Exogenous Information, Endogenous Information and Optimal Monetary Policy

Transcription

1 Exogenous Information, Endogenous Information and Optimal Monetary Poliy Luigi Paiello Einaudi Institute for Eonomis and Finane Mirko Wiederholt Northwestern University November 2010 Abstrat Most of the analysis of optimal monetary poliy is onduted with the Calvo model. This paper studies optimal monetary poliy when the slow adjustment of the prie level is due to imperfet information by deision-makers in firms. We onsider two models: a model with exogenous dispersed information and a rational inattention model. In the model with exogenous dispersed information, omplete stabilization of the prie level is optimal after aggregate produtivity shoks but not after markup shoks. By ontrast, in the rational inattention model, omplete stabilization of the prie level is optimal both after aggregate produtivity shoks and after markup shoks. Moreover, in the model with exogenous dispersed information, there is no value from ommitment to a future monetary poliy. By ontrast, in the rational inattention model, there is value from ommitment to a future monetary poliy beause then the private setor an trust the entral bank that not paying attention to ertain variables is optimal. JEL: E3,E5,D8. Keywords: dispersed information, rational inattention, optimal monetary poliy We thank Fernando Alvarez, Andy Atkeson, Larry Christiano, Jordi Gali, Christian Hellwig, Jonathan Parker, Alessandro Pavan, Aleh Tsyvinski, Pierre-Olivier Weill and seminar partiipants at CREI, EIEF, IAE, Maryland, NBER Summer Institute 2010, Northwestern, SED 2010 and UCLA for helpful omments. 1

2 1 Introdution This paper studies optimal maroeonomi poliy when deision-makers in firms deide how muh attention they devote to aggregate onditions. It seems like a good desription of reality that deision-makers in firms have a limited amount of attention and deide how muh attention they devote to aggregate onditions. What are the impliations for the optimal ondut of monetary poliy? To the best of our knowledge, this paper is the first paper that studies this question. We model an eonomy with many firms, a representative household, and a government. Firms supply differentiated goods. These goods are produed with labor. The representative household onsumes the different goods, supplies labor, and holds nominal government bonds and money. Money demand is derived from a ash-in-advane onstraint. The eonomy is hit by aggregate produtivity shoks and markup shoks (i.e., shoks that hange the desired markup by firms). In eah period, all firms set pries for their goods. Pries respond slowly to shoks either due to exogenous dispersed information by firms or due to rational inattention by deision-makers in firms. In the rational inattention model, deision-makers in firms hoose the preision of their signals about aggregate produtivity and the desired markup, subjet to a ost of information flow. The entral bank sets the money supply in response to shoks. The entral bank aims to maximize expeted utility of the representative household. We derive optimal monetary poliy under ommitment. The main results are the following. First, in the model with exogenous dispersed information and in the rational inattention model, the optimal poliy response to aggregate produtivity shoks is to fully stabilize the prie level in response to aggregate produtivity shoks. The reason is the following. By offsetting fully the effet of aggregate produtivity shoks on the profit-maximizing prie (i.e., by inreasing the money supply in response to a positive produtivity shok so that the profit-maximizing prie does not respond to an aggregate produtivity shok), the entral bank an repliate the response of the eonomy to aggregate produtivity shoks under perfet information. Furthermore, the response of the eonomy to aggregate produtivity shoks under perfet information is effiient. Hene, this poliy is the optimal monetary poliy response to aggregate produtivity shoks. One feature of this poliy is that pries do not respond to aggregate produtivity shoks. Seond, in the model with exogenous dispersed information, it is not optimal to fully stabilize the prie level in response to markup shoks. By offsetting fully the effet of markup shoks on the profit-maximizing prie, the entral bank an in priniple repliate the 2

3 response of the eonomy to markup shoks under perfet information. However, the response of the eonomy to markup shoks under perfet information is ineffiient. In partiular, there is ineffiient onsumption variane. By offsetting only partially the effet of markup shoks on the profit-maximizing prie, the entral bank inreases ineffiient prie dispersion but redues ineffiient onsumption variane (relative to the perfet-information solution). Aepting some ineffiient prie dispersion in exhange for redued onsumption variane turns out to be the optimal monetary poliy. At the optimal monetary poliy, the profit-maximizing prie, atual pries, and the prie level respond to markup shoks. By ontrast, in the rational inattention model, it is optimal to fully stabilize the prie level in response to markup shoks. By ounterating the effet of markup shoks on the profit-maximizing prie (i.e., by reduing the money supply in response to a positive markup shok), the entral bank redues the variane of the profit-maximizing prie due to markup shoks, whih now redues both ineffiient prie dispersion and the attention that deision-makers in firms devote to markup shoks. The latter redues the response of the prie level to markup shoks and thereby redues onsumption variane due to markup shoks. Hene, the trade-off between prie dispersion and onsumption variane due to markup shoks disappears. Reduing the money supply more in response to a positive markup shok now redues both prie dispersion and onsumption variane. The optimal monetary poliy is to ounterat the effet of markup shoks on the profitmaximizing prie until the variane of the profit-maximizing prie due to the markup shok is suffiiently small so that deision-makers in firms pay no attention to markup shoks. Thus, at the optimal monetary poliy, pries do not respond to markup shoks. In summary, in the rational inattention model, the trade-off between prie dispersion and onsumption variane due to markup shoks disappears and therefore omplete prie level stability is optimal. This is important beause the trade-off between prie dispersion and onsumption variane due to markup shoks has been emphasized a lot in the literature on optimal monetary poliy in the New Keynesian model. Third, in the model with exogenous dispersed information, there is no value from ommitment to a future monetary poliy. By ontrast, in the rational inattention model, there is value from ommitment to a future monetary poliy beause then the private setor an trust the entral bank that not paying attention to ertain variables is optimal. This paper is related to four papers that also study optimal monetary poliy in models in whih prie setting firms have imperfet information. First, Ball, Mankiw and Reis (2005) study optimal 3

