Exogenous Information, Endogenous Information and Optimal Monetary Policy

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1 Exogenous Information, Endogenous Information and Optimal Monetary Poliy Luigi Paiello Einaudi Institute for Eonomis and Finane Mirko Wiederholt Northwestern University January 2011 Abstrat This paper studies optimal monetary poliy when deision-makers in firms hoose how muh attention they devote to aggregate onditions. When the amount of attention that deisionmakers in firms devote to aggregate onditions is exogenous, omplete prie stabilization is optimal only in response to shoks that ause effiient flutuations under perfet information. When deision-makers in firms hoose how muh attention they devote to aggregate onditions, omplete prie stabilization is optimal also in response to shoks that ause ineffiient flutuations under perfet information. Hene, reognizing that deision-makers in firms an hoose how muh attention they devote to aggregate onditions has major impliations for optimal poliy. JEL: E3,E5,D8. Keywords: dispersed information, rational inattention, optimal monetary poliy Paiello: Einaudi Institute for Eonomis and Finane, Via Sallustiana 62, Rome, Italy ( Wiederholt: Department of Eonomis, Northwestern University, 2001 Sheridan Road, Evanston, IL ( We thank Fernando Alvarez, Andy Atkeson, Larry Christiano, Jordi Gali, Christian Hellwig, Olivier Loisel, Bartosz Maćkowiak, Jordi Mondria, Jonathan Parker, Alessandro Pavan, Aleh Tsyvinski, Pierre-Olivier Weill and seminar and onferene partiipants at Booni, CREI, EIEF, HEC Montreal, IAE, Maryland, NAWMES 2011, NBER Summer Institute 2010, Northwestern, SED 2010, Toulouse and UCLA for helpful omments. An earlier version of this paper was irulated under the title Imperfet Information and Optimal Monetary Poliy. Mirko Wiederholt thanks CREI and EIEF for hospitality. 1

2 1 Introdution Deision-makers in firms have a limited amount of attention and they an hoose how muh attention they devote to aggregate onditions. What are the impliations for optimal eonomi poliy? What are the impliations for optimal monetary poliy? To address this question formally, we derive optimal monetary poliy in two models. In the first model the amount of attention that deision-makers in firms alloate to aggregate onditions is exogenous. In the seond model deision-makers in firms hoose how muh attention they devote to aggregate onditions. Our main findings onerning optimal monetary poliy are the following. In the model with an exogenous alloation of attention by deision-makers in firms, omplete prie stabilization is the optimal poliy only in response to shoks that ause effiient flutuations under perfet information. In the model with an endogenous alloation of attention by deision-makers in firms, omplete prie stabilization is the optimal poliy also in response to shoks that ause ineffiient flutuations under perfet information. Hene, reognizing that deision-makers in firms an hoose how muh attention they devote to aggregate onditions has major impliations for optimal monetary poliy. The optimality of omplete prie stabilization beomes a muh more general result. There is a large literature on optimal monetary poliy. Most of this literature studies optimal monetary poliy in the New Keynesian framework (see Woodford (2003) or Gali (2008) for a detailed summary of the results). 1 To make our results omparable to this benhmark in the literature on optimal monetary poliy, we maintain several assumptions of the standard New Keynesian model: We assume that there is a large number of ex-ante idential firms supplying differentiated produts and setting pries for these produts; the monetary poliy instrument is a nominal variable; and the entral bank an affet real variables with the nominal monetary poliy instrument beause pries adjust slowly. The eonomy is subjet to different types of shoks. The main poliy question is how the entral bank should adjust the monetary poliy instrument in response to these shoks. We make two hanges to the standard New Keynesian model. First, we assume that slow adjustment of pries to hanges in aggregate onditions is due to limited attention by deision-makers in firms rather than prie stikiness à la Calvo (1983). Seond, we endogenize the amount of attention that deision-makers in firms devote to aggregate onditions. We then address two questions: (1) 1 For reent work on optimal monetary poliy in New Keynesian models, see, e.g., Giannoni and Woodford (2010). 2

