2 +A3 andB =B2+B3. yx yx yx y y y 5Likewise, lineartermsinX andU couldbeaddedtothetransitionequationforthebackwardlookingvarit|t t|t ables.

TECHNICALAPPENDIX v TheSimpleNKModel Equations (1), (2), (3) and (5) describe the class of LQ models for which we seek a solution to the optimal policy problem under discretion and hidden information. The simple NK model of Section 2 in the paper can be represented in the general framework as follows: The output gap equalsthepolicycontrol,U = x andisalsoidenticaltothemeasurementvectorZ = U suchthat t t t t C = 1 and C = 0. Furthermore, the backward and forward looking variables are X = [τ ε ]0, u x t t t respectivelyY = π . t t In this model, the backward looking variables are purely exogenous, A = 0 and B = 0, xy x which considerably simplifies the solution under symmetric information (Svensson 2007, p. 24). However, in the hidden information problem the state vector will be augmented by public beliefs and the state vector will be endogenous. So no additional complication arises from allowing the backwardlookingvariablesin(1)tobepartlyendogenous,too. Thebackwardlookingvariablesare       τ ρ 0 σ 0 t+1 η X =   =  X + w t+1     t   t+1 ε 0 0 0 σ t+1 ε | {z } | {z } Axx D withA = 0andB = 0. xy x Inflation is the only forward looking variable of the private sector, Y = π , and the Phillips t t Curve corresponds to the associated forward looking constraint with A1 = β, A = 1, A = 0, yy yy yx andB = −κ. y

TECHNICALAPPENDIX vi B Private Sector Equilibrium The policymaker is constraint by the beliefs and the behavior of the private sector. The private sector is atomistic and takes policies as given. Before turning to optimal policy, it is useful to considernotionsofprivatesectorequilibriumforagivenpolicy. Thisgeneralizesthediscussionin Section2.2ondetermininginflationforagivenpolicyfunction. Attention is limited to time-invariant, Markov-perfect equilibria. Policies will depend only on current levels of backward-looking variables and prior beliefs about those. In equilibrium, policy isafunctionoftheMarkovstates: U = F0X +F0X (6) t 1 t 2 t|t−1 for some F0, F0. Notice that this does not presuppose a commitment of the policymaker to such 1 2 a rule. Discretion will rather require that this policy is ex-post optimal, such that the policymaker hasnoincentivetodeviateoncetheprivatesectorhasformedbeliefsconsistentwiththepolicy. For the time being, the discussion adopts now the perspective of the private sector who takes the policy (6) as given when forming beliefs and making choices. This gives rise to a fairly strong notion of private sector equilibrium which can be applied to the simple NK model in the paper. As will be seen shortly, such an equilibrium need not always be unique. As will be shown below, a weaker notion of “temporary equilibrium” will in general be sufficient to constrain the discretionarypolicyproblem. Definition (Private Sector Equilibrium). Given the policy in (6), the private sector equilibrium is a sequence of observations {Z }, perceived states {X }, perceived policies {U } and private t t|t t|t sectorchoices{Y }suchthat t • Expectations and beliefs are rational. In this linear framework, they are formed using the KalmanfilterwithmeasurementsZ . t • Choicesareoptimal,thatistheysatisfytheforwardlookingconstraint(3).