4 monetary poliy in the stiky information model of Mankiw and Reis (2002). The main differene between their paper and our paper is that in their paper the information struture is exogenous. In partiular, in their paper the probability with whih firms update their information sets is independent of monetary poliy. Seond, Adam (2007) studies optimal monetary poliy in a model in whih firms pay limited attention to aggregate variables. In his model the amount of attention that firms devote to aggregate variables is exogenous; whereas in the rational inattention model presented below the amount of attention that firms devote to aggregate variables is endogenous (and depends on monetary poliy). We show that this hanges optimal monetary poliy in a fundamental way. Third, Lorenzoni (2010) and Angeletos and La O (2008) study optimal monetary poliy in models with dispersed information. In Lorenzoni (2010), prie setting firms observe the history of the eonomy up to the previous period, the sum of aggregate and idiosynrati produtivity, and a noisy publi signal about aggregate produtivity. There are several differenes between his paper and our paper: (i) in his paper the noise in the private signal onerning aggregate produtivity is idiosynrati produtivity, while in our paper the noise arises from limited attention, (ii) in his paper the information struture is exogenous, while in our paper the information struture is endogenous, and (iii) in his paper the entral bank has imperfet information, while we assume that the entral bank has perfet information about the state of the eonomy. We make this assumption to derive the optimal monetary poliy response to hanges in fundamentals. Afterwards, we study whether the entral bank an also implement this optimal monetary poliy response with less information. Like in Lorenzoni (2010), agents in Angeletos and La O (2008) observe the history of the eonomy up to the previous period, the sum of aggregate and idiosynrati produtivity, and a noisy publi signal about aggregate produtivity. In addition, in Angeletos and La O (2008) agents observe noisy signals onerning endogenous variables with exogenous variane of noise. This reates an informational externality beause a stronger response of agents to their private signals makes the signals onerning endogenous variables more informative. Angeletos and La O (2008) study how this informational externality affets optimal fisal and monetary poliy. In summary, this paper is the first paper that studies optimal monetary poliy in a model in whih agents hoose the attention that they alloate to aggregate variables. This paper is also related to the literature on the soial value of publi information, for example, Morris and Shin (2002), Hellwig (2005), and Angeletos and Pavan (2007). In this literature, the 4

5 main monetary poliy question is whether the entral bank should provide information about eonomi fundamentals. We instead ask how the entral bank should set the money supply or a nominal interest rate in response to fundamentals. In addition, in the literature on the soial value of publi information the information struture (i.e., what agents observe) is typially exogenous. The rest of the paper is organized as follows. Setion 2 presents the model setup. Setion 3 speifies the objetive of the entral bank. Setion 4 states the optimal monetary poliy problem under ommitment in the model with an exogenous information struture and in the model with an endogenous information struture. Setion 5 shows that there is a monetary poliy that repliates the alloation under perfet information. Setion 6 derives the optimal monetary poliy response to aggregate produtivity shoks. Setion 7 derives the optimal monetary poliy response to markup shoks. Setion 8 ontains additional results. Setion 9 onludes. 2 Model setup The eonomy is populated by a representative household, firms, and a government. Household: The household s preferenes in period zero over sequenes of onsumption and labor supply {C t,l t } t=0 are given by E 0 " X t=0 β t à C 1 γ t 1 1 γ!# L1+ψ t, (1) 1+ψ where C t is omposite onsumption and L t is labor supply in period t. Here E 0 denotes the expetation operator onditioned on information of the household in period zero. The parameter β (0, 1) is the disount fator. The parameter γ>0 is the inverse of the intertemporal elastiity of substitution and the parameter ψ 0 is the inverse of the Frish elastiity of labor supply. Composite onsumption in period t is given by a Dixit-Stiglitz aggregator C t = à 1 I IX C i=1 1 1+Λ t i,t! 1+Λt, (2) where C i,t is onsumption of good i in period t. ThereareI different onsumption goods and the elastiity of substitution between those different onsumption goods equals (1 + 1/Λ t ) in period t. The variable Λ t will equal the desired markup by firms in period t. Therefore, we all Λ t the desired markup. We assume that the log of the desired markup follows a stationary Gaussian first-order 5

6 autoregressive proess ln (Λ t )=(1 ρ λ )ln(λ)+ρ λ ln (Λ t 1 )+ν t, (3) where the parameter Λ > 0, the parameter ρ λ [0, 1), and the innovation ν t is i.i.d.n 0,σ 2 ν.we all the innovation ν t a markup shok. We introdue the markup shok in the model as an example of a shok that has the following property: the response of the eonomy to the shok under perfet information is ineffiient. 1 We all this property of a shok the ineffiieny property. In Setion 6 we derive the optimal monetary poliy response to a markup shok. In Setion 8 we show that our results onerning the optimal monetary poliy response to a markup shok extend to other shoks that have the ineffiieny property. The flow budget onstraint of the representative household in period t reads Ã! IX M t + B t = R t 1 B t 1 + W t L t + D t T t + M t 1 P i,t 1 C i,t 1. (4) The right-hand side of the flow budget onstraint is pre-onsumption wealth in period t. HereB t 1 are the household s holdings of nominal government bonds between periods t 1 and t, R t 1 is the nominal gross interest rate on those bond holdings, W t is the nominal wage rate in period t, D t are nominal aggregate profits in period t, T t are nominal lump sum taxes in period t, andthe term in brakets are unspent nominal money balanes arried over from period t 1 to period t. Therepresentativehouseholdantransform his pre-onsumption wealth in period t into money balanes, M t, and bond holdings, B t. The purpose of holding money is to purhase goods. We assume that the representative household faes the following ash-in-advane onstraint IX P i,t C i,t = M t. (5) i=1 The representative household also faes a no-ponzi-sheme ondition. We introdue the ash-in-advane onstraint beause it allows us to explain the intuition behind our results onerning optimal monetary poliy in a very simple way. In Setion 8 we show that our results onerning optimal monetary poliy extend to a ashless eonomy à la Woodford (2003). Furthermore, reall that there are different formulations of the ash-in-advane onstraint and note that in the formulation of the ash-in-advane onstraint that we use there are no monetary fritions beause wage inome an be immediately transformed into ash and ash an be immediately spent 1 We define effiieny formally in Setion 3. i=1 6