3 Does it matter for optimal monetary poliy that slow adjustment of pries to hanges in aggregate onditions is due to limited attention by deision-makers in firms rather than prie stikiness à la Calvo (1983)? (2) Does it matter for optimal monetary poliy whether the amount of attention that deision-makers in firms devote to aggregate onditions is exogenous or endogenous? The answer to the seond question is the fous of this paper. This paper is the first paper solving a Ramsey optimal poliy problem for an eonomy where deision-makers in firms hoose how muh attention they alloate to aggregate onditions. We model the attention deision by deision-makers in firms following the literature on rational inattention (see Sims (2003)). Paying limited attention to aggregate onditions is modeled as reeiving a noisy signal onerning aggregate onditions. 2 Paying more attention to aggregate onditions inreases the preision of the signal. We assume that the noise in the signal is idiosynrati beause this aords well with the idea that the soure of noise is limited attention by individual deision-makers rather than lak of publily available information. Deision-makers in firms hoose the amount of attention that they alloate to aggregate onditions faing an opportunity ost of alloating attention to aggregate onditions. The main predition onerning the alloation of attention is that when the benefit of paying attention to aggregate onditions is larger, deision-makers in firms pay more attention to aggregate onditions. In the model there are two types of shoks ausing aggregate flutuations: shoks that ause effiient flutuations under perfet information and shoks that ause ineffiient flutuations under perfet information. In the benhmark model setup, these two types of shoks are aggregate tehnology shoks and markup shoks (i.e., shoks to the elastiity of substitution between goods). We start with aggregate tehnology shoks and markup shoks beause studying the optimal monetary poliy response to these shoks is ommon in the literature on optimal monetary poliy. We then show that our results extend to other shoks. Furthermore, in the benhmark model setup, we assume that deision-makers in firms reeive independent signals onerning aggregate tehnology and the desired markup. We also show that optimal monetary poliy is idential when deision-makers in firms an deide to reeive signals onerning any linear ombination of aggregate tehnology and the desired markup (e.g., they an deide to reeive signals/pay attention to endogenous variables). 2 Think of the noise in the signal as the noise in the answers you get when you ask a sample of eonomists what the offiial CPI inflation rate for the United States was last year. 3

4 We derive optimal monetary poliy under ommitment assuming that the entral bank aims to maximize expeted utility of the representative household. In the model with an exogenous information struture, omplete prie stabilization is optimal in response to aggregate tehnology shoks but not in response to markup shoks. The reason for the result about aggregate tehnology shoks is as follows. The response of the eonomy to aggregate tehnology shoks is effiient under perfet information. Furthermore, by stabilizing pries the entral bank an repliate the perfet-information response of the eonomy to aggregate tehnology shoks. Thus, omplete prie stabilization is optimal in response to aggregate tehnology shoks. To understand the result onerning markup shoks, note firstwhathappenswhenthemonetary poliy instrument (i.e., the money supply or nominal interest rate) remains onstant after a markup shok. In this ase, a positive markup shok (i.e., a shok that raises the desired markup) inreases the profit-maximizing prie. Prie setters in firms therefore put a positive weight on their signals onerning the desired markup whih auses ineffiient prie dispersion due to noise in the signal ( ross-setional ineffiieny ). Furthermore, the prie level inreases whih given the onstant monetary poliy instrument auses a fall in onsumption ( aggregate ineffiieny ). To redue ross-setional ineffiieny, the entral bank an ounterat the effet of the markup shok on the profit-maximizing prie with a ontrationary monetary poliy (i.e., by lowering the money supply or raising the nominal interest rate). The profit-maximizing prie then inreases by less in response to a markup shok and prie setters in firms therefore put less weight on their noisy signals onerning the desired markup, whih redues ineffiient prie dispersion. Unfortunately, the redution in ross-setional ineffiieny omes at the ost of inreased aggregate ineffiieny: The ontrationary monetary poliy amplifies the fall in onsumption. Hene, there is a trade-off between ineffiient prie dispersion and ineffiient onsumption variane. Furthermore, the marginal benefit of reduing ineffiient prie dispersion goes to zero as ineffiient prie dispersion goes to zero. Therefore, omplete prie stabilization in response to markup shoks is never optimal. This tradeoff between ross-setional ineffiieny and aggregate ineffiieny in the presene of markup shoks is emphasized a lot in the literature on optimal monetary poliy and the result that omplete prie stabilization is not optimal in response to these shoks is a lassi result in monetary eonomis. In the model with an endogenous information struture, omplete prie stabilization is optimal in response to aggregate tehnology shoks and in response to markup shoks. This result is inde- 4