TECHNICALAPPENDIX vii UsingtheKalmanfilter,beliefsthenevolveas X = X +K0(Z −Z ) (7) t|t t|t−1 t t|t−1 U = F0X +F0X (8) t|t 1 t|t 2 t|t−1 Amongst others, the Kalman gain K0 depends on the policy coefficients F0 = [F0 F0] in (6). 1 2 Before turning to conditions for existence and uniqueness of the private sector equilibrium, some detailsarepresentedfortheKalmanFilter. KalmanFilter Forthepolicygivenin(6),theprivatesector’sKalmanfiltercombines(6)with(1)and(2)toobtain thestateandmeasurementequations X = (A +B F0)X +A Y +B F0X +Dw (9) t+1 xx x 1 t xy t|t x 2 t|t−1 t+1 | {z } ≡A Z = (C +C F0)X +C F0X (10) t x u 1 t u 2 t|t−1 | {z } ≡C andbeliefsevolveas X = X +K0(Z −Z ) (11) t|t t|t−1 t t|t−1 withKalmangainK K ≡ Cov(X ,Z −Z ) Var(Z −Z )−1 (12) t t t|t−1 t t|t−1 TheKalmangainisidenticaltothecoefficientsofaleastsquaresprojectionofX onZ −Z . t t t|t−1 K0 = ΣC0(CΣC0)−1 (13)

TECHNICALAPPENDIX viii whereΣsolvestheRiccatiequation Σ = AΣA0 +DD0 −AΣC0(CΣC0)−1CΣA0 (14) The Kalman filter depends only on the policy coefficients F0, via which policy reacts to X , and 1 t is independent of the reaction coefficients associated with the predetermined state variable X . t|t−1 The presence of private sector controls Y and predetermined variables X does not affect the t|t t|t−1 Kalmangain. The above assumes that the N × N matrix C has full row rank.6 In principle (and also in z x practice) it can happen that C is collinear for some F0. Numerically it is already critical if C 1 is nearly collinear. This corresponds to situations when there are multiple observables7 which are (almost) perfectly correlated such that VarZ ˜ = CΣC0 is ill-conditioned. Economically, this t means that a candidate policy F0 tries to mimic other signals in Z . I have never observed such t mimickingstrategiesinequilibrium,butdependingoninitialconditionsitcanoccuralongthepath of the policy improvement algorithm. In these cases, the Kalman filter is implemented by pruning the redundancies in the set of observable variables via a singular value decomposition of C. To obtainnumericallystablesolution,thisisdoneforsingularvaluesofC smallerthan10−8. ConditionsforExistenceandUniqueness Optimal choices of the private sector solve the forward-looking constraint (3) given the policy (6) and private sector beliefs about X . Based on the Kalman filter, (3) and (1), this can be written as t asystemofexpectationaldifferenceequationsdrivenbytheiiddisturbanceZ ˜ .8: t X = (A +B F ˆ0)X +A Y +(A +B F0)K0Z ˜ t+1|t xx x t|t−1 xy t|t xx x 1 t A1 Y = (A +B F ˆ0)X +A Y +(A +B F0)K0Z ˜ yy t+1|t yx y t|t−1 yy t|t yx y 1 t 6RecallthatN <N ≤N . z w x 7ThisisthecaseinthemodelwithbeliefshocksinSection3,butnotinthesimplemodelofSection2. 8Innovations Z˜ are defined relative to the public’s prior belief Z˜ ≡ Z − Z . By construction they are t t t t|t−1 orthogonaltopriorinformationoftheprivatesectorandareiidunderthepublic’sprobabilitymeasure.