7 on goods. We deided to abstrat from monetary fritions in our benhmark eonomy for two reasons: (i) abstrating from monetary fritions is ommon in the New Keynesian literature on optimal monetary poliy and therefore abstrating from monetary fritions failitates omparison of optimal monetary poliy in the two models that we onsider to optimal monetary poliy in the standard New Keynesian model, and (ii) we think it is useful to study in isolation the impliations of different fritions for optimal monetary poliy. In Setion 8 we onsider an extension with monetary fritions and there we study how monetary fritions affet optimal monetary poliy in the two models that we onsider. In every period, the representative household hooses a onsumption vetor, labor supply, nominal money balanes, and nominal bond holdings. The representative household takes as given the nominal interest rate, the nominal wage rate, nominal aggregate profits, nominal lump sum taxes, and the pries of all onsumption goods. Firms: There are I firms. Firm i supplies good i. The tehnology of firm i is given by Y i,t = A t L α i,t, (6) where Y i,t is output and L i,t is labor input of firm i in period t. A t is aggregate produtivity in period t. The parameter α (0, 1] is the elastiity of output with respet to labor input. The log of aggregate produtivity follows a stationary Gaussian first-order autoregressive proess ln (A t )=ρ a ln (A t 1 )+ε t, (7) where the parameter ρ a [0, 1) and the innovation ε t is i.i.d.n 0,σ 2 ε. The proesses {At } and {Λ t } are assumed to be independent. We all the innovation ε t an aggregate produtivity shok. We introdue the aggregate produtivity shok in the model as an example of a shok that has the following property: the response of the eonomy to the shok under perfet information is effiient. We all this property of a shok the effiieny property. In Setion 6 we derive the optimal monetary poliy response to an aggregate produtivity shok. In Setion 8 we show that the results onerning the optimal monetary poliy response to an aggregate produtivity shok extend to other shoks that have the effiieny property. Nominal profits of firm i in period t equal (1 + τ p ) P i,t Y i,t W t L i,t, (8) 7

8 where τ p is a prodution subsidy paid by the government. In every period, eah firm sets a prie and ommits to supply any quantity at that prie. Eah firm takes as given omposite onsumption by the representative household, the nominal wage rate, and the following prie index 2 P t = Ã 1 I IX i=1 P 1 Λ t i,t! Λt I. (9) Government: There is a monetary authority and a fisal authority. The monetary authority ommits to set the money supply aording to the following rule ln (M s t )=F t (L) ε t + G t (L) ν t, (10) where Mt s denotes the money supply in period t. F t (L) and G t (L) are infinite-order lag polynomials whih an depend on t. The last equation simply says that the log of the money supply in period t an be any linear funtion of the sequene of shoks up to and inluding period t. Wewillaskthe question whih linear funtion is optimal. Two remarks may be useful. First, the reader may wonder how the money market lears at a given money supply. In equilibrium, the endogenous variables (e.g., the prie level, the nominal interest rate, and onsumption) adjust suh that the demand for money balanes by the representative household equals the supply of money balanes by the monetary authority (i.e., M t = Mt s ). Seond, in equation (10) we assume that the entral bank an ommit to a money supply rule. In Setion 8 we show that the set of attainable alloations is idential when the entral bank an ommit to an interest rate rule of the form ln (R t )=F t (L) ε t + G t (L) ν t. (11) Thedrawbakofaninterestrateruleisthatmultipliityofequilibriaatagivenmonetarypoliy arises more easily. Therefore, we assume in the benhmark eonomy that the entral bank an ommit to a money supply rule and we postpone the disussion of unique implementation in the ase of an interest rate rule to Setion 8. 2 Dixit and Stiglitz (1977), in their seminal artile on monopolisti ompetition, also assume that there is a finite number of physial goods and that firmstaketheprieindexasgiven. Moreover,itseemstobeagooddesription of the U.S. eonomy that there is a finite number of physial onsumption goods and that firms take the onsumer prie index as given. 8

9 Next, onsider fisal poliy. The government budget onstraint in period t reads à IX! T t + B t = R t 1 B t 1 + τ p P i,t Y i,t. (12) i=1 The government has to finane maturing nominal government bonds and the prodution subsidy. The government an ollet lump sum taxes or issue new one-period nominal government bonds. We assume that the fisal authority pursues a Riardian fisal poliy. For ease of exposition, we assume that the fisal authority fixes nominal government bonds at some non-negative level B t = B 0. (13) Furthermore, we assume that the fisal authority sets the prodution subsidy so as to orret, in the non-stohasti steady state, the distortion arising from monopolisti ompetition. Formally, τ p = Λ. (14) Alternatively, one ould assume that the fisal authority sets the prodution subsidy so as to orret perfetly at eah point in time the distortion arising from monopolisti ompetition. Formally, τ p,t = Λ t. (15) However, sine in the United States fisal poliy has to be approved by Congress while monetary poliy deisions are implemented diretly by the Federal Reserve, we find it more realisti to assume that the fisal authority annot adjust the prodution subsidy quikly while the monetary authority an adjust the money supply quikly. Information: We onsider two models, one with an exogenous information struture and one with an endogenous information struture. In both models, the information set in period t of the deision-maker who is responsible for setting the prie of good i is I i,t = I i, 1 {s i,0,s i,1,...,s i,t }, (16) where I i, 1 ontains any initial information that the prie setter of firm i has in period minus one and s i,t is the signal that the prie setter of firm i reeives in period t. We assume that the struture of the eonomy is ommon knowledge in period minus one. Furthermore, in the model with an exogenous information struture, we assume that the prie setter of firm i reeives in every period 9