5 pendent of parameter values. The reason for the result about markup shoks is the following. In the model with an endogenous information struture, deision-makers in firms pay more attention to aggregate onditions when the benefit of paying attention to aggregate onditions is larger. Consider again what happens when the entral bank ounterats the effet of a positive markup shok on the profit-maximizing prie with a ontrationary monetary poliy. There are two effets that are already present in the model with an exogenous information struture: The profit-maximizing prie inreases by less after a positive markup shok, implying that deision-makers in firms put less weight on their noisy signals, whih redues ineffiient prie dispersion; but the ontrationary monetary poliy by itself amplifies the fall in onsumption. In addition, there is a new effet due to the endogenous alloation of attention. When the profit-maximizing prie responds less to markup shoks, deision-makers in firms hoose to pay less attention to the desired markup. Therefore, the prie level inreases by less after a positive markup shok, whih by itself mutes the fall in onsumption. It turns out that the new effet on onsumption dominates for all parameter values. Thus, so long as deision-makers in firms pay some attention to the desired markup, the entral bank an redue both ineffiient prie dispersion and ineffiient onsumption variane by ounterating markup shoks more strongly. The lassi trade-off between ross-setional ineffiieny and aggregate ineffiieny disappears. The unique optimal monetary poliy is the one that makes deision-makers in firms pay just no attention to the desired markup. Hene, at the optimal monetary poliy, prie setters in firms pay no attention to random variation in the desired markup and therefore pries do not respond to markup shoks. Complete prie stabilization inresponseto markup shoks is optimal. These results on optimal monetary poliy in the model with an endogenous information struture generalize in important ways. Most importantly, the result that omplete prie stabilization is optimal in response to markup shoks extends to other shoks that ause ineffiient flutuations under perfet information and the result that omplete prie stabilization is optimal in response to aggregate tehnology shoks extends to other shoks that ause effiient flutuations under perfet information. Aggregate tehnology shoks and markup shoks are just simple examples. Furthermore, these results on optimal monetary poliy extend to more general signal strutures. In the benhmark model setup, we assume that deision-makers in firms reeive independent signals onerning aggregate tehnology and the desired markup and hoose the preision of these two 5

6 signals. In an extension, we assume that deision-makers in firms an hoose to reeive signals onerning any linear ombination of aggregate tehnology and the desired markup (e.g., signals onerning endogenous variables). Optimal monetary poliy is idential in these two model setups. To summarize, let us answer the question that is the fous of this paper: Does it matter for optimal monetary poliy whether the amount of attention that deision-makers in firms devote to aggregate onditions is exogenous or endogenous? Our answer is: a lot. When the amount of attention that deision-makers in firms devote to aggregate onditions is exogenous, omplete prie stabilization is the optimal poliy only in response to shoks that ause effiient flutuations under perfet information. When deision-makers in firms hoose how muh attention they devote to aggregate onditions, omplete prie stabilization is the optimal poliy also in response to shoks that ause ineffiient flutuations under perfet information. This paper is related to four reent papers studying optimal monetary poliy in models with information fritions. The most losely related paper is Adam (2007). He studies optimal monetary poliy in a model in whih prie setters in firms pay limited attention to aggregate onditions, but the amount of attention that prie setters devote to aggregate onditions is exogenous. He shows that omplete prie stabilization is optimal in response to labor supply shoks but not in response to markup shoks. Ball, Mankiw and Reis (2005) study optimal monetary poliy in the stiky-information model of Mankiw and Reis (2002). In this model prie setters in firms update their information sets with an exogenous probability. They show that omplete prie stabilization is optimal in response to aggregate tehnology shoks but not in response to markup shoks. Finally, Lorenzoni (2010) and Angeletos and La O (2008) study optimal monetary poliy in models with dispersed information due to an island struture. In Lorenzoni (2010) prie setters in firms observe the omplete history of the eonomy up to the previous period, the sum of aggregate and idiosynrati produtivity, and a noisy publi signal onerning aggregate produtivity. There are several differenes between his paper and our paper. 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; in his paper the information struture is exogenous whereas in our paper the information struture is endogenous; and in his paper the entral bank has imperfet information. We initially assume that the entral bank has perfet information about the state of the eonomy to derive the optimal monetary poliy response to shoks, and we then study whether 6

