Imperfect Information and Monetary Models: Multiple Shocks and their Consequences

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Imperfect Information and Monetary Models: Multiple Shocks and their Consequences Leon W. Berkelmans 2008-58 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.

Imperfect Information and Monetary Models: Multiple Shocks and their Consequences Leon Berkelmans1 Board of Governors of the Federal Reserve System October 2008 1Thanks go to Alberto Alesina, Thomas Baranga, Robert Barro, Davin Chor, Eyal Dvir, Emmanuel Farhi,BenjaminFriedman,GitaGopinath,KaiGuo,OlegItskhoki,KennethKasa,MikeKiley,DavidLaibson, Guido Lorenzoni, Greg Mankiw, Emi Nakamura, Kristoffer Nimark, Christina Romer, David Romer, GiacomoRondina, PhilippaScott, RossScott, Jo´nSteinsson, AlehTsyvinskiandparticipantsatnumerous seminars. This paper is a revised version of the second chapter of my Ph.D. dissertation at Harvard University and was partly written while I was a visiting scholar at the University of Sydney. All errors are my own. Theanalysisandconclusionssetfortharethoseoftheauthoranddonotindicateconcurrencebyother membersofthestaff,bytheBoardofGovernors,orbytheFederalReserveBanks.

Abstract This paper examines the role of multiple aggregate shocks in monetary models with imperfect information. Because agents can draw mistaken inferences about which shock has occurred, the existenceofmultipleaggregateshocksprofoundlyinfluencesmacroeconomicdynamics. Inparticular,afteracontractionarymonetaryshockthesemodelscangenerateaninitialincreaseininflation (the “price puzzle”) and a delayed disinflation (a “hump”). A conservative numerical illustration exhibits these patterns. In addition, the model shows that increased price flexibility is potentially destabilizing.

1 Introduction This paper incorporates multiple aggregate shocks into monetary models with imperfect information. Afteracontractionarymonetaryshockthesemodelsexhibit: a)aninitialincreaseininflation, i.e. the “price puzzle”; and b) a delayed disinflation, i.e. a “hump”. These patterns are found in many empirical studies, for example see Christiano et al. (2005). With multiple aggregate shocks, imperfect information implies that agents are not only unsure of the magnitude of the aggregate shock, but they are also unsure which shock has hit. In the New Keynesian variant of the models I present,whenacontractionarymonetaryshockoccursagentsseetheinterestrateincreasing. They conclude that this is either due to a monetary disturbance or due to rising inflationary pressures, and so put some weight on both of these explanations. The weight they place on the “inflationary pressures” explanation then acts as a force inducing firms to increase their price. As time passes agents are better able to determine what has happened and start to cut prices in response to falling demand,generatingahump. Igeneratethispatternwithveryconservativeparametervalues. It is important to explain what happens after a monetary shock because it influences how we view the monetary transmission mechanism and its associated delays. Empirically, the existence ofthepricepuzzleiscontentious. Invectorautoregressionexercisesitisoftensuggestedthataugmentation of models with commodity prices eliminates the problem, see for example Christiano et al. (1999). Hanson (2004) argued that this is most likely a coincidence. Hanson also pointed that out that some authors have proposed informal specification tests where a model is rejected if the price puzzle is found. Such specification tests are flawed if the price puzzle is indeed a structural part of the economy. By contrast the existence of the hump is widely accepted (eg. Romer and Romer (2004), Bernanke et al. (2005)) but how it is modeled often lacks microfoundations. Typicallyitisachievedbyassumingsomesortofindexingbehavior(seeWoodford(2003b)). In addition, I show that increased price flexibility can lead to more monetary non-neutrality. This runs counter to the results of traditional models, where the very source of non-neutrality is pricestickiness. TherecentcontroversyoverthedegreeofstickinessintheUnitedStates(Bilsand Klenow (2004), Nakamura and Steinsson (Forthcoming)) has certainly been viewed through this 1

conventionalprism. Thefindingsinthispaperquestionthatpresumption. Monetary models with imperfect information have experienced a revival of late. Initially they werepopularizedbyLucas(1972),whoexpandeduponsomeideasofFriedman(1968)andPhelps (1969), but their use waned after the 70s. Woodford (2003a) noted that this was, at least partially, because the simplifying assumptions needed to solve the models implied short-lived fluctuations. ForexampleLucas(1972)assumedeverythingbecamecommonknowledgeafteroneperiod’sduration. However Woodford (2003a) was able to get around such problems, obtaining long-lived responses to shocks. This, along with Sims’s work on rational inattention (Sims (2003)), sparked much of the revival. One notable characteristic of this new work is the introduction of situations where agents have different information sets. This can introduce higher order expectations into models, where agents’ actions depend upon what they believe other agents believe. This was the crucialmechanisminWoodford(2003a). Othertopicshavebeeninvestigatedinthisliterature. For example Mac´kowiak and Wiederholt (Forthcoming) looked at how agents allocate their attention between aggregate and idiosyncratic shocks. Gorodnichenko (2008) analyzed a menu cost model where information gets “trapped” when firms do not reset their price. I have a different focus. Uncertainty regarding which aggregate shocks have hit the economy drives the results. It also illustratesabroadermethodologicalpoint. Inimperfectinformationmodelsallsignificantaggregate shocksshouldbeincludedinordertogetaccuratemodelpredictions. The paper proceeds as follows. Section 2 introduces a simple static model that demonstrates someofthemainpointsofthepaper. Section3,introducesimperfectinformationintoatraditional New Keynesian framework. Sharply different results are generated than would be the case if information were perfect, even though agents are very good at determining what is occurring. Section 4 considersthe stabilizing role of sticky prices, showing that greater flexibilitycan induce morevolatility. Section5concludes. 2

2 Simple Static Model The model presented in this section is static and hence cannot replicate the delayed disinflation. However, it is usually the case that anything that enhances the price puzzle enhances the hump in themodelofSection3. ThereforeIviewthestaticexerciseasinformative. Thereareacontinuum offirms,indexedby jontheinterval[0,1]. Afirm’spriceisapositivefunctionoftheirexpectation oftheaggregatepricelevel,thelevelofeconomicactivity,andacommonmarkupshock. Formally, eachfirmsetsthelogoftheirpriceaccordingtothefollowingrule: p =E (p+y+ν) j j where E is the expectation of agent j, p is the aggregate price level (p= R1p dj), y is the log of j 0 j output, and ν is the markup shock. This could be derived using a model of imperfect competition. Indeed pricing rules of this type are quite familiar in the New Keynesian literature (see Woodford (2003b)). Monetary policy is conducted in the form of a money supply rule. Because the central bank dislikestheinflationaryconsequencesofmarkupshocks,themoneysupplyiscutwhentheyoccur. Themoneysupplyisalsosubjecttoitsownshocks. Explicitly: 1 m=ψ− ν φ where m is the log of money, ψ is a monetary shock, and φ>1 determines how aggressively the central bank reacts to a markup shock.1 The response of the central bank to markup shocks is the key difference between this model and Lucas (1972). There is a fixed velocity of money, so that withappropriatenormalization: m= p+y 1φ>1 is necessary to ensure that the central bank’s reaction is not too strong. If φ>1 and information were perfect,priceswouldfallafterpositivemarkupshockduetotheseverityofthemonetaryresponse. 3