10 t 0 a two-dimensional signal onsisting of a noisy signal onerning aggregate produtivity and a noisy signal onerning the desired markup: s i,t = ln (A t)+η i,t ln (Λ t /Λ)+ζ i,t, (17) where the noise terms have the following properties: (i) the stohasti proesses ª ª η i,t and ζi,t are independent of the stohasti proesses {A t } and {Λ t }, (ii) the stohasti proesses ª η i,t and ª ζi,t are independent aross firms and independent of eah other, and (iii) the noise term ηi,t is i.i.d.n ³ 0,σ 2 η and the noise term ζi,t is i.i.d.n 0,σ 2 ζ. In the model with an exogenous information struture, the varianes of noise σ 2 η and σ 2 ζ are strutural parameters. By ontrast, in the model with an endogenous information struture, the varianes of noise σ 2 η and σ 2 ζ are endogenous. Following the literature on rational inattention, we assume that the deision-maker who is responsible for setting the prie of good i hooses the varianes of noise subjet to an information flow onstraint. Formally, the prie setter of firm i solves the following deision problem in period minus one: ( " # X E i, 1 β t π (P i,t,p t,c t,w t,a t, Λ t ) max (1/σ 2 η,1/σ 2 ζ) R 2 + t=0 subjet to equations (16)-(17) and in every period t 0 ) 1 β κ, (18) P i,t =arg max x R ++ E i,t [π (x, P t,c t,w t,a t, Λ t )], (19) and 1 2 log σ2 a s t 1 i 2 σ log σ2 λ s t 1 i 2 σ 2 = κ. (20) a s t i λ s t i Here E i,t denotes the expetation operator onditioned on the information of the prie setter of firm i in period t, π denotes the real profit funtion defined as the nominal profit funtion times the marginal utility of onsumption of the representative household divided by P t, σ 2 denotes the a s t i onditional variane of a t ln (A t ) given information of the prie setter of firm i in period t, and σ 2 λ s t i denotes the onditional variane of λ t ln (Λ t /Λ) given information of the prie setter of firm i in period t. Thevariableκ is the per-period information flow onerning aggregate onditions and the parameter >0is the per-period marginal ost of information flow. The deision-maker hooses the preision of the two signals in period minus one so as to maximize the expeted disounted sum 10

11 of profits net of the ost of information flow. The deision-maker takes into aount how his/her hoie of signal preision affets future prie setting behavior. Furthermore, equation (20) measures the information flow onerning aggregate onditions and objetive (18) states that information flow onerning aggregate onditions is ostly. We interpret the ost as the prie setter s opportunity ost of devoting attention to aggregate onditions. Three remarks are in plae before we proeed. First, we interpret the noise in the signal (17) as arising from the limited attention of the prie setter of firm i. Therefore, we find it reasonable to assume that the noise is idiosynrati. Seond, in equation (17) we assume that the prie setter of firm i reeives independent signals onerning aggregate produtivity and the desired markup. In Setion 8 we show that this assumption has no effet on optimal monetary poliy in the model with an endogenous information struture. We show that optimal monetary poliy in the model with an endogenous information struture is exatly the same when the deision-maker an hoose to reeive signals onerning any linear ombination of a t and λ t. Third, in the deision problem given in the previous paragraph the prie setter of firm i hooses signal preision one and for all. In Setion 8 we show that our propositions onerning optimal monetary poliy in the model with an endogenous information struture also hold when the deision-maker hooses signal preision period by period. We assume that the monetary authority has perfet information (i.e., in every period t 0, the monetary authority knows the entire history of the eonomy up to and inluding period t). We make this assumption beause we are interested in the optimal ondut of monetary poliy. We think this is an interesting benhmark. In Setion 8 we also onsider an extension where the monetary authority only knows the prie level and omposite onsumption. In addition, we assume that the representative household has perfet information. We make this assumption (i) to isolate the impliations of information fritions on the side of prie setters for optimal monetary poliy, and (ii) to failitate the omparison to optimal monetary poliy in the simplest New Keynesian model, where the only frition apart from monopolisti ompetition is prie stikiness. Aggregation: When we ompute the prie index terms will appear that are linear in 1 I X I X I η i=1 i,t and 1 I ζ i=1 i,t. These averages are random variables with mean zero and variane 1 I σ2 η and 1 I σ2 ζ, respetively. We will neglet these terms beause these terms have mean zero and a variane that an be made arbitrarily small by hoosing a suffiiently high number of firms I. For example, one 11

12 ould set I = Weworkwithafinite number of firms rather than a ontinuum of firms beause we find that it makes the derivation of the entral bank s objetive in the next setion more transparent. 3 Objetive of the entral bank We assume that the entral bank s aim is to maximize expeted utility of the representative household, given by equations (1)-(2). We now derive a simple expression for expeted utility by using the fat that one an express period utility at a feasible alloation as a funtion only of the onsumption vetor at time t, aggregate produtivity at time t, and the desired markup at time t. First, at any feasible alloation the representative household has to supply the labor that is needed to produe the onsumption vetor L t = IX i=1 µ Ci,t A t 1 α. (21) Furthermore, equation (2) for the onsumption aggregator an be written as 1= 1 I IX Ĉ i=1 1 1+Λ t i,t, where Ĉi,t (C i,t /C t ) denotes relative onsumption of good i in period t. Rearranging yields à XI 1 Ĉ I,t = I Ĉ i=1 1 1+Λ t i,t! 1+Λt. (22) Substituting equations (21) and (22) into the period utility funtion in (1) yields the following expression for period utility at a feasible alloation U ³C t, Ĉ1,t,...,ĈI 1,t,A t, Λ t = C1 γ t 1 1 γ 1 µ 1 Ã! Ct α (1+ψ) XI 1 Ĉ 1 XI 1 1 (1+Λt) 1 α α 1+Λ 1+ψ i,t + I Ĉ t 1+ψ i,t (23). A t Hene, expeted utility at a feasible alloation equals " X ³ E β t U C t, Ĉ1,t,...,ĈI 1,t,A t, Λ t #. (24) t=0 12 i=1 i=1