7 the entral bank an also implement the optimal monetary poliy with less information. Angeletos and La O (2008) study optimal monetary poliy when agents observe signals onerning endogenous variables with exogenous variane of noise. There is an information externality beause a stronger response of agents to their private signals inreases the signal-to-noise ratio of the signals onerning endogenous variables. Angeletos and La O (2008) study how this information externality affets optimal fisal and monetary poliy. To reapitulate, the main differene between this paper and the papers ited above is that we derive optimal monetary poliy when deision-makers in firms hoose the amount of attention that they alloate to aggregate onditions. This paper is also related to the literature on rational inattention. See, for example, Sims (2003, 2006, 2010), Luo (2008), Maćkowiak and Wiederholt (2009, 2010), Van Nieuwerburgh and Veldkamp (2009, 2010), Woodford (2009), Matejka (2010), Mondria (2010) and Paiello (2010). However, none of these papers studies optimal poliy. 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 derives the equilibrium alloation under perfet information as a benhmark. Setion 6 derives the optimal monetary poliy response to aggregate tehnology shoks. Setion 7 derives the optimal monetary poliy response to markup shoks. Setion 8 ontains several additional results, inluding the results about more general shoks and more general signal strutures. Setion 9 onludes. 2 Model setup The eonomy is populated by firms, a representative household, 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 " X E 0 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. 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. E 0 denotes 7

8 the expetation operator onditioned on information of the household in period zero. Composite onsumption in period t is given by a Dixit-Stiglitz aggregator Ã! 1+Λt 1 IX 1 1+Λ C t = C t i,t, (2) I i=1 where C i,t is onsumption of good i in period t. ThereareI different onsumption goods and the elastiity of substitution between onsumption goods in period t equals (1 + 1/Λ t ). We all the variable Λ t the desired markup beause Λ t equals the desired markup by firms in period t. We assume that the log of the desired markup follows a stationary Gaussian first-order 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. 3 In Setion 7 we derive the optimal monetary poliy response to markup shoks. In Setion 8.4 we show that our results onerning markup shoks extend to other shoks that ause ineffiient flutuations under perfet information. 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. B t 1 are the household s holdings of government bonds between period t 1 and period 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, and the term in brakets are unspent money balanes arried over from period t 1 to period t. The representative household an transform 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) 3 We define effiieny formally in Setion 3. i=1 i=1 Here 8

9 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 for our results about optimal monetary poliy in a simple way. In Setion 8.6 we show that our results about optimal monetary poliy extend to a ashless eonomy à la Woodford (2003). In addition, in Setion 8.6 we study optimal monetary poliy in a version of the eonomy with monetary transation fritions. The formulation of the ash-in-advane onstraint given above implies that there are no monetary transation fritions beause wage inome an be transformed immediately into ash and ash an be spent immediately on goods. We deided to abstrat from monetary transation fritions in the benhmark eonomy for two reasons. First, abstrating from monetary transation fritions is ommon in the New Keynesian literature on optimal monetary poliy and thus abstrating from monetary transation fritions failitates omparison of our results to results about optimal monetary poliy in the New Keynesian literature. Seond, we think it is useful to study in isolation the impliations of different fritions for optimal monetary poliy. Therefore, we first abstrat from monetary transation fritions and we then add monetary transation fritions in Setion 8.6 by hanging the timing of the ash-in-advane onstraint, i.e., by assuming that ash has to be held for one period before it an be spent on goods. In every period, the representative household hooses a onsumption vetor, labor supply, money balanes and 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. We all the innovation εt an aggregate tehnology shok. The proesses {A t } t=0 and {Λ t} t=0 are assumed to be independent. We introdue the aggregate tehnology shok in the model as an example of a shok that has the 9