Thetwoshocksthathittheeconomy,ψandν,areuncorrelatedandnormallydistributed:   ψ X = ∼N(0,Σ)   ν   σ2 0 Σ= ψ    0 σ2 ν whereX canbeinterpretedasthestateoftheeconomy. Each firm receives an idiosyncratic noisy signal, H , of the money supply, m, and markup j shock,ν,wherethenoiseassociatedwitheachobservationisnormallydistributedanduncorrelated withtheother,i.e.: H =BX+µ j j where:   1 −1 B= φ    0 1 and:   σ2 0 n,m µ∼N(0,Σ ), Σ =  n n   0 σ2 n,ν These noisy signals could be motivated by the kind of information frictions that are introduced in Section3. Agentsalsoknowtheparametersofthemodelandallfirmshaverationalexpectations,so: Z E (x)= xf (x|H )dx j x j for any variable x, where f is the probability density function implied by the model. The equilibx riumintheeconomyisdefinedtobethefollowing: Definition. An equilibrium is a set of beliefs for firms (for any variable x, a pdf f (: | :)) prices x 4

(∀j, p ),output(y),andgovernmentpolicies(m)suchthat: j 1. f (:|:)arerational x R 2. GivenH ,E x= xf (x|H )dx,and p =E (p+y+ν) j j x j j j 3. m=ψ−1ν φ 4. m= p+ywhere p= R1p dj 0 j Appendix A shows that the equilibrium is unique. If the price level after a monetary shock is considered,thefollowingresultsareobtained: Claim 1. After a unitary contractionary monetary shock, i.e. ψ = −1 and ν = 0, the aggregate pricelevel, p ,satisfies: c • p >−1 c • dpc >0 dσ2 ν • p >0ifσ−2 > φ2 (σ−2+σ−2)whereσ−2 =(σ2)−1 c ψ φ−1 n,ν ν x x Proof. SeeAppendixA The first result, p > −1, indicates that p fails to fall by as much as the money supply. This c shouldnotcomeasasurprise. Asawhole,agentsarenotsureifthedecreaseinthemoneysupply that they see is due to their individual noise, or due to a movement in the aggregate, i.e. they are uncertain of the magnitude of the shock. This is a standard signal extraction result. Average expectations of the change in the money supply, and hence price, will be less than the actual change in the money supply. However, there is a further element to the inference problem. Firms areunsurewhichaggregateshockhashittheeconomy. Onaverage,theyseethatthemoneysupply has fallen, but this could be due to the response of monetary policy to a positive markup shock. They therefore put some weight on this possibility, a situation I shall call confusion. This places upwardpressureonprices. 5

This mechanism relies on agents using their signal of the central bank’s action to update their beliefs. Thiswouldimplythatthecentralbankhassomeseparateinformationthatthepublicdoes not posses. This information structure finds empirical support in the study of Romer and Romer (2000), who conclude that “the Federal Reserve has considerable information about inflation beyondwhatisknowntocommercialforecasters”. If the variance of the markup shock is increased then firms will attribute more of what they see to a markup shock. Therefore firms will set a higher price than they otherwise would, which explains why dpc > 0. Indeed, the confusion about which shock has occurred can be so severe dσ2 ν that prices may go up after a contractionary monetary shock. This price puzzle result occurs if σ−2 > φ2 (σ−2+σ−2),i.e. iftherelativevarianceofthemonetaryshocksissufficientlysmall. ψ φ−1 n,ν ν 3 A New Keynesian Model The model above abstracts from many different issues. For example, central banks rarely use money supply as a policy instrument. Also, in the model firms get signals of exogenous variables, m and ν, and not endogenous variables such as prices, p. I address these and other issues in the moresophisticatedmodelthatfollows. 3.1 The Model The core components of this model follow a standard New Keynesian model, in the tradition of Woodford (2003b) and Clarida et al. (1999). The point of departure is that each agent has an idiosyncraticinformationset. 3.1.1 PrivateAgents There are a continuum of private agents indexed by j∈[0,1]. In what follows, T andt index time periods. Each agent in the economy can be thought of as a yeoman farmer, producing a unique 6

good,alsoindexedby j. Theonlyinputintoproductionislaborwithfunctionalform: γ Y =L , 0<γ≤1 j,T j,T whereY is the amount of good j produced and L is the amount of the agent’s labor used to j,T j,T produceit. TheythensellY forpriceP . j,T j,T Each period the agent consumes a finite number of goods, N. The identity of these goods is chosen randomly from the continuum in the economy. This is intended to limit the amount of information that the agent can infer from their consumption basket about the overall price level.2 The per period utility function of agent j is a linearly separable function of a consumption index andlabor: C1−σ j,T U = −L j,T j,T 1−σ where: ! 1 N 1 θj,T C = ∑ C θj,T j,T N k,j,T k=1 and{C ,...C }isconsumptionoftheN goodsthatagent jcanconsume. θ isapreference 1,j,T N,j,T j,T parameter that determines the elasticity of substitution between goods. There is a random, persistent,aggregatecomponenttoθ alongwithatransitionary,idiosyncraticcomponentfollowing: j,T lnθ =lnθ +µ , µ ∼N(0,σ2 ) j,T T j,θ,T j,θ,T µ,θ lnθ −lnθ =ρ (lnθ −lnθ )+ν , ν ∼N(0,σ2) T ss θ T−1 ss θ,T θ,T θ Arandomelasticityofsubstitutionbetweenproductsservesthesamepurposeasthemarkupshock in Section 2. A low elasticity of substitution (low θ) will lead to higher markups because the elasticity of demand will be low. Therefore shocks to θ will be referred to as markup shocks. 2Asfarastheauthorisaware,thismethodofdecouplingtheconsumptionbasketfromtheoveralleconomyhasnot beenusedbefore. Lorenzoni(2008)achievedasimilarendbyassumingthatagentsconsumedacontinuumofgoods, butthattheshockaffectingthosegoodswascorrelatedforeachagent. 7

There are other ways to generate markup shocks in these models. For example, in a model with a labormarket,employeescouldbargainoverwagessobargainingshockscouldbeintroduced. That pathwasn’tfollowedbecauseitintroducesanothermarket,addingcomplexitytothemodel. The only asset in the economy is a nominal one-period bond, so the budget constraint at time T is: N ∑C P +Q B ≤B +P Y (1) k,j,T k,j,T T j,T j,T−1 j,T j,T k=1 where P is the price of the kth good in agent j’s consumption basket, Q is the price of the k,j,T T nominalbond,andB isthenumberofbondspurchased. Ponzischemesareruledout: T lim E Q B ≥0 (2) j,t tT j,T T→∞ forallpointsalongaplannedconsumptionpath,whereQ tT =∏ T i=t Q i .3 AppendixBshowsthatthisimplies: (cid:18) (cid:19) 1 P k,j,T θj,T−1 C = C (3) k,j,T j,T P j,T and: N ∑C P =NP C (4) k,j,T k,j,T j,T j,T k=1 withpriceindex: θj,T−1 N 1 θj,T ! θj,T P = ∑ P θj,T−1 (5) j,T N k,j,T k=1 TheagentwillselltoN otheragentswhicharerandomlyallocatedeachtimeperiod. Thereforethe 3AsnotedinSims(2004c),whenassetmarketsareincomplete,thisconditionseemstoimplythatrunningariskthat somethingwillhappenthatmakesdebtbecomelargeisacceptableaslongasitisoffsetbythepossibilitythatassets willbebuiltupinotherstatesoftheworld. Suchasituationcouldoccurifthecreditor,inthiscasethegovernment, couldexamineagents’accountstoensurethattheyareabidingbythisconstraint, andanypreviouscommitmentsto accumulate assets. The government would not need to do this for every agent in every period. It could do so with someprobability,inflictingaseverepunishmentiftheagentwerenotdoingso. 8