13 In summary, by substituting the tehnology and the onsumption aggregator into the period utility funtion one an express period utility at time t as a funtion only of omposite onsumption, the onsumption mix, aggregate produtivity, and the desired markup at time t. Next,westudytheeffiient alloation. The effiient alloation in period t, defined as the feasible alloation in period t that maximizes utility of the representative household, is and, for all i =1,...,I 1, C t = ³ α I 1+ψ 1 1 α (1+ψ) γ 1+ α 1 (1+ψ) γ 1+ α A 1 (1+ψ) t, (25) Ĉ i,t =1. (26) Effiient omposite onsumption in period t is stritly inreasing in aggregate produtivity at time t. The effiient onsumption mix in period t is to onsume an equal amount of eah good. Note that the effiient onsumption vetor is independent of the desired markup. In the following setions, we work with a log-quadrati approximation of expeted utility (24) around the non-stohasti steady state. In the rest of the paper, variables without time subsript denote values in the non-stohasti steady state and small variables denote log-deviations from the non-stohasti steady state (e.g., t =ln(c t /C) and ĉ i,t =ln³ĉi,t /Ĉi ). Due to the prodution subsidy (14), the non-stohasti steady state is effiient. Expressing the funtion U given by equation (23) in terms of log-deviations from the non-stohasti steady state and using C = C and Ĉi = Ĉ i yields the following expression for period utility at a feasible alloation u ( t, ĉ 1,t,...,ĉ I 1,t,a t,λ t ) = C1 γ e (1 γ) t 1 1 γ Ã C1 γ e 1 α (1+ψ)( t a t ) 1 XI 1 e 1 α ĉi,t + 1 XI 1 I (1 + ψ) I I 1 α i=1 i=1 1 eĉi,t 1+Λe λ t! α(1+λe 1 λ t ) 1+ψ. (27) Proposition 1 (Objetive of the entral bank) Let ũ denote the seond-order Taylor approximation to the period utility funtion u at the origin. Let E denote the unonditional expetation operator. 13

14 Let x t, z t,andω t denote the following vetors ³ 0 x t = t ĉ 1,t ĉ I 1,t, (28) ³ 0 z t = a t λ t, (29) ³ 0 ω t = x 0 t zt 0 1. (30) Let ω n,t denote the nth element of ω t. Suppose that there exist two onstants δ<(1/β) and R suh that, for eah period t 0 and for all n and k, E ω n,t ω k,t <δ t. (31) Then " # " X # X X 1 E β t ũ (x t,z t ) = E β t ũ (x t,z t ) + β t E 2 (x t x t ) 0 H (x t x t ), (32) t=0 t=0 t=0 where the matrix H is given by γ 1+ α 1 (1 + ψ) Λ α 1+Λ α 1+Λ α I(1+Λ)α I(1+Λ)α I(1+Λ)α H = C 1 γ 1+Λ α I(1+Λ)α Λ α I(1+Λ)α 1+Λ α 1+Λ α 0 I(1+Λ)α... I(1+Λ)α 2 1+Λ α I(1+Λ)α, (33) and the vetor x t is given by and t = 1 α (1 + ψ) γ 1+ 1 α (1 + ψ)a t, (34) ĉ i,t =0. (35) Proof. See Appendix A. After the log-quadrati approximation of the period utility funtion (23), expeted utility of the representative household is given by equation (32). The effiient onsumption vetor in period t is given by equations (34)-(35) and the utility loss in the ase of a deviation from the effiient onsumption vetor in period t is given by the quadrati form in square brakets on the right-hand side of equation (32). The upper-left element of the matrix H determines the utility loss in the 14

15 ase of ineffiient omposite onsumption, while the lower-right blok of the matrix H determines theutilitylossintheaseofanineffiient onsumption mix. Finally, ondition (31) ensures that in the expression on the left-hand side of equation (32) one an hange the order of integration and summation and the infinite sum onverges. In the models that we onsider, ondition (31) is always satisfied. 4 The Ramsey problem In this setion, we state the problem of the entral bank that aims to ommit to the money supply rule that maximizes expeted utility of the representative household. In the model with an exogenous information struture, the problem of the entral bank is " X ³ max E β t U C t, Ĉ1,t,...,ĈI 1,t,A t, Λ t #, (36) {F t (L),G t (L)} t=0 subjet to t=0 C i,t = P t = P t C t = M t, (37) Ã W t P i,t 1 I P t! 1+ 1 Λ t C t, (38) = L ψ t P Cγ t, (39) t! Λt IX I, (40) Ã 1 I i=1 P 1 Λ t i,t π (Pi,t,P t,c t,w t,a t, Λ t ) E P i,t I i,t =0, (41) I i,t = I i, 1 {s i,0,s i,1,...,s i,t }, (42) s i,t = ln (A t)+η i,t ln (Λ t /Λ)+ζ i,t, (43) L t = IX i=1 µ Ci,t A t 1 α, (44) ln (A t )=ρ a ln (A t 1 )+ε t, (45) ln (Λ t /Λ) =ρ λ ln (Λ t 1 /Λ)+ν t, (46) 15

16 and ln (M t )=F t (L) ε t + G t (L) ν t. (47) Equations (37)-(40) are the household s optimality onditions. 3 Equation (41) is the firms optimality ondition and equations (42)-(43) speify the information set of the prie setter of firm i in period t. Equation (44) is the labor market learing ondition, equations (45)-(46) speify the laws of motion of the exogenous variables, and equation (47) is the equation for the money supply. 4 The funtion U defined by equation (23) gives period utility at a feasible alloation, F t (L) and G t (L) are infinite-order lag polynomials whih an depend on t, and the innovations ε t, ν t, η i,t,andζ i,t have the properties speified in Setion 2. In the model with an exogenous information struture, the varianes of noise σ 2 η and σ 2 ζ poliy. are strutural parameters. They do not depend on monetary By ontrast, in the model with an endogenous information struture, the varianes of noise σ 2 η and σ 2 ζ are given by the solution to the problem (18)-(20) and the entral bank understands that the hoie of the money supply rule affets the firms hoie of signal preision. Next, in the model with an exogenous information struture, a log-quadrati approximation of the entral bank s objetive (36) around the non-stohasti steady state and a log-linear approximation of the equilibrium onditions (37)-(41) and (44) around the non-stohasti steady state yields the following linear quadrati Ramsey problem " X β t E ( t t ) 2 + δ 1 I min {F t (L),G t (L)} t=0 t=0 # IX (p i,t p t ) 2, (48) i=1 subjet to t = a a t, (49) t = m t p t, (50) p t = 1 I IX p i,t, (51) i=1 p i,t = E p i,t I i,t, (52) 3 We do not state the onsumption Euler equation beause here the onsumption Euler equation is only a priing equation that determines the equilibrium nominal interest rate. 4 The requirement that eah firm produes the quantity demanded is embedded in the profit funtion π and the money market learing ondition M t = M s t isembeddedinequation(47). 16