10 following property: The response of the eonomy to the shok under perfet information is effiient. In Setion 6 we derive the optimal monetary poliy response to aggregate tehnology shoks. In Setion 8.4 we show that our results onerning aggregate tehnology shoks extend to other shoks that ause effiient flutuations under perfet information. Nominal profits of firm i in period t equal (1 + τ p ) P i,t Y i,t W t L i,t, (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 the laws of motion for omposite onsumption, the nominal wage rate and the following prie index 4 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. To study the optimal monetary poliy response to shoks, we initially assume that the entral bank has perfet information. In Setion 8.7 we show that the entral bank has a high inentive to be informed about the aggregate state of the eonomy and that the entral bank an implement the optimal monetary poliy also with less information. In addition, in Setion 8.5 we show that the set of equilibria that the entral bank an implement with a money supply rule of the form (10) equals the set of equilibria that the entral bank an implement with an interest rate rule of 4 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 onsumption goods and that firms take the onsumer prie index as given. 10

11 the form ln (R t )=F t (L) ε t + G t (L) ν t. (11) The drawbak of ommitting to an interest rate rule rather than a money supply rule is that multipliity of equilibria at a given monetary poliy 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 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. In partiular, 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 the distortion arising from monopolisti ompetition in the non-stohasti steady state: 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: 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. 5 A remark about the money market may be useful. In the model, the money market lears in the usual way. In equilibrium, endogenous variables (e.g., the prie level, onsumption and the nominal interest rate) adjust suh that the demand for money balanes by the representative household equals the supply of money balanes by the monetary authority. 11

12 Information: We now speify the assumptions about the information struture. We onsider two models. In the model with an exogenous information struture, the amount of attention that deision-makers in firms alloate to aggregate onditions is exogenous. In the model with an endogenous information struture, deision-makers in firms hoose how muh attention they alloate to aggregate onditions. In both models, the information set of the prie setter in firm i in period t is I i,t = I i, 1 {s i,0,s i,1,...,s i,t }, (16) where I i, 1 is the initial information set of the prie setter in firm i and s i,t is the signal that he or she reeives in period t. The latter is a two-dimensional vetor onsisting of a noisy signal onerning aggregate tehnology 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) We assume that the noise in the signal is due to limited attention by the deision-maker. 6 The noise in the signal has the following properties: (i) the proesses ª η i,t t=0 and ª ζ i,t are independent t=0 of the proesses {A t } t=0 and {Λ t} t=0, (ii) the proesses ª η i,t t=0 and ª ζ i,t are independent t=0 aross firms and independent of eah other, and (iii) η i,t and ζ i,t follow Gaussian white noise proesses with varianes σ 2 η and σ 2 ζ. The assumption that the noise in the signal is idiosynrati aords well with the idea that the soure of noise is limited attention by individual deision-makers rather than lak of publily available information. The assumption that deision-makers in firms reeive independent signals onerning aggregate tehnology and the desired markup is only for ease of exposition. In Setion 8.3, we show that optimal monetary poliy in the model with an endogenous information struture is idential when deision-makers in firms an deide to reeive signals onerning any linear ombination of aggregate tehnology and the desired markup (e.g., they an pay attention to endogenous variables). In the model with an exogenous information struture, the varianes of noise σ 2 η and σ 2 ζ are exogenous. In the model with an endogenous information struture, deision-makers in firms hoose the preision of the signals faing an opportunity ost of alloating attention to aggregate onditions. Following the literature on rational inattention (see Sims (2003)), we quantify the amount 6 See Footnote 2. 12