demandforagent j’sproductis: N (cid:18) P j,T (cid:19) θk,T 1 −1 Y = ∑ C (6) j,T j,k,T P k=1 j,k,T where C , and P are the consumption and price indices of the k’th consumer in agent j’s j,k,T j,k,T customerbaseforthatperiod. Prices are sticky using the Calvo (1983) specification. Specifically, there is a probability equal to α that the agent will have the opportunity to reset their price in any given period. The discount factorisβ. At the end of each time period, the agent observes their own θ ; how much they sell, Y ; j,T j,T the prices in their consumption basket, {P ,...,P }; and the price of the nominal bond, Q . 1,j,T N,j,T T Agents get to see these only after they have made the decision on P and C .4 Therefore the j,T j,T informationusedwhenmakingdecisionsonP andC is: j,T j,T I ={θ ,Y ,{P ,...,P },Q ,I } (7) j,T j,T−1 j,T−1 1,j,T−1 N,j,T−1 T−1 j,T−1 However, when deciding on C the agent has information on the prices of the goods in their k,j,T consumption basket. This arrangement could be thought of as having a consumer and a producer in the household. At the beginning of the period they decide upon P and C , and then the j,T j,T consumergoesoutintothemarketplacetodecidehowbesttoachieveC .5 j,T As a result, using the production function, the agent’s problem can be written as choosing a planforC andP undereachstateoftheworldthatmaximizes: j,T j,T ∞ " C1−σ # ! E t ∑ βT j,T −(Y j,T ) 1 γ |I j,t (8) 1−σ T=t 4The results are not sensitive to this assumption, but there is a timing issue of it is not made. If agents a and b aresettingtheirpricessimultaneously,andagentaconsumesagentb’sproduct,itisthenunclearhowagent’sb’snew pricecouldbeina’sinformationset. 5AnalternativeistoassumethattheconsumeralsodecidesuponC sothedecisiononC ismadewithmore j,T j,T informationthanthedecisiononP . j,T 9

subjecttoEquations1,2,6,and7withstickypricesa` laCalvo(1983),wherethehazardisα. Agents are rational, so that the underlying probability density functions used to calculate expectationsarethoseimpliedbythemodel. 3.1.2 TheCentralBank TheeasiestandmostconventionalcharacterizationofmonetarypolicyisaTaylorRule: i =ρi +(1−ρ)(φ π +φ y )+ν , ν ∼N(0,σ2) (9) T T−1 π T y T i,T i,T i where i is the nominal interest rate (which sets the price of the nominal bond), π is the inflation t t rate, y is the output gap and ν is a monetary policy shock. The output gap is defined to be the t i,t logdeviationofaggregateoutputfromthesteadystatelevel: y =ln (cid:18)Z 1 YθTdj (cid:19) θ 1 T ! −lnY T j,T ss 0 where: Y = (cid:18)Z 1 Yθssdj (cid:19) θ 1 ss , Y =Thesteadystatelevelofindividual j0soutput ss j,ss j,ss 0 Inflationisdefinedastherateofchangeoftheaggregateidealizedpriceindex:6  !θT−1  θT θT  R1PθT−1dj  (cid:18) (cid:19)  0 j,T  P T   π T =100 −1 =100 −1 (10) P T−1   θT−1 ! θT θT − − 1− 1 1    R1PθT−1−1dj  0 j,T−1 The addition of the lagged interest rate on the right hand side of Equation 9 reflects the large 6The variability of θ will not affect the level of output or the inflation rate directly because equations are log T linearizedaroundthesymmetric,perfectinformation,zeroinflationsteadystatewhereallpricesandoutputlevelsare thesame. Thatis,atthosepoints dPT = dyT =0. dθT dθT 10

and significant value for ρ found in many studies, for example see Clarida et al. (1998). This dependenceuponpastratesisusuallyinterpretedas“interestratesmoothing”. 3.2 Equilibrium Inowdefineequilibrium: Definition. An equilibrium is a set of beliefs for agents (for any variable z, a pdf f (:|:)), prices z (∀j,T, P ), allocations for consumers (∀j,T, {C ,...,C }), and government policies (i ) j,T 1,j,T N,j,T T suchthat: 1. f (:|:)arerational. z 2. EachagentisrandomlyassignedN consumptiongoodsandN customers. 3. Pricesandallocationssatisfytheagent’sproblem,i.e. Equation8. 4. i =ρi +(1−ρ)(φ π +φ y )+ν . T T−1 π T y T i,T The model is log linearized around a perfect information, zero inflation steady state (AppendixC). ThesolutionmethodisdescribedinAppendixD. 3.3 Numerical Illustration I begin my exposition of this model with the parameter values specified in Tables 1 and 2. I will arguethattheseparametervaluesareconservative,inthesensethatparametervaluesthatleadtoa largerpricepuzzleandamoredelayeddisinflationcanbejustified. Table1coversparametervalues that would affect the economy’s reaction to a monetary shock even if information were perfect. A time period is specified to be a quarter, so the Calvo hazard parameter of 0.25 implies an average price duration of a year. This accords with some older estimates, see for example Blinder et al. (1998), but is slightly longer than some more recent work. For example, Nakamura and Steinsson (Forthcoming) find a median duration for prices of 8-11 months. The intertemporal elasticity of 11

Table1: TraditionalParameters Parameter Symbol Value Calvohazardparameter α 0.25 Int. elasticityofsubstitution σ 1 TaylorRuleinflationcoeff. φ 1.5 π TaylorRuleoutputgapcoeff. φ 0.125 y Interestratesmoothing ρ 0.7 DiscountFactor β 1 Steadystateθ θ 3 ss 4 ElasticityofY w.r.tL γ 0.9 substitutionis1,followingtherealbusinesscycleliterature(Woodford(2003b),p. 165). Ibasethe coefficients of the Taylor Rule on well known estimates (eg. Clarida et al. (2000)). The discount factor,β,isequalto1.7 Thiseliminatessomewealtheffectsthatwouldotherwiserenderthemodel difficulttosolve,asdescribedinAppendixC. Theparametersmentionedthusfardonotaffecttheresultssignificantlyifvariedoverplausible ranges. That is not true for the last two parameters in Table 1. The steady state value of θ is 3, 4 which corresponds to an elasticity of substitution between goods of 4. This is on the low side of existing calibrations, see for example Chari et al. (2000), but it is consistent with the recent empirical work of Broda and Weinstein (2006). The elasticity of output with respect to the labor input,γ, is0.9,so themodel’s laborshareof incomeis 0.6.8 Aspointed outin Woodford(2003b), strategiccomplementarityinpricing,i.e. thesituationwhereanagentwishestoincreasetheirprice inresponsetoanincreaseinthepriceofothers,isincreasinginθ anddecreasinginγ. Therefore ss my values correspond to low levels of strategic complementarity relative to other work, see again Charietal.(2000)andalsoRotembergandWoodford(1997). Thisisconservativeinthesensethat strongerstrategiccomplementarityleadstoalargerpricepuzzleandalongerdelayindisinflation. ThisisdiscussedinSection3.5. 7Toavoidissuesofinfinitevaluefunctions,thelimitofagents’behaviorasβ→1istaken. SeeAppendixC. 8Thereisnolabormarketinthismodel,butahouseholdcanbethoughttobemadeupofalaborerandamanager, withthelaborerpaidawageequaltotheirmarginalproduct. Thesteadystateratioofmarginalcosttopriceis θss− N 1 . 1−1 (cid:18) (cid:19) N 3−1 Table2specifiesN=4. Thereforethelaborshareofincomeis0.9 4 4 =0.6. 1−1 4 12