17 p i,t = p t + t a a t + λ λ t, (53) I i,t = I i, 1 {s i,0,s i,1,...,s i,t }, (54) s i,t = a t + η i,t λ t + ζ i,t, (55) a t = ρ a a t 1 + ε t, (56) λ t = ρ λ λ t 1 + ν t, (57) and m t = F t (L) ε t + G t (L) ν t, (58) where = a = λ = δ = ψ α + γ + 1 α α 1+ 1 α 1+Λ α Λ ψ α + α α 1+Λ α Λ Λ 1+Λ 1+ 1 α 1+Λ α Λ 1+Λ α (1+Λ)α 1+ 1 Λ > 0, (59) > 0, (60) > 0, (61) 2 γ 1+ α 1 > 0. (62) (1 + ψ) Here t is effiient omposite onsumption in period t, p i,t is the profit-maximizing prie of firm i in period t,, a and λ are the oeffiients in the equation for the profit-maximizing prie, and δ is the relative weight on prie dispersion in the entral bank s objetive. In the model with an exogenous information struture, the varianes of noise are exogenous. By ontrast, in the model with an endogenous information struture, the varianes of noise are given by the solution to problem (18)-(20) and the entral bank understands that the money supply rule affets firms alloation of attention. After a log-quadrati approximation of the profit funtion π the attention problem (18)-(20) reads ( X subjet to min (1/σ 2 η,1/σ 2 ζ) R 2 + t=0 ) β t ω h pi,t 2 E i, 1 p i 2 i,t + 1 β κ, (63) p i,t = E p i,t I i,t, (64) 17

18 and 1 2 log σ2 a s t 1 i 2 σ log σ2 λ s t 1 i 2 σ 2 = κ, (65) a s t i λ s t i where ω = C γ WL i P 1+Λ Λ α µ 1+ 1 α α 1+Λ. (66) Λ Here p i,t is the profit-maximizing prie of firm i in period t given by equation (53), I i,t is the information set of the prie setter of firm i in period t given by equations (54)-(55), and the oeffiient ω determines the loss in profit in the ase of a suboptimal prie. 5 Perfet information solution and the entral bank s ability to repliate it In this setion, we derive the solution of the model under perfet information, and we show that the entral bank an always repliate this solution under imperfet information. This intermediate result will be useful when we derive the optimal monetary poliy response to aggregate produtivity shoks in the next setion. Suppose that deision-makers who set pries have perfet information. Then, eah firm harges the profit-maximizing prie, and equations (50)-(53) imply that and t = a a t λ λ t, (67) p i,t p t =0, (68) p t = m t t. (69) The eonomy s response to aggregate produtivity shoks under perfet information is effiient, while the eonomy s response to markup shoks under perfet information is ineffiient. To see this, note that prie dispersion equals zero under perfet information, and ompare equilibrium omposite onsumption given by equation (67) to effiient omposite onsumption given by equation (49). The reason for the ineffiient response to markup shoks is that the effiient alloation is independent of the desired markup but under perfet information firms atual markup varies with the desired markup. Finally, note that monetary poliy has no effet on the equilibrium alloation under perfet 18

19 information. Monetary poliy only affets nominal variables. For example, the entral bank an ompletely stabilize the prie level by setting m t = a a t λ λ t but this has no effet on welfare. The entral bank an repliate the perfet information solution under imperfet information with a partiular monetary poliy rule. More preisely, when the entral bank sets m t = a a t λ λ t, then the equilibrium alloation under perfet information is also the unique equilibrium alloation under imperfet information. The proof is as follows. First, substituting equation (50) into equation (53) yields the following equation for the profit-maximizing prie of firm i in period t µ p i,t =(1 ) p t + m t a a t + λ λ t. (70) Seond, when m t = a a t λ λ t then equations (51), (52), and (70) imply p t =(1 ) 1 I IX E [p t I i,t ]. (71) i=1 The unique solution to the last equation is p t =0. Thus, when m t = a a t λ λ t then p t =0is the unique equilibrium prie level. Finally, substituting m t = a a t λ λ t and p t =0into equation (50) yields t = a a t λ λ t, (72) and substituting m t = a a t λ λ t and p t =0into equations (52) and (70) yields p i,t =0, (73) implying p i,t p t =0. (74) Hene, when the entral bank sets m t = a a t λ λ t, then the equilibrium alloation under perfet information is also the unique equilibrium alloation under imperfet information. Moreover, the same arguments also apply shok by shok. By setting F t (L) ε t = a a t the entral bank an repliate the response of the eonomy to aggregate produtivity shoks under perfet information. By setting G t (L) ν t = λ λ t the entral bank an repliate the response of the eonomy to markup shoks under perfet information. We have derived the solution of the model under perfet information, and we have shown that the entral bank an repliate this solution under imperfet information with a partiular monetary 19