TECHNICALAPPENDIX ix whereF ˆ0 ≡ F0 +F0. Thematrices 1 2     I 0 (A +B F ˆ0) A A¯ =   and B¯ =  xx x xy      0 A1 (A +B F ˆ0) A yy yx y yy collectthecoefficientsontheendogenousvariables. ThisisthekindoflinearsystemsstudiedbyKingandWatson(1998)andKlein(2000),where A1 is allowed to be singular. And their “counting rules” for stable and unstable roots can be yy appliedtoderiveconditionsforexistenceanduniquenessoftheprivatesectorequilibrium. Proposition 1 (Existence and Uniqueness of Private Sector Equilibrium). Existence and uniquenessofaprivatesectorequilibriumdependontherootsz of|A¯z−B¯| = 0formatricesA¯andB¯ defined above. A unique equilibrium exists only if there are N roots inside the unit circle and N x y outside. ThematricesA¯andB¯,and thusalsothe conditionfor existence anduniqueness, depend on the policy rule (6) but not on the Kalman gain K0. This is an instance of certainty equivalence inlinearrationalexpectationssystems. Proof. TheresultfollowsfromapplyingthesolutionmethodsofKingandWatson(1998)orKlein (2000)tothelinearrationalexpectationssystemabove. Applyingtheirmethodsyieldsthecounting ruleinthepropositionandthesolutionhastheform ¯ ˜ Y = GX +H Z t|t t|t−1 y t ¯ ˜ X = AX +H Z t+1|t t|t−1 x t whereG ¯ andA ¯ dependonlyonA¯ andB¯ butnotonK0 (forgivenpolicies,F0 andF0.) 1 2 In the simple NK model of Section 2 in the paper, the condition is trivially met9 but in general this needs not be the case. A pertinent example is the nominal indeterminacy of Sargent and 9Itisstraightforwardtocheckthattherearetwostableroots(ρand0)associatedwiththeexogenoustargetvariables andoneunstableroot(1/β)associatedwithinflation,whichistheonlyforwardlookingvariable.

TECHNICALAPPENDIX x Wallace(1975),whichholdsforanyexogenouspolicylike(6)whentheinterestrateisthecontrol variable. ThisappliesalsototheNewKeynesianmodel,asdiscussedforexamplebyGali(2008). Ifauniquesolutionexists,theconstructionoftheprivatesectorequilibriumisusefultoanalyze outcomes under different candidate policies as in Section 2. But for the purpose of constraining the discretionary policy problem, the above equilibrium notion is actually too strong. In this equilibrium, private sector expectations treat (6) as a time-invariant policy rule, carried out forever. And even though this will resemble the equilibrium outcome, it misrepresents the nature of the discretion problem where the policymaker can reoptimize his plans at each period. Therefore, non-uniqueness of a private sector equilibrium does not foreclose uniqueness of a discretionary equilibrium. To constrain the discretion problem, a weaker form of private sector equilibrium is sufficient. ItisatemporaryequilibriuminthespiritofGrandmont(1977): Definition (Temporary Private Sector Equilibrium). At a given point in time, the private sector has given beliefs about current policy according to (6). They are embodied in a Kalman gain K0 usedtoupdatebeliefsaboutX asin(7). Furthermore,peopleholdpossiblydifferentbeliefsabout t future policies. They are embodied in a mapping G0 which leads to expectations about future privatedecisions: Y = G0X (15) t+1|t t+1|t The temporary equilibrium then reduces to optimal choices which satisfy the forward looking constraint(3)giventhebeliefsin(15). In a temporary equilibrium, private sector expectations of future choices are given. It is then straightforward to substitute the forward-looking variables by a linear combination of publicly

TECHNICALAPPENDIX xi perceivedpoliciesandstates: Y = G0X +G0U (16) t|t x t|t u t|t where G0 = (A1 G0A −A )−1(A −A1 G0A ) x yy xy yy yx yy xx and G0 = (A1 G0A −A )−1(B −A1 G0B ) u yy xy yy y yy x The construction his temporary equilibrium is not a special feature of this hidden information setup. Similar computations are performed for example by So¨derlind (1999) in his derivation of optimalMarkovperfectpoliciesundersymmetricinformation. C Discretion Policy and Equilibrium Discretionary policy is time-consistent. At each point in time the policymaker can reoptimize while taking his future optimizations as given. This leads to a recursive representation of the optimization problem as a dynamic program. The state variables of the policy problem are the backwardlookingvariablesandpriorbeliefs,thereisnofurtherhistorydependence. Furthermore, the policymaker must account for the rational expectations and optimal choices of the private sector. Thisissummarizedinthefollowingdefinition: Definition (Discretionary Policy). At each point in time, for given private beliefs embodied in F0 andG0,thepolicymakerchoosesU tominimize t (cid:8) (cid:9) V = min L +βE V0 t t t t+1 Ut,Yt,Xt+1 s.t. X = A X +A Y +B U +Dw t+1 xx t xy t x t t+1 Y = G0X +G0U t|t x t|t u t|t where G0 and G0 are defined as in (16) above. The constraints correspond to the transition x u equation for X (1), and the private sector’s temporary equilibrium (16). The continuation value t