Table2: Non-TraditionalParameters Parameter Symbol Value Markupshockvariance σ2 1 θ θautocorr. ρ 0.95 θ Policyshockvariance σ2 0.07 i Varianceofindividualθ σ2 7 µ,θ Numberofgoodsconsumed N 4 Table 2 specifies parameter values that would not affect the economy’s reaction to a monetary shockifinformationwereperfect.9 Icannormalizethevarianceofthemarkupshockto1because ratios of variances determine agents’ signal extraction calculations. The autocorrelation of θ is 0.95. I base this on Smets and Wouters (2007). This study estimated a structural model of the US economyusingBayesianmethodsandfoundthatmarkupshocksareextremelypersistent,withan autocorrelationofatleast0.9.10 I choose the final three parameters, namely the variance of the monetary shock, the number of goods in the consumption basket, and the variance of the idiosyncratic θ shock, to give the price puzzle and a delayed disinflation. I shall discuss these parameters in Section 3.4. The resulting impulseresponseofinflationtoamonetaryshockisshowninFigure1. Bydesign,theinflationrate initially increases. This increase results from the same forces discussed in Section 2. Agents see the interest rate increasing and conclude that this is partly because the central bank is responding toapositivemarkupshock. Agentsthereforeincreasetheirprice. The period of peak disinflation occurs 4 quarters after the shock. This hump is generated by the sameforces generating theprice puzzle. Agents arenot sure which shock has hit theeconomy and as a result they do not cut prices initially. As time passes they gather more information and form a better picture of the shock. They then act upon the fall in demand and begin to cut their 9This is not strictly true of N, but over the parameter values I consider the dominant effect of N is through its determinationofagents’signalprecision. 10Notethatkeepingthevarianceofθ fixed, i.e. keeping σ2 θ constant, ahigherρ meanstheroleofconfusion T 1−ρ2 θ θ isreduced. Increasingρ meansσ2 hastodecrease,soanyshortrunvariationswillthenbemorelikelyascribedtoa θ θ monetaryshockthanamarkupshock. Thereforethisparameterisnotdrivinganyoftheresults. 13

Figure1: ContractionaryMonetaryShockImpulseResponse(ν =1) i 0.06 0.04 0.02 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 -0.02 -0.04 -0.06 noitalfnI prices. ThissourceofthehumpisdifferenttothesourceinWoodford(2003a)andotherimperfect information work, for example Nimark (2008). That vein of literature produces a hump through a high autocorrelation of the shock and inertia in higher order expectations–beliefs of what other agentsbelieve. Inthatframework,whentheshockoccursagentsformanopinionabouttheshock, but they believe other agents are less perceptive of it. Strategic complementarity then dictates that priceswillnotmovequickly. However,astimepasseshigherorderexpectationsadjustandagents begin to change their price more aggressively. This delayed response is aided by the autocorrelation of the shock. In contrast, the mechanism I emphasize has no reliance on these higher order expectationsorautocorrelationofthemonetaryshock. There is also another period of inflation beginning 9 quarters after the shock. This is part of some oscillatory behavior that dies down eventually. It appears to be driven by beliefs. At that pointagentsarestillupdatingtheirbeliefsofwhathappenedinthepast,butalsoofwhathappened relativelyrecently. Intheperiodsleadinguptothe9quartermark,agentsarerevisingtheirbeliefs ofrecentmarkupshocksupward. 14

Table3: VarianceDecomposition: ForecastErrorDuetoMonetaryShocks 4periods 8periods 16periods 51% 22% 17% (36,71) (11,38) (6,35) Notes: These are the results from variance decompositions estimated from 10,000 vector autoregressions using simulated data that was 100 periods in length. The variables used were the output gap, inflation and the interest rate. Identification of the shocks was performed recursively, ordered: 1) inflation, 2) output gap, and 3) interest rate. Thisidentifiesmonetaryshocksbecause,inthemodel,thecurrentmonetaryshockhasnoeffectontheother variables. Theaverageoftheforecasterrorduetointerestrateshocksoverthe10,000simulationsisreported,and thebracketsgivethe5thand95thpercentiles. 3.4 The Final Three Parameters InowfocusattentionontheparametersIchoseinordertogeneratethepatternsinFigure1,namely the variance of the monetary shock, σ2; the number of goods in the consumption basket, N; and i the variance of the idiosyncratic θ shock, σ2 . While in some ways, estimating these parameters µ,θ wouldhavebeendesirable,computationalissuesrenderedthisimpractical. Section 2 demonstrated that the relative variance of the two shocks influences the price level. Table 2 specifies σ2 = 0.07. I conduct a variance decomposition from simulated data as a guide i to determine how this relates to the real world importance of this shock. The results are shown in Table3. In the model, monetary shocks account for over half of the four period ahead output forecast error. However, such a significant role for monetary shocks has little empirical support, see for exampleKim(1999)andKimandRoubini(2000). Thesestudiesconcludedthatmonetaryshocks are only responsible for 10-20 per cent of forecast errors. In the estimated model of Smets and Wouters (2007), the contribution of monetary shocks was either on the low side of this 10-20 per cent range or below it at all horizons. While the numbers from Table 3 are in the right ballpark in thelongerrun,itistheshorthorizonfiguresheavilyinfluencelearninginthefirstfewperiodsafter the shock, and so should be emphasized. Therefore even lower values of σ2 can be justified, so in i thissenseσ2=0.07isconservative. However,themodelhasonlytwoshocks. Ifothershockswere i 15

Table4: MeanSquareErrorofOutputGapForecastoverOutputGapVariance Data Model 0.186 0.0026 Notes: “Data”iscalculatedfromthemeansquareerroroftheaverageforecastofcurrentGDPfromtheFederal Reserve Bank of Philadelphia’s Survey of Professional Forecasters from 1990 Quarter 1 to 2007 Quarter 1. To avoid issues such as changing definitions, etc, the first revision numbers were used to determine this error. This iscommonintheliteratureonevaluatingforecasts(forexampleseeZarnowitzandBraun(1993)andBaueretal. (2003)). TheoutputgapwascalculatedusingtheCongressionalBudgetOffice’sestimatesofpotentialoutput. added, then to keep the numbers in Table 3 stable, the variance of the markup shocks would have to be reduced. The rejoinder to this point is that most other major shocks that would be replacing themarkupshockwouldinducethesamesortofresponse. Takeforexampleanaggregatedemand shock. If added to the model, then when interest rates increase agents will attribute this partly to rising demand, which will place upward pressure on their price. Specifications that included demand shocks have been experimented with and the qualitative features of the results are not changed. The other two parameters left to consider are σ2 = 7 and N = 4, i.e. the variance of the µ,θ individual shock to θ and the number of goods consumed by each agent. These determine how well agents perceive the state of the economy because they govern the precision of their signals. While errors of beliefs regarding the actual shocks can’t be measured, errors of beliefs regarding aggregates can from the Survey of Professional Forecasters. This survey includes a question on respondents’expectationsofGDPforthecurrentquarter. Modelsofperfectinformationimplythat agents should know this exactly. Table 4 shows that they do not. The statistic used to show this is the ratio of the mean square error of the average forecast of the output gap to its variance. This is avariablethatisnegativelyrelatedtohowwellagentsperceivecurrentconditions. Thisstatisticis 0.186 in the data, which is more than a factor of 70 greater than the model implies. Therefore, the resultsarenotdependentuponassuminganunrealisticdegreeofuncertainty. Inthissenseσ2 =7 µ,θ andN =4areconservative. 16