13 of attention alloated to aggregate onditions by unertainty redution. The timing is as follows. In period minus one, deision-makers in firms hoose the preision of the signals so as to maximize expeted profits net of the opportunity ost of devoting attention to aggregate onditions. In the following periods, deision-makers in firms reeive the signals and take the optimal prie setting deisions given the signals that they have reeived. Formally, the prie setter in 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/σ2 ζ) R 2 + subjet to equations (16)-(17) and t=0 ) 1 β κ, (18) and Consider first objetive (18). P i,t =arg max P i R ++ E[π (P i,p t,c t,w t,a t, Λ t ) I i,t ], (19) κ = h (A t, Λ t I i,t 1 ) h (A t, Λ t I i,t ). (20) information of the prie setter in firm i in period minus one. Here E i, 1 denotes the expetation operator onditioned on the The funtion π denotes the real profit funtion defined as the nominal profit funtion divided by P t times the marginal utility of onsumption by the representative household. The variable κ is the amount of attention that the deision-maker devotes to aggregate onditions. The parameter >0 is the per-period marginal ost of devoting attention to aggregate onditions. ost. We interpret the ost as an opportunity Paying more attention to the prie setting deision means paying less attention to some other ativity. Following Sims (2003), we quantify the amount of attention devoted to aggregate onditions by unertainty redution, where unertainty is measured by entropy. See equation (20). Here h (A t, Λ t I i,t 1 ) denotes the onditional entropy of A t and Λ t given I i,t 1 and h (A t, Λ t I i,t ) denotes the onditional entropy of A t and Λ t given I i,t. The differene between the two quantifies the information reeived in period t. Finally, equation (19) speifies the prie setting behavior. The basi trade-off is the following. A higher preision of the signals improves the prie setting behavior but requires paying more attention to aggregate onditions. Finally, we make a simplifying assumption. To abstrat from transitional dynamis in onditional seond moments, we assume that at the end of period minus one (i.e., after the deision-maker has hosen the preision of the signals), the deision-maker reeives information suh that: (i) the 13

14 onditional distribution of (ln (A 0 ), ln (Λ 0 )) given information at the end of period minus one is normal, and (ii) the onditional ovariane matrix of (ln (A 0 ), ln (Λ 0 )) given information at the end of period 1 equals the steady-state onditional ovariane matrix of (ln (A t ), ln (Λ t )) given informationinperiodt 1. Before we proeed, it is useful to point out two features of the deision problem (18)-(20) that one may expet to be important for our results about optimal monetary poliy but that turn out to be irrelevant for our results about optimal monetary poliy. First, in equation (17) we assume that deision-makers in firms reeive independent signals onerning aggregate tehnology and the desired markup. In Setion 8.3, we show that optimal monetary poliy in the model with an endogenous information struture is idential when deision-makers in firms an deide to reeive signals onerning any linear ombination of aggregate tehnology and the desired markup (e.g., signals onerning endogenous variables). Seond, in the deision problem (18)-(20) we assume that deision-makers in firms hoose a onstant signal preision one and for all. Our propositions about optimal monetary poliy in the model with an endogenous information struture also hold when deision-makers in firms hoose signal preision period by period or when deision-makers in firms hoose signal preision as a funtion of time in period minus one. We assume that the representative household has perfet information. We make this assumption for two reasons. First, this assumption failitates the omparison of our results about optimal monetary poliy to the results about optimal monetary poliy in the basi New Keynesian model, where the only frition apart from monopolisti ompetition is prie stikiness. Seond, this assumption allows us to isolate the impliations of limited attention by prie setters in firms for optimal monetary poliy. Aggregation: When omputing 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 setting the number of firms suffiiently high. For example, one an set I = Alternatively, one ould work with a ontinuum of firms and apply the law of large numbers in Uhlig (1995). We work with a finite 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. 14

15 3 Objetive of the entral bank We assume that the entral bank aims to maximize expeted utility of the representative household, given by equations (1)-(2). We now derive a simple expression for expeted utility of the representative household 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 ) is 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 α (1+Λ t) α 1+Λ 1+ψ i,t + I Ĉ t 1+ψ i,t (23). Thus, expeted utility at a feasible alloation equals " X ³ E β t U C t, Ĉ1,t,...,ĈI 1,t,A t, Λ t #. t=0 A t To summarize, by substituting the tehnology and the onsumption aggregator into the period utility funtion 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. i=1 i=1 15

16 We define the effiientalloationinperiodt as the feasible alloation in period t that maximizes utility of the representative household. The effiient alloation in period t is given by 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) Ĉ i,t =1. t, The effiient onsumption level in period t is stritly inreasing in aggregate produtivity in period t and is independent of the desired markup. The effiient onsumption mix in period t is to onsume an equal amount of eah good. In the following setions, we work with a log-quadrati approximation to the period utility funtion (23) around the non-stohasti steady state. In the following, 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 (i.e., C = C and Ĉi = Ĉ i ). Expressing the period utility funtion U defined by equation (23) in terms of log-deviations from thenon-stohastisteadystateandusingc = 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 at) 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. (24) A seond-order Taylor approximation to this funtion at the non-stohasti steady state yields the result stated in Proposition 1. 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. 16