TECHNICALAPPENDIX xii of this dynamic optimization problem, V0 , is a given function of future, Markov perfect state t+1 variables   X t+1 S ≡   t+1   X t+1|t Since the problem is linear quadratic, the value function can be taken to be linear quadratic as well(Bertsekas2005): V0 = S0 V 0S +v0 (17) t+1 t+1 t+1 The solution is then based on iterating between a conventional linear regulator problem and the Kalman Filter.10 The regulator problem has the following form and is described in more detail in AppendixD: (cid:8) (cid:9) S0V ∗S +v∗ = min S0Q0S +2S0N0U +U0R0U +βE S0 V 0S +v0 (18) t t t t t t t t t t+1 t+1 Ut s.t. S = A0S +B0U +Dw (19) t+1 t t t+1 forgivenF0,G0,apositivedefiniteV0 andascalarv0. ThematricesQ0,N0,R0,B0 andD are derivedinthenextsection.11 Theoptimalpolicyis U = −(R0 +βB00 V 0B0)−1(N0 +βB00 V 0A0) S (20) t t ≡ F∗S t Theoptimalpolicyislinearashasbeenanticipatedin(6). Thepolicyappearscertaintyequivalent since it is independent of the shock loadings D.12 But in fact, the setup of the regulator itself is 10ThedefinitionofthediscretionproblemtakesthematrixV0andthescalarv0asgiven.Inthepolicyimprovement algorithmusedtoimplementthesolution,theywillbecalculatedsuchastobeconsistentwithcontinuingthepolicy F0andthebeliefsG0forever. Thisisshownattheendofthissection. 11ExceptforV0 andv0 matriceswithsuperscript“0”dependonF0 andG0. Aswillbeseebelow,alsoV0 andv0 canbecomputedtobeconsistentwithcarryingoutpoliciesF0andG0forever. 12Certaintyequivalenceisawell-knownresultoflinearregulatorproblems(Bertsekas2005).

TECHNICALAPPENDIX xiii notcertaintyequivalentsinceitdependsontheprivatesector’sKalmanfilter. Policiesarethusnot certaintyequivalent. Definition(EquilibriumunderDiscretion). Equilibriumunderdiscretionarypolicymakingconsists ofsequences{U },{X },{Y }and{Z }suchthateach t t t t • U solvesthepolicymaker’sproblem t • Y is the solution to a temporary equilibrium whose underlying beliefs are consistent with t theoptimallychosenpoliciesU t • X andZ evolveaccordingto(1)and(2) t t wherepoliciesareatime-invariantfunctionofthestates. Formally, this requires that F0 = F∗, and G0 = G∗ = G0 +G0(F∗ +F∗), where F∗ and F∗ x u 1 2 1 2 partitionF∗ conformablywithX andX . (K0 isthenconsistentwithF0 = F∗.) Furthermore, t t|t−1 1 1 thevaluefunctionsatisfies V 0 = V ∗ = Q0 +N0F∗ +F∗0R0F∗ +β(A0 +B0F∗)0V 0(A0 +B0F∗) (21) This equilibrium concept is similar to the self-confirming equilibria of Fudenberg and Levine (1993) and Sargent (1999) in that both are a fixed point of mutual beliefs and actions in multiplayergames. However,inaself-confirmingequilibrium,playersholderroneousbeliefsaboutthe structure of the economy, which are justified by observable outcomes. A similar fixed point of beliefs and outcomes is used in the limited-information rational expectations equilibria of Marcet andSargent(1989a,1989b)andSargent(1991). Thisisdifferenthere,wherethepubliccompletely knowsandunderstandsthestructureoftheeconomy.