Figure2: IncreasingStrategicComplementarity θ = 6 andγ=0.8 ss 7 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 -0.1 noitalfnI 3.5 Strategic Complementarity The previous section made it clear that the degree of error in agents’ beliefs is small. However, theconsequencesarelarge. Apricepuzzleisproducedandthereisadelayedresponseofinflation that would otherwise not occur. A reason for the significant consequences of a small amount of uncertainty is the strategic complementarity in pricing. Intuitively, strategic complementarity acts as a multiplier. If an agent thinks that a particular disturbance has pushed other agents’ prices higher, more strategic complementarity induces this agent to set a higher price themselves. This multiplier role of strategic complementarity is familiar, see for example Cooper and John (1988). Inmymodel,themultiplicativeeffectislargedespitethelowerlevelofstrategiccomplementarity comparedtootherparameterizationsintheliterature. Ifparametersarealteredinordertoresemble thisotherwork,withθ = 6,andγ=0.8,11 thenFigure2showsthatthepricepuzzleandthedelay ss 7 11These are approximately the same as the Rotemberg and Woodford (1997) estimates. Chari et al. (2000) use parametersthat,inthissetting,wouldimplyevenmorestrategiccomplementarity. 17

becomeevenmorepronounced.12 3.6 Comparative Statics I will not document the sensitivity of the results to changes in all of the parameters from Section 3.3. I shall instead focus on the most interesting cases. These are σ2 , N and σ2 . Variations i µ,θ in the remaining paramters over plausible values have little effect on the results. Figure 3 shows thatincreasingthevarianceoftheinterestrateshockandincreasingtheprecisionofthesignalsreducesthepricepuzzleandbringsthepeakeffectoninflationforward. Theleftpanelshowsvalues for which the price puzzle is eliminated. The right panel shows the points at which the hump is eliminated. Increasing σ2 from 0.07 to 0.09 eliminates the price puzzle. This increase pushes the four i period ahead forecast error due to monetary shocks up from 51% to 54%. Increasing the variance furtherto0.13eliminatesthehump. Inthiscasetheforecasterrorduetomonetaryshocksis60%. Increasing N and decreasing σ2 in integer steps eliminates the price puzzle at N = 5 and µ,θ σ2 =3. TheseparameterscorrespondtoratioscalculatedinTable4of0.0025and0.0023respec- µ,θ tively. Continuing this process means the hump is eliminated when N =12 and σ2 =1. These µ,θ parameterscorrespondtoaratiocalculatedinTable4of0.0018and0.0015respectively. Itcanthenbeconcludedthatthepresenceofthepricepuzzleisquitesensitivetotheparameters chosen. However,thehumpismorerobust. 4 Is Price Flexibility Stabilizing? The degree of price stickiness in the United States has recently been subject to some debate. Bils and Klenow (2004) stimulated this debate by looking at data provided by the Bureau of Labor Statistics, arguing that the median length of price duration was a surprisingly short 4.3 months. 12LinEquation20isafunctionofθ ,γandNwhichareallchangedinthissectionandinSection3.6.Thischanges ss thevarianceofLθ˜ ,whichgiventheresultsfromSection2,willaffecttheimpulseresponsesinwaysunrelatedtothe T channels being considered. To avoid this effect clouding the results, σ is normalized so as to keep the variance of θ Lθ˜ constant. T 18

Figure3: Robustness PricePuzzleDisappears PeakEffectinFirstPeriod σ2 =0.09 σ2 =0.13 i i 0 0 2 4 6 8 10 12 14 16 -0.02 -0.04 -0.06 noitalfnI 0 0 2 4 6 8 10 12 14 16 -0.02 -0.04 -0.06 noitalfnI N =5 N =12 0.02 0 0 2 4 6 8 10 12 14 16 -0.02 -0.04 -0.06 -0.08 noitalfnI 0 0 2 4 6 8 10 12 14 16 -0.03 -0.06 -0.09 -0.12 -0.15 noitalfnI σ2 =3 σ2 =1 µ,θ µ,θ 0 0 2 4 6 8 10 12 14 16 -0.02 -0.04 -0.06 -0.08 noitalfnI 0 0 2 4 6 8 10 12 14 16 -0.03 -0.06 -0.09 noitalfnI 19

Figure4: ChangesinPriceFlexibility 0.06 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 -0.06 -0.12 noitalfnI 6 5.5 5 0.1 0.15 0.2 0.25 0.3 0.35 α α=0.1 α=0.2 α=0.35 noitaiveD tuptuO derauqS SincethisresearchNakamuraandSteinsson(Forthcoming)havetakenadifferentlookatthedata, generating longer estimates. This subject has received so much attention because more stickiness traditionally leads to more monetary non-neutrality. However, models have been constructed where this is not necessarily the case. For example, in a Keynesian model, DeLong and Summers (1986) showed that increased price flexibility can be destabilizing due to a “Mundell” effect. In their model increased price flexibility implied future prices were expected to fall further after a contractionary shock. The dependence of the forward looking IS curve on the real interest rate thendictatedthatagentswouldreducetheirdemandfurther,exacerbatingtheshock. Similarlyinthis model,increasedpriceflexibilitycan bedestabilizingafteramonetaryshock, butfordifferentreasons. Figure4showsthat,withthebaselineparametersfromSection3.3,output stabilityisincreasinginpriceflexibilityuptothepointofα=0.2. Thereasonissimple,increased priceflexibilitymeansthattheinitialpricepuzzleismorepronouncedandthisdestabilizesoutput. For price flexibility greater than α=0.2, the initial price puzzle is mitigated when α is increased. This is because a different influence dominates pricing decisions arising from the dynamic nature ofpricesetting. Pricesinthefuturewillbelowerifpriceflexibilityisincreased. Therefore,agents’ newpriceswillfallimmediatelyaftertheshock,becausetheymaybestuckwiththatpriceforsome time. 20