20 poliy rule. Hene, if the entral bank onduts optimal monetary poliy, welfare under imperfet information has to be weakly larger than welfare under perfet information. 6 Optimal monetary poliy response to aggregate produtivity shoks In this setion, we derive the optimal monetary poliy response to aggregate produtivity shoks. We begin with the model with an exogenous information struture and we then onsider the model with an endogenous information struture. Equipped with the results of the previous setion, the proofs are straightforward. Proposition 2 (Exogenous information struture) Consider the Ramsey problem (48)-(62), where the varianes of noise σ 2 η and σ 2 ζ are strutural parameters. If σ2 η > 0, the unique optimal monetary poliy response to aggregate produtivity shoks is F t (L) ε t = a a t. (75) At this poliy, the unique equilibrium response of the eonomy to aggregate produtivity shoks equals the effiient response to aggregate produtivity shoks, and the prie level does not respond to aggregate produtivity shoks. Proof. First, when F t (L) ε t = a a t, the unique equilibrium response of omposite onsumption to aggregate produtivity shoks equals the perfet information response of omposite onsumption to aggregate produtivity shoks, and the perfet information response of omposite onsumption to aggregate produtivity shoks is effiient. See the previous setion. Seond, when F t (L) ε t = a a t, there is no prie dispersion aused by the noise in the signal onerning aggregate produtivity. See again the previous setion. Third, if σ 2 η > 0, any monetary poliy rule with F t (L) ε t 6= a a t yields an ineffiient response of omposite onsumption to aggregate produtivity shoks or prie dispersion aused by the noise in the signal onerning aggregate produtivity: if prie setters put weight on the signal onerning aggregate produtivity then there is prie dispersion aused by the noise in the signal onerning aggregate produtivity, and if prie setters put no weight on the signal onerning aggregate produtivity then the response of omposite onsumption to aggregate 20

21 produtivity shoks is ineffiient. Finally, the hoie of F t (L) affets neither the equilibrium response of omposite onsumption to markup shoks nor the extent to whih there is prie dispersion aused by the noise in the signal onerning the desired markup. Next, we onsider the model with an endogenous information struture, that is, the model in whih prie setters hoose how muh attention they devote to aggregate onditions. Compared to the model with an exogenous information struture, the result is the same and the proof is similar. Proposition 3 (Endogenous information struture) Consider the Ramsey problem (48)-(66), where the varianes of noise σ 2 η and σ 2 ζ are given by the solution to problem (63)-(66). If >0, the unique optimal monetary poliy response to aggregate produtivity shoks is F t (L) ε t = a a t. (76) At this poliy, the unique equilibrium response of the eonomy to aggregate produtivity shoks equals the effiient response to aggregate produtivity shoks, deision-makers in firms who set pries devote no attention to aggregate produtivity, and the prie level does not respond to aggregate produtivity shoks. Proof. First, when the entral bank sets F t (L) ε t = a a t, then for any signal preision 1/σ 2 η 0 the unique equilibrium response of omposite onsumption to aggregate produtivity shoks equals the effiient response of omposite onsumption to aggregate produtivity shoks and there is no prie dispersion aused by the noise in the signal onerning aggregate produtivity. See the previous setion. Seond, when F t (L) ε t = a a t, the prie level and the profit-maximizing prie do not respond to aggregate produtivity shoks. See again the previous setion. If >0, this implies that deision-makers in firms who set pries devote no attention to aggregate produtivity. Third, if >0, anypoliyf t (L) ε t 6= a a t yields an ineffiient response of omposite onsumption to aggregate produtivity shoks or prie dispersion aused by the noise in the signal onerning aggregate produtivity: if prie setters pay no attention to aggregate produtivity then the response of omposite onsumption to aggregate produtivity shoks is ineffiient, and if prie setters devote attention to aggregate produtivity then there is prie dispersion aused by the noise in the signal onerning aggregate produtivity. Finally, the hoie of F t (L) affets neither the equilibrium response of omposite onsumption to markup shoks nor the extent to whih there is prie dispersion aused by the noise in the signal onerning the desired markup. 21

22 7 Optimal monetary poliy response to markup shoks In this setion, we derive the optimal monetary poliy response to markup shoks. The main result is that omplete prie stabilization in response to markup shoks is never optimal in the model with an exogenous information struture, whereas omplete prie stabilization in response to markup shoks is always optimal in the model with an endogenous information struture. In Setion 8 we show that this result for markup shoks extends to other shoks that have the ineffiieny property (i.e., the property that the response of the eonomy to the shok under perfet information is ineffiient). Hene, whether the information struture is exogenous or endogenous has a major impliation for the optimal ondut of monetary poliy. For ease of exposition, we assume in the rest of this setion that there are no aggregate produtivity shoks. This assumption simplifies the notation in Propositions 4 and 5 and has no impat on the optimal monetary poliy response to markup shoks. 7.1 Exogenous information struture The next proposition speifies the optimal monetary poliy response to markup shoks in the model with an exogenous information struture when the desired markup follows a white noise proess. Proposition 4 (Exogenous information struture) Consider the Ramsey problem (48)-(62), where the varianes of noise σ 2 η and σ 2 ζ are strutural parameters. Suppose that ρ λ =0, σ 2 ν > 0 and a 1 = σ 2 ε =0. Consider poliies of the form G t (L) ν t = g 0 ν t and equilibria of the form p t = θλ t. The unique equilibrium at any monetary poliy g 0 R is p t = g 0 + λ λ t, (77) + σ2 ζ σ 2 λ t = σ 2 ζ g σ 2 0 λ λ λ t, (78) + σ2 ζ σ 2 λ p i,t p t = g 0 + λ ζ i,t. (79) + σ2 ζ σ 2 λ Furthermore, if σ 2 ζ > 0, the unique optimal monetary poliy g 0 R is g0 = (1 δ ) λ. (80) σ 2 ζ + δ 2 σ 2 λ 22