TECHNICALAPPENDIX xiv D Regulator for Discretion Problem To set up the linear regulator problem shown in (18) and (19), the temporary equilibrium (16) and the Kalman filter (7) can be used to substitute Y out of the loss function (5) and the transition t|t equation(1)forX . TheKalmanfilteryieldsthetransitionequationforX . t t|t−1 The derivation proceeds by using the temporary equilibrium (16) and the Kalman filter (7) to substituteY outofthelossfunction(5)andtransitionequation(1)forX . TheKalmanfilteralso t|t t yields the transition equation for X . The Kalman filter also depends on a prior belief about t|t−1 observables Z = C X + C U and thus on a prior belief on policy. To simplify the t|t−1 x t|t−1 u t|t−1 regulator,itisassumedthatthisbeliefisconsistentwithF0 (asitwillbeinequilibrium),suchthat Z = (C +C (F0 +F0))X t|t−1 x u 1 2 t|t−1 | {z } Cˆ TheKalmanupdatecanbewrittenas ˆ X = KC X +(I −KC)X +KC U t|t x t t|t−1 u t Togetherwiththetemporaryequilibrium(16)thisyields Y = Γ0X +Γ ˆ0X +Γ0U t|t x t x t|t−1 u t with Γ0 = (G0 +G0F0)KC x x u 1 x Γ ˆ0 = (G0 +G0F0)(I −KC ˆ )+G0F0 x x u 1 u 2 Γ0 = (G0 +G0F0)KC u x u 1 u

TECHNICALAPPENDIX xv LossFunction Thelossfunction(5)canberewrittenintermsoftheregulator’sstatesandcontrolusing      X I 0 0 X t t           Y  = Γ0 Γ ˆ0 Γ0X   t|t  x x u t|t−1      U 0 0 I U t t | {z } H0 suchthat  0   X X  0   t t     S S L =  Y   Q  Y   =  t  H00 QH0 t  t  t  t         U t U t U U t t = S0Q0S +2S0N0U +U0R0U t t t t t t whereQ0,N0 andR0 conformablypartitiontheabovequadraticformas:   Q0 N0 H00 QH0 =     N00 R0 StateTransition Likewise,thestatetransitionsforX andX canbederivedas t t|t−1 X = (A +A Γ0)X +A Γ ˆ0X +(A Γ0 +B )U +Dw t+1 xx xy x t xy x t|t−1 xy u x t t+1 (cid:16) (cid:17) X = A KC X + A (I −KC ˆ )+(A G0 +B )F0 X +A KC U t+1|t xx x t xx xy u x 2 t|t−1 xx u t where A = A +A (G0 +G0F0)+B F0 xx xx xy x u 1 x 1

TECHNICALAPPENDIX xvi ThematricesA0,B0 andD in(19)arethusgivenby:   (A +A Γ0) A Γ ˆ0 A0 =  xx xy x xy x   (cid:16) (cid:17) A KC A (I −KC ˆ )+(A G0 +B )F0 xx x xx xy u y 2     A Γ0 +B D B0 =  xy u x  and D =       A KC 0 xx u ValueFunctionconsistentwithF0 andG0 The policy improvement algorithm described in Section E uses a continuation value consistent withcarryingoutthepolicyF0 forever. ThecontinuationvalueislinearquadraticinS asin(17). t Itiscomputedfromtheclosedlooprepresentationoftheregulatorobtainedbypluggingthepolicy F0 into(18)and(19). V 0 solvestheLyapunovequation n o V 0 = Q0 +N0F0 +F00 R0F0 +β(A0 +B0F0)0V 0(A0 +B0F0) Theequationhasauniquesolutionifthematrixincurlybracesispositivedefiniteandiftheclosed loop transition matrix (A0 + B0F0) has all eigenvalues inside the unit circle. The former is assured by the form of the original loss function13 and the latter holds if a stationary equilibrium exists.14 Optimal policies are certainty equivalent (for given F0) and do not depend on v0.15 Still, the scalarv0 canbecomputedfrom: β v0 = tr(V 0DD0) 1−β wheretristhetraceoperator. 13PleaserecallthatQin(5)isassumedtobepositivedefinite. 14EfficientmethodsforsolvingLyapunovequationsareavailableforexampleviatheLAPACKroutinesencodedin MATLABorbyusingthedoublingalgorithmsofAndersonetal.(1995). 15SargentandLjungqvist(2004)orSvensson(2007)givefurtherdetails.