It should be noted however, that the price puzzle is not a necessary precondition for a positive relationshipbetweenpriceflexibilityandoutput. Thereareotherparameterizationswhereaninitial fallinpricesistemperedwithgreaterpriceflexibility,whichinturnleadstomoreoutputinstability. The dominant force driving this comparative static is that flexibility leads to higher prices after a positive markup shock. In equilibrium, this means that more of the variability of prices and therefore interest rates is driven by markup shocks. This leads agents to place more weight on theirmarkupshockbeliefsafteramonetaryshock,whichactsasaforcetemperingtheinitialprice decrease. 5 Conclusion This paper presented models where agents mistook a contractionary monetary shock for the endogenous response of monetary policy. As a result, they initially increase their own price, and by more if strategic complementarity in pricing is larger. However, the models illustrate a much broader point. Imperfect information models require a complete description of the important shocks that buffet the economy in order to get accurate model predictions. Agents don’t know whichshockhashittheeconomyandthereforewillattributesomeofwhattheyseetoallpossible disturbances. Thiscanhavelargeeffectsandservesasanoteofcautionforfurtherwork. The paper has shown that the mechanism that generates the price puzzle can also generate a hump in inflation’s response to a monetary shock. Hump-shaped impulse responses turn up in many places in macroeconomics, presenting theoretical challenges. Indeed, the hump-shaped impulse response is what motivated Woodford (2003a). The mechanism creating the hump in Figure3isqualitativelydistinctfromanythattheauthorhasseenbefore. WhileWoodford(2003a) had a simple model that shared some features in common with the model employed here, the explanationforthehumpisdifferent. The New Keynesian model developed could be enhanced further in order to investigate other challengespresentedbytheempiricalliterature. Apotentialfindingofinterestisthehump-shaped 21

response of output to monetary shocks. While the model as it stands cannot replicate this, there aremodificationsthatcan. Forexample,iftheinterestrateshockisseriallycorrelated,thenhumpshapedresponsesarepossible. 22

A Proof of Claim 1 Let f(Y|Z)denotetheprobabilitydensityfunctionofY givenZ. Itiseasytoshowthat f(X|H )is j symmetricaroundX =(B0Σ−1B+Σ−1)−1B0Σ−1H ,so n n j E (X)=(B0Σ−1B+Σ−1)−1B0Σ−1H (11) j n n j Thereforeintegratingoverallindividualprices, p =E m+E ν,gives: j j j Z 1 ⇒ p= p =[ 1 1 ]B(B0Σ−1B+Σ−1)−1B0Σ−1BX j n n 0 As a result p is unique, so the equilibrium is unique. Computing this expression after a monetary shockgives: ! (φ−1)σ−2σ−2 p =−D σ−2(σ−2+σ−2)− n,m ψ (12) c n,m n,ν ν φ2 where: 1 D= >0 ( σ− n, 2 m +σ−2+σ−2)(σ−2)+(σ−2+σ−2)(σ−2) φ2 n,ν ν ψ n,ν ν n,m Therefore p >−1,and p >0ifσ−2 > φ2 (σ−2+σ−2). DifferentiatingEquation12gives: c c ψ 1−φ n,ν ν σ−2 σ− n, 2 mσ−2+(σ−2+σ−2) (φ−1)σ− n,m 2σ− ψ 2 dp c n,m φ2 ψ ψ n,m φ2 dp c =− <0⇒ >0 dσ− ν 2 (( σ φ − n 2 , 2 m +σ− n, 2 ν +σ− ν 2)(σ− ψ 2)+(σ− n, 2 ν +σ− ν 2)(σ− n, 2 m ))2 dσ2 ν B Derivation of Equation 3 IfB istakenasgiventhentheLagrangianfortheproblemis: j,T C1−σ N ! j,T L = −L +λ ∑C P +Q B ≤B +P Y j,T j,T k,j,T k,j,T t j,T j,T−1 j,T j,T 1−σ k=1 DifferentiatingwithrespecttoC gives: k,j,T 1 C θj,T−1 C 1−θj,T−σ +λP N k,j,T j,T l,j,T (cid:18) C (cid:19)θj,T−1 P k,j,T k,j,T ⇒ ∀k,l = C P l,j,T l,j,T N 1 N θj,T ⇒ ∑C P =C P 1−θj,T ∑P θj,T−1 (13) k,j,T k,j,T l,j,T l,j,T k,j,T k=1 k=1 23

and: N N ∑C P =C 1−θj,TP ∑C θj,T (14) k,j,T k,j,T l,j,T l,j,T k,j,T k=1 k=1 DividingEquation13byEquation14andrearranginggives: (cid:18) (cid:19) 1 P l,j,T θj,T−1 C = C l,j,T j,T P j,T SubstitutingintoEquation14thengivesEquation4. C Log-linearized first order conditions for the private agent SubstitutinginEquation6andEquation4,theLagrangianis:13 L =E j,t ∑ ∞ βT−t   C j 1 , − T σ − ∑ N (cid:18) P j,T (cid:19) θk,T 1 −1 C j,k,T !1 γ   1−σ P T=t k=1 j,k,T +λ NP C +Q B −B +P ∑ N (cid:18) P j,t−1 (cid:19) θk,t− 1 1−1 C ! j,t j,t−1 j,t−1 t−1 j,t−1 j,t−2 j,t−1 j,k,t−1 P k=1 j,k,t−1 +E ∑ ∞ βT−t+1λ (cid:18) NP C +Q B −B +P ∑ N (cid:18) P j,T (cid:19) θk,T 1 −1 C (cid:19) j,t j,T+1 j,T j,T T j,T j,T−1 j,T j,k,T P T=t k=1 j,k,T ThefirstorderconditionwithrespecttoC yields: j,t C−σ =−NβE λ P j,t j,t j,t+1 j,t Loglinearizingthisaroundthesteadystateofperfectinformationwithnoshocksgives: 1 ˜ c =− E (λ +p ) (15) j,t j,t j,t+1 t σ where: c isthelogdeviationofC fromsteadystate(thatthereisasteadystateisverifiedlater), j,t j,t λ ˜ =lnλ and p = R1p dj, p =lnP .14 j,s j,s t 0 j,t j,t j,t ThefirstorderconditionwithrespecttoB yields: j,t−1 λ Q =βE λ j,t t−1 j,t j,t+1 Rearrangingandloglinearizingaroundthesteadystategives: ˜ ˜ i =E (λ −λ ) t j,t+1 j,t+1 j,t+2 13The method for solving this problem is taken from Kushner (1965), Sims (2004a), and Sims (2004b). In the notationofSims(2004a)(P ,C ,B )=C¯. j,t j,t j,t−1 t 14Everythingisobservedwithaoneperiodlag,sotheexpectationofthepricelevelfacedbytheagenttodayisthe expectationoftheaggregatepricelevel. 24