17 Let x t, z t,andω t denote the following vetors ³ 0 x t = t ĉ 1,t ĉ I 1,t, ³ 0 z t = a t λ t, ³ 0 ω t = x 0 t zt 0 1. 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. (25) Then " # " X # X E β t ũ (x t,z t ) E β t ũ (x t,z t ) = t=0 t=0 X t=0 1 β t E 2 (x t x t ) 0 H (x t x t ), (26) 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, (27) and the vetor x t is given by and t = 1 α (1 + ψ) γ 1+ 1 α (1 + ψ)a t, (28) ĉ i,t =0. (29) Proof. See Appendix A. After the log-quadrati approximation to the period utility funtion (23) around the nonstohasti steady state, the effiient onsumption vetor in period t is given by equations (28)-(29) and the loss in expeted utility in the ase of deviations of the atual onsumption vetor from the effiient onsumption vetor is given by equation (26). The upper-left element of the matrix H determines the loss in utility in the ase of an ineffiient onsumption level. The lower-right blok of 17

18 the matrix H determines the loss in utility in the ase of an ineffiient onsumption mix. Condition (25) ensures that in the expressions on the left-hand side of equation (26) one an hange the order of integration and summation and the infinite sum onverges. ondition (25) is always satisfied. In the models that we onsider, 4 The Ramsey problem In this setion, we state the maximization problem of the entral bank that aims to ommit to the poliy 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 #, (30) {F t(l),g t(l)} t=0 subjet to and t=0 C i,t = P t = P t C t = M t, (31) Ã! 1+ 1 Λ P t i,t 1 I P C t, t (32) W t = L ψ t P Cγ t, (33) t! Λt IX I, (34) Ã 1 I i=1 P 1 Λ t i,t P i,t =arg max E[π (P i,p t,c t,w t,a t, Λ t ) I i,t ], P i R ++ (35) I i,t = I i, 1 {s i,0,s i,1,...,s i,t }, (36) s i,t = ln (A t)+η i,t ln (Λ t /Λ)+ζ i,t, (37) L t = IX i=1 µ Ci,t A t 1 α, (38) ln (A t )=ρ a ln (A t 1 )+ε t, (39) ln (Λ t /Λ) =ρ λ ln (Λ t 1 /Λ)+ν t, (40) ln (M t )=F t (L) ε t + G t (L) ν t. (41) 18

19 The objetive (30) is expeted utility of the representative household. The funtion U defined by equation (23) gives period utility at a feasible alloation. Equations (31)-(34) are the household s optimality onditions. 7 Equation (35) states that prie setters in firms take the optimal prie setting deisions given their information. Equations (36)-(37) speify the information set of the prie setter in firm i in period t. Equation (38) is the labor market learing ondition. Equations (39)-(40) speify the laws of motion for the exogenous variables. Finally, equation (41) is the equation for the money supply, where F t (L) and G t (L) are infinite-order lag polynomials whih an depend on t. 8 The innovations ε t, ν t, η i,t and ζ 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 are strutural parameters. They do not depend on monetary poliy. 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 attention problem (18)-(20) and the entral bank understands that the hoie of the poliy rule affets the firms alloation of attention. In the literature on optimal monetary poliy it is ommon pratie to study the Ramsey problem after a log-quadrati approximation of the entral bank s objetive and a log-linear approximation of the equilibrium onditions. See Woodford (2003), Gali (2008), Adam (2007) and Ball, Mankiw and Reis (2005). We follow this ommon pratie. This makes our results omparable to their results. A log-quadrati approximation of the entral bank s objetive and a log-linear approximation of the equilibrium onditions around the non-stohasti steady state yields the following linear quadrati Ramsey problem in the model with an exogenous information struture: " # X β t E ( t t ) 2 + δ 1 IX (p i,t p t ) 2, (42) I subjet to min {F t (L),G t (L)} t=0 t=0 i=1 t = a a t, (43) t = m t p t, (44) p t = 1 IX p i,t, I (45) i=1 7 We do not state the onsumption Euler equation beause here the onsumption Euler equation is only a priing equation determining the equilibrium nominal interest rate. 8 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 is embedded in equation (41). See Footnote 5. 19