TECHNICALAPPENDIX xvii Unconditionally expectedlossesarecomputedfromtheunconditionalvariancecovariancematrixofthestates: E(V0) = tr(V 0ES S0)+v0 t t t ES S0 = (A0 +B0F0) (ES S0) (A0 +B0F0)0 +DD0 t t t t E Policy Improvement Algorithm Theequilibriumisafixedpointofpublicbeliefsandpolicyactionsandmaps(F∗,G∗,V ∗)intoitself. Anintuitiveandefficientwaytocomputethisfixedpointisthefollowingpolicyimprovement algorithm. It is efficient, since policy improvement methods converge faster than value function iterations (Whittle 1996; Bertsekas 2005).16 It is intuitive, since the algorithm uses the regulator (18) to seek for a one-period deviation from a candidate equilibrium. Non-existence of such a deviationisthedefiningpropertyofequilibrium. Formally, the algorithm starts with a candidate policy F0 and beliefs G0 and computes the Kalman gain K0 and continuation value V 0 associated with continuing this policy forever. If the conditions for a private sector equilibrium are met (Proposition 1), one can even compute the G0 consistent with F0. The solution (20) to the above regulator problem then yields the optimal oneperiod deviation. As long as F0 6= F∗ and G∗ 6= G0 there is no equilibrium. In this case, a new iterationstartsusing(F∗,G∗)asnewcandidatepolicies. The difference with a value function iteration is that at each step, the regulator uses a continuation value consistent with carrying out the candidate policy forever whereas a value function iteration would update V 0 = V ∗ at the j-th step. In contrast, the policy improvement algoj+1 j rithm solves at each step an infinite horizon problem, where Kalman gain K0 and continuation valueV 0,andifpossiblealsoG0,areconsistentwiththecandidatepolicy. 16So¨derlind(1999)solvesforoptimaldiscretionarypoliciesundersymmetricinformationwithvaluefunctioniterationsandcommentsontheslowperformanceofthealgorithm.

TECHNICALAPPENDIX xviii UniquenessofEquilibrium Above I argued for uniqueness of the equilibrium in steady state, since the model then collapses to a full information setting with a unique steady state under simultaneous move timing. But offsteady state, the above equilibrium is an intricate fixed point between optimal one-period policies (F∗),andpublicbeliefs(F0,G0). Formally,itisafixedpointbetweentwoRiccatiequations,one from the policymaker’s regulator problem, combining (20) and (21), the other associated with the public’s Kalman Filter, see equation (14). Under suitable regularity conditions (Bertsekas 2005), both solve well-defined problems with unique solutions given the other’s solution. However, to the best of my knowledge there exist no results on the existence and uniqueness of such nested systems. This is also the conclusion of Hansen and Sargent (2007, Chapter 15) who solve multiplayerequilibriawithsimilarlystackedRiccatiequations. However, in my practical experience, the algorithm typically converges, and if so always to the same equilibrium from arbitrary starting values for (F0,G0). In particular, over a wide range of calibrations (see Figure 4 in the paper), each equilibrium has been checked by drawing 50 times initial values from a mean zero Normal distribution with variance 10. Given equilibrium coefficients between zero and one, this is basically a flat prior. Each time, when the algorithm convergesitconvergestothesameequilibrium.17 17Occasionallyanequilibriummightnotbefoundforaparticularinitialguess. Inthiscase,anotherdrawismade untilthealgorithmhasconverged50times.

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