23 At this poliy, the prie level stritly inreases in response to a positive markup shok, omposite onsumption stritly falls in response to a positive markup shok, and there is ineffiient prie dispersion. Proof. See Appendix B. The most important result for the rest of this setion is that, in the model with an exogenous information struture, omplete prie stabilization is not optimal. The intuition is the following. Suppose for a moment that the entral bank does not hange the money supply in response to markup shoks. Then the profit-maximizing prie inreases after a positive markup shok, implying that deision-makers who are responsible for setting pries put a positive weight on their signals onerning the desired markup. Thus, the prie level inreases in response to a positive markup shok and there is prie dispersion aused by the noise in the signal onerning the desired markup. Furthermore, omposite onsumption falls in response to a positive markup shok beause the money supply is onstant and the prie level inreases after a positive markup shok. To redue prie dispersion, the entral bank would have to lower the money supply in response to a positive markup shok. On the other hand, to redue onsumption variane, the entral bank would have to inrease the money supply in response to a positive markup shok. Hene, there is a trade-off between prie dispersion and onsumption variane. It depends on the value of δ whether it is optimal for the entral bank to inrease, derease or not hange the money supply in response to a positive markup shok. However, it is never optimal to drive prie dispersion to zero (by ompletely stabilizing the profit-maximizing prie and thereby pries) beause as prie dispersion goes to zero the benefit of further reduing prie dispersion goes to zero while the ost of further reduing prie dispersion inreases. For this reason, omplete prie stabilization is never optimal. By ontrast, in the model with an endogenous information struture, we will find that omplete prie stabilization is always optimal. Next, we onsider the ase of an autoorrelated desired markup to know whether or not omplete prie stabilization is optimal in the model withanexogenousinformationstruturewhenρ λ > 0. When ρ λ > 0, we solve the Ramsey problem (48)-(62) numerially. We turn this infinite-dimensional problem into a finite-dimensional problem by restriting G t (L) tobethesameineahperiodand by restriting G t (L) to be the lag-polynomial of an ARMA(2,2) proess. Following the proedure in Woodford (2002), one an then ompute an exat linear rational expetations equilibrium of the 23

24 model (49)-(62) for a given monetary poliy by solving a Riati equation. We then run a numerial optimization routine to obtain the optimal monetary poliy. 5 Figure 1 shows the optimal monetary poliy response to a markup shok in the model with an exogenous information struture for the following parameter values: β =0.99, γ =1, ψ =0, α =(2/3), Λ =(1/3), σ ν =0.2, andσ ζ =0.4. The upper panel of Figure 1 shows optimal monetary poliyintheaseofρ λ =0. The lower panel of Figure 1 shows optimal monetary poliy in the ase of ρ λ =0.9. For omparison, Figure 1 also shows optimal monetary poliy in the Calvo model with perfet information and an average prie duration of 2.5 quarters. The parameter value Λ =(1/3) implies a steady-state prie elastiity of demand of four, whih is within the range of estimates of the prie elastiity of demand in the Industrial Organization literature. The parameter values ρ λ =0.9 and σ ν =0.2 are within the range of estimates of markup shoks in the New Keynesian literature. The Calvo parameter is taken from Nakamura and Steinsson (2008). The standard deviation of noise is set suh that the model with imperfet information and the Calvo model yield the same response of the prie level to a markup shok when the omponent of the profit-maximizing prie driven by markup shoks is a random walk. The idea is that we aim to ompare the model with imperfet information and the Calvo model for parameter values that imply the same degree of stikiness of the prie level. All impulse responses are to a positive one standard deviation markup shok. A response equal to one means a one perent deviation from the non-stohasti steady state. Time is measured in quarters along the horizontal axis. We obtain the following results. First of all, in the model with an exogenous information struture, omplete prie stabilization is still suboptimal when ρ λ > 0. At the optimal monetary poliy, the prie level stritly inreases on impat of a positive markup shok and omposite onsumption stritly falls on impat of a positive markup shok. Figure 1 shows this result for our benhmark parameter values. We solved the Ramsey problem (48)-(62) for many sets of parameter values with ρ λ > 0 and we always obtained this result. Seond, whether the optimal monetary poliy response to markup shoks is similar in the model with exogenous imperfet information and in the Calvo model depends on ρ λ. When ρ λ =0.9, wefind that optimal monetary poliy is roughly the same in the two models. When ρ λ =0, optimal monetary poliy is quite different in the two models. 5 We hoose an ARMA(2,2) parameterization beause it is well known from time series eonometris that an ARMA(p,q) parameterization is a very flexible and parsimonious parameterization. 24

25 Speifially, when ρ λ =0, the optimal poliy in the model with exogenous imperfet information is to respond to the markup shok only in the period of the shok, while the optimal poliy in the Calvo model is to respond to the markup shok also in the periods after the shok. The reason is the following. In the model with exogenous imperfet information, firms set pries period by period implying: (i) future monetary poliy has no effet on today s prie setting, and (ii) when ρ λ =0today s markup shok reates no ineffiienies in future periods so long as the entral bank does not respond to today s markup shok in future periods. Hene, when ρ λ =0,theoptimal monetary poliy in the model with exogenous imperfet information is to respond to the markup shok only in the period of the shok. In the Calvo model, prie setting is forward looking and firms adjusting pries today but not tomorrow arry the markup shok forward. For these reasons, even when ρ λ =0, the optimal monetary poliy in the Calvo model is to respond to the markup shok also in the periods after the shok. Despite these differenes between the model with exogenous imperfet information and the Calvo model, the optimal monetary poliy is roughly the same in the two models when ρ λ =0.9. We think this is an interesting result, but this is not the main result of this paper. We now turn to the main result of this paper. 7.2 Endogenous information struture In this subsetion, we speify the optimal monetary poliy response to markup shoks in the model with an endogenous information struture. We begin with the ase of an i.i.d. desired markup. In this ase, we have a losed-form solution for the optimal monetary poliy. It turns out that omplete prie stabilization in response to markup shoks is optimal. In fat, we find that in the model with an endogenous information struture omplete prie stabilization in response to markup shoks is optimal for all parameter values. Proposition 5 (Endogenous information struture) Consider the Ramsey problem (48)-(66), where the varianes of noise σ 2 η and σ 2 ζ are given by the solution to problem (63)-(66). Suppose that ρ λ =0, σ 2 ν > 0, anda 1 = σ 2 ε =0. Consider poliies of the form G t (L) ν t = g 0 ν t and equilibria of the form p t = θλ t. Assume that >0 and define s ω ( b g 0 + λ ) 2 σ 2 λ ln (2). (81) 25