where i is the negative of the log deviation of Q from steady state, i.e. the nominal interest t t rate. Substituting this into Equation 15 and then substituting in Equation 15 that has been iterated forwardgives: 1 c =E (c − (i −π )) (16) j,t j,t j,t t t+1 σ whereπ = p −p . t+1 t+1 t ThefirstorderconditionwithrespecttoP givesthefollowing: j,t   2−θk,T 1  ∞ (β(1−α))T−t  N  P j θ , k t ,T−1 P j θ , k t ,T−1 dP j,k,T −E t|s∈Ω T ∑ =t γ (cid:0) θ k,T −1 (cid:1)   k ∑ =1   P j θ , k k , , T 1 T −1 C j,k,T − P θk θ , k T ,T −1 C j,k,T dP j,t     j,k,T N (cid:18) P j,t (cid:19) θk,T 1 −1 !1 γ −1 ∞ N (β(1−α))T−t+1λ j,T+1 C j,k,T ∑ C =E ∑ ∑ P j,k,T t|s∈Ω (1−α)(θ −1) k=1 j,k,T T=tk=1 k,T   (cid:18) (cid:19) 1 (cid:18) (cid:19) θk,T P j,t θk,T−1 P j,t θk,T−1 dP k,T θ k,T −  (17) P P dP j,k,T j,k,T j,t where Ω denotes states of the world where the agent has not had the opportunity to change their price. Eachtimeperiodreceivesweight(1−α)T−t becauseagentsgettochangepriceeachperiod withprobabilityα. Equation5implies: (cid:18) (cid:19) 1 dP j,k,T 1 P j,t θk,T−1 = dP N P j,t j,k,T SubstitutingthisintoEquation17andmultiplyingthroughbyP inordertogetanexpressionthat j,t hasasteadystate,andthenloglinearizinggives: E " ∑ ∞ (β(1−α))T−t ∑ N 1 (cid:18)(cid:18) −θ ss θ˜ k,T (cid:19) + c j,k,t + (1 γ −1)(N−1)+N−(θ ss −1) ! t|s∈Ω N θ −1 γ (N−1)(θ −1) T=t k=1 ss ss !# " (cid:0) p∗ −p (cid:1) (cid:19) =E ∑ ∞ (β(1−α))T−t ∑ N 1 λ ˜ +c +p∗ + −θ ss θ˜ j,t j,k,T t|s∈Ω N j,T+1 j,k,T j,t θ −1 k,T T=t k=1 ss !# (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19) + N θ θ˜ + 1 (cid:0) p∗ −p (cid:1) − 1 θ ss +1(cid:0) p∗ −p (cid:1) (18) Nθ −1 ss k,T θ −1 j,t j,k,T N θ −1 j,t j,k,T ss ss ss where θ˜ is the log deviation of θ from steady state, c is the log deviation of consumption of j,k,t j’s kth customer, p is the log of the price index of j’s kth customer and p∗ is the log of j’s j,k,T j,t optimalprice. 25

P p∗ −p =ln j,t j,t j,k,T θk,T−1 θk,T ! θk,T ∑ N 1P θk,T−1 l=1 N l,k,j,T where P is the price of the lth good of j’s kth’s consumer’s consumption basket. It must be l,k,j,T thatP =P foranl. Withoutlossofgeneralitysupposeitisforl =N,so: l,k,j,T j,t   p∗ j,t −p j,k,T =− θ k θ ,T −1 ln N 1 + N ∑ −1 N 1 (cid:18) P l P ,k,j,T (cid:19) θk θ , k T ,T −1 ≈ N 1 N ∑ −1 (cid:0) p∗ j,t −p l,k,j,T (cid:1) (19) k,T l=1 j,t l=1 FromthispointIlookatthesolutionasβ→1. Thisimpliesthewealtheffectarisingfromresetting prices approaches zero. Therefore consumption becomes independent of whether the agent has had the opportunity to reset prices, i.e. c¯ =0 so there is a steady state. All aggregates are also j,t independentofwhethertheagenthasresetprices. Therefore,substitutingEquations15and19into Equation 18 and recognizing that c , p and θ˜ are i.i.d draws from the population j,k,T,s l,k,j,T,s j,k,T,s yields: ∞ p∗ =αE ∑ (1−α) T−t(cid:0) p +Kc +Lθ˜ +Mc (cid:1) j,t t T T T j,T T=t where: 1−1 γ K = D Nθ ss L= D(Nθ −1) ss −σ M = D (cid:16) (cid:17)  1−1 (N−1)+N−θ −1 (cid:18) (cid:19) γ ss (Nθ ss −θ ss −1)(N−1) D= − −1 N(θ −1) (Nθ −1)(θ −1)(N) ss ss ss Ensuringsupplyequalsdemandgives: Z 1 Z 1 c = c dj= y =y T j,T j,T T 0 0 ∞ ⇒ p∗ =αE ∑ (1−α) T−t(cid:0) p +Ky +Lθ˜ +Mc (cid:1) (20) j,t t T T T j,T T=t 26

D Solution Method D.1 The initial guess of impulse responses Denote the initial guess for the impulse response as M, a (n+1,8) matrix.15 Each column is the impulse response of a variable for the first n+1 periods to one of the shocks. It is assumed that the impulse response is flat after n+1 periods. Let the first two columns of M be the response of output (which equals consumption) to a θ˜ shock and an interest rate shock. Let the following 2 columns be the response of price, then the next 2 be the response of interest rates and the next 2 theresponseofthepreferenceparameterθ˜ =lnθ −lnθ . T ss D.2 The state space Thetruestatespaceoftheeconomyispostulatedtobe:   ν ν θ,t i,t X˜ t =  ν θ,t−1 ν i,t−1   . . . . . . If it is assumed that impulse responses are flat after n+1 periods, then each of the shocks further inthepastthann+1hasthesameeffectontheaggregates. Thismeansthatthestatespacecanbe writtenas:   ν θ,t ν  θ,t−1   .   . .     υ  X = θ,t  (21) t  ν   i,t   ν   i,t−1   .  .  .  υ i,t Whereυ =υ +ν andthestatespacefollowsthetransitionequation: x,t x,t−1 x,t−n X =FX +ξ (22) t t−1 t whereF containsadiagonalvectorof1’sdirectlybelowthemaindiagonal,exceptat(n+1,n+2). 1’sarealsolocatedat(n+1,n+1)and(2n+2,2n+2),and: ξ0 = (cid:2) ν 0 ... ν 0 ... 0 (cid:3) t θ,t i,t wherethenon-zeroentriesareplacedatpositions. (1,1)and(1,n+2). 15n=200waschosenforthecalculationsinthispaper. 27

D.3 What is observed FromEquation7,thefollowingvariablesareusedtoupdateanagent’sbeliefsbeforemakingtheir decisions(convertingtheseobservationsintotheirloglinearcounterparts): {θ˜ ,y ,{p ,...,p },i } j,t−1 j,t−1 1,j,t−1 N,j,t−1 t−1 θ˜ isnormallydistributedaroundθ˜ ,andi isobservedperfectly. {p ,...,p }are j,t−1 t−1 t−1 1,j,t−1 N,j,t−1 N independent observations of the price level, but they are not necessarily normally distributed. This presents a problem for inference. It is overcome by assuming that for the purposes of filtering, each agent gets N independent signals of p that are normally distributed around p t−1 t−1 with a variance equal to the variance of the newly set prices, denoted σ2. These N independent p σ2 observationscanbeaveragedsothatonesignalisreceivedwithavarianceequalto p. N LoglinearizingEquation6aroundsteadystategives: 1 N (cid:18) 1 (cid:19) y = ∑ c + (p −p ) j,t−1 j,k,t−1 j,t−1 j,k,t−1 N θ −1 k=1 ss InsertingtheapproximationfromEquation19gives: !! 1 N 1 1 N−1 y = ∑ c + ∑(p −p ) j,t−1 j,k,t−1 j,t−1 l,k,j,t−1 N θ −1 N k=1 ss l=1 wherec and p areindependentdrawsfromthepopulationofconsumptionandprices. j,k,t−1 l,k,j,t−1 Therefore the signal received is as if they are receiving N separate signals, each of which is c t−1 plus N−1 signals of p t−1 . c is normally distributed around c 16 with variance σ2 and N(1−θss) j,k,t−1 t−1 c thesameassumptionismadeasaboveregardingthesignalsof p . Theseseparatesignalscanbe t−1 averagedso,forthepurposesofinference,itisasiftheagentreceivesasignalof: 1 N−1 c − p t−1 t−1 θ −1 N ss withavarianceequalto 1 multipliedbythevarianceofconsumptionintheeconomyplus( 1 )2N−1 N 1−θss N3 multipliedbythevarianceofthenewlysetpricesintheeconomy. Asaresult,ifM¯ isdefinedtobe: (cid:20) (cid:21) M(1) M(3) M(5) M(7) M¯ = M(2) M(4) M(6) M(8) whereM(n)isthenthcolumnofM,theneachagentobserves: h =BM¯0X +η j,t t−1 j,t 16Thisispostulatedatthemoment,butcanbeverifiedfromthesolutionofthemodel. 28

where:  1 − 1 N−1 0 0  θss−1 N  0 1 0 0  B=   0 0 1 0  0 0 0 1 andη istheobservationnoisewithvariance:17 t  1σ2+( 1 )2N−1σ2 0 0 0  N c 1−θss N3 p σ2 =   0 N 1σ2 p 0 0   η  0 0 0 0  0 0 0 σ2 ν,θ If this is the start of the recursion, any values of σ2 and σ2 can be assumed. If it is not, denote c p previous mean square error matrix used in the Kalman filter as Σ. In the algorithm, calculations were made for the mean beliefs of the population for the state space. Therefore the mean square errormatrixcanbecalculatedforaveragebeliefs,denotedΣ . LetE X betheaverageexpectation a a t ofthestatespaceX . Itiseasytoshow: t Σ=Σ +E((E X −E X )2) a a t j t The last term is the disagreement of the population, which forms the basis for σ2 and σ2. Denote c p thisV =Σ−Σ . IteratingEquation16forwardgives: a ! ∞ 1 c =E ∑ (i −π ) +c¯ j,t j,t T T+1 j,t σ T=t where c¯ is period t’s expected deviation of consumption from steady state as T approaches inj,t finity. This is affected by changes in permanent income. From this point I take the solution as β→1. Taking the limit rather than setting β=1 avoids issues relating to infinite value functions. Thisimpliesthatchangesthroughtimeinc¯ becomearbitrarilysmall,sowecansetc¯ =0∀j,t. j,t j,t Therefore: ∞ ∞ c =E ∑(−σ(i −(p +p ))=E ∑φX =E ΦX (23) j,t j,t t+s t+1+s t+s j,t t+s j,t t s=0 s=0 where:18 ∞ φ=−σ (cid:2) 0 1 1 0 (cid:3) M¯ +σ (cid:2) 0 1 0 0 (cid:3) M¯F, Φ=φ∑Fs s=0 Thereforethevarianceofc isgivenby: j,t σ2 =ΦVΦ0 c 17Thecovariancesarezerobecauseeachobservationisunrelatedtotheother 18Truncationsofnperiodswereusedtogetaroundissuesofinfinitesums. 29

Pricesetters’optimalbehaviorisgivenbyEquation20. Thiscanbewrittenintheform ∞ p∗ =αE ∑(1−α)s+1φ˜X =E ΨX (24) j,t j,t t+s j,t t s=0 where φ˜ = (cid:2) K 1 0 L (cid:3) +MΦ, Ψ = ∑ ∞ (1−α)s+1φ˜Fs, and K,L,M are defined as in Equas=0 tion20andΦcomesfromEquation23. Therefore: σ2 =ΨVΨ0 p D.4 Proof that the aggregate economy follows the same path as that chosen by an agent that observes no noise BeliefsaredeterminedbytheKalmanfilter19 so: E X = E X +κ(BM¯0X +η −BM¯0FE X ) j,t t j,t−1 t−1 t j,t j,t−1 t−1 ∞ = ∑(I−κBM¯0F)sκ(BM¯0X +η ) (25) t−s j,t−s s=0 whereκistheKalmangainsmatrixandI istheidentitymatrix. SubstitutingthisintoEquation23 or24gives: ∞ x =Λ∑(I−κBM¯0F)sκ(BM¯0X +η ), Λ=Φ,Ψ (26) j,t t−s j,t−s s=0 For the agent that observes no noise, η is a zero matrix for all v. If we integrate Equation 26 i,t−v overallindividualsthenη cancelsoutalso. j,t−s D.5 Calculating a new impulse response function From the result of Appendix D.4 only the stand in agent who observes no noise needs to be considered. Ifirstconsideraninterestrateshock. Fortheperiodaftertheshockthestandinagenthas seenallvariablesattheirbaselinelevelinthepast(whichwaszero)andobserves: h =BM¯0X 1 1 where X has a 1 at (n+2,1) (corresponding to the interest rate shock) and zeros everywhere 1 else. The agent then forms beliefs about the state space of the economy, E X =E FX using the 2 2 2 1 Kalman filter and makes their decisions using Equations 23 and 24. This gives the first entry for 19GivenanM theremayormaynotbelongtermeffectsonthevariables. Iflessthantwoofthemhavelongterm effects,thenthereislessthantwolinearlyindependentcontributionsoftheinfinitesumpartsofEquation21towhat agentssee. Ifthatisthecase,thenthemeansquareerrormatrixusedinthecalculationoftheKalmanfilterwillnot converge. Whatcanbedoneinthiscaseistoremovefromthestatespacetheseinfinitesumsandreplacethemwith thelinearcombinationofinfinitesumsthatdoesaffectthevariablesobserved(ifthereisone,ifthereisnotone,then allcanbeignored). 30

price/outputgapinthenewimpulseresponse,withaggregatepricegivenby: p =(1−α)p +αp∗ (27) t t−1 t where the last term is given by Equation 24 for the stand in agent. This follows from Appendix E. ThenewvaluefortheinterestratewillbegivenbyEquation9. Thisprocessisthenrepeatedforthenextn−1periodsandallshocks. Thisisthemappingthat takes us from M to the new impulse response M∗, and is iterated until convergence. Sometimes to preventdestabilizingoscillationspartialadjustmentisrequired. The adequacy of the state space postulated in Appendix D.2 is proven as Equations 23 and 24 arelienarfunctionsofthepostulatedstatespace. D.6 Proof that the fixed point is an equilibrium Points 2 and 4 from the definition are trivially true because they are used in the construction at each iteration. Point 1 follows from the use of the Kalman filter by the agents, and that the actual motionoftheeconomyfollowstheaggregatethattheagentschooseasaresultofthefixedpointof the iterative procedure. Point 3 follows because the new impulse responses are constructed using thesolutiontothisproblemateachiteration. E Evolution of the aggregate price level Thepriceindex,P,usedtocomputetheinflationinEquation10iscanbetransformedtoyield t θt Z θt Z Z 1 θt Pθt−1 = Pθt−1dj+ Pθt−1dj (28) t j,t j,t j∈R j∈/R 0 whereRisthesetoffirmswhogettoresettheirpriceinperiodt. θt θt R Pθt−1dj =(1−α) R1Pθt−1 dj because the agents that get to reset their prices are chosen j∈/R j,t 0 j,t−1 randomly. ThereforeEquation28becomes P t θt θ − t 1 =αe ( θt θ − t 1 )pR,t +(1−α)P t θ − t θ − 1 t 1 (29) If we approximate around the point where, p = p and θ = θ , and where information is R,t t−1 t ss perfect,thenwegetwithsomerearrangingoftheabove: Z p =αp +(1−α)p =(1−α)p + p∗ t R,t t−1 t−1 j,t j∈R 31

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