20 p i,t = E p i,t I i,t, (46) p i,t = p t + t a a t + λ λ t, (47) I i,t = I i, 1 {s i,0,s i,1,...,s i,t }, (48) s i,t = a t + η i,t λ t + ζ i,t, (49) a t = ρ a a t 1 + ε t, (50) λ t = ρ λ λ t 1 + ν t, (51) and where m t = F t (L) ε t + G t (L) ν t, (52) = a = λ = 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 2 > 0, (53) > 0, (54) > 0, (55) δ = γ 1+ α 1 > 0. (56) (1 + ψ) The objetive (42) follows from substituting the log-linear demand funtion for good i into equation (26) and by using equation (45). The variable t is effiient omposite onsumption in period t and the parameter δ is the relative weight on ross-setional ineffiieny versus aggregate ineffiieny in the entral bank s objetive. The variable p i,t is the profit-maximizing prie of good i in period t. In the model with an exogenous information struture, the varianes of noise σ 2 η and σ 2 ζ are exogenous. In the model with an endogenous information struture, prie setters in firms hoose the preision of the signals. After a log-quadrati approximation of the real profit funtion π around the non-stohasti steady state, the attention problem (18)-(20) reads: ( " # X E i, 1 β t ω pi,t p 2 i,t β κ subjet to min (1/σ 2 η,1/σ 2 ζ) R 2 + t=0 ), (57) p i,t = E p i,t I i,t, (58) 20

21 and Ã! Ã! κ = 1 σ 2 2 log a t σ 2 2 log λ t 1 2, (59) σ 2 a t where the oeffiient ω determining the profit loss in the ase of a deviation of the atual prie from the profit-maximizing prie is given by ω = C γ WL i P 1+Λ Λ α µ 1+ 1 α α σ 2 λ t 1+Λ. (60) Λ Here p i,t denotes the profit-maximizing prie of good i in period t given by equation (47) and I i,t denotes the information set of the prie setter in firm i in period t given by equations (48)-(49). Equation (20) redues to equation (59) beause the onditional distribution of (a t,λ t ) is Gaussian both given I i,t 1 and given I i,t and beause a t and λ t are onditionally independent both given I i,t 1 and given I i,t. In the following, σ 2 λ, σ2 λ t 1 and σ2 λ t denote the unonditional variane of λ t, the onditional variane of λ t given I i,t 1 and the onditional variane of λ t given I i,t, respetively. 5 Perfet information solution As a benhmark, we now derive the response of the eonomy to aggregate shoks under perfet information. Suppose that prie setters in firms have perfet information. Eah firm then harges the profitmaximizing prie and equations (44)-(47) imply and t = a a t λ λ t, (61) p i,t p t =0, (62) p t = m t t. (63) Under perfet information, the response of the eonomy to aggregate tehnology shoks is effiient while the response of the eonomy to markup shoks is ineffiient. To see this, note that there is no ineffiient prie dispersion under perfet information and ompare equations (43) and (61). The response of the eonomy to markup shoks under perfet information is ineffiient beause under perfet information firms vary the atual markup with the desired markup whih auses ineffiient onsumption flutuations. 21

22 6 Optimal monetary poliy response to tehnology shoks In this setion, we derive the optimal monetary poliy response to aggregate tehnology shoks in the model with an exogenous information struture and in the model with an endogenous information struture. We show that in both models omplete prie stabilization is the optimal poliy in response to aggregate tehnology shoks. 6.1 Exogenous information struture Proposition 2 (Exogenous signal preision) Consider the Ramsey problem (42)-(56), where the varianes of noise σ 2 η and σ 2 ζ are exogenous. Consider equilibria with the property that the prie level p t is a linear funtion of the shoks. If σ 2 η > 0, the unique optimal monetary poliy response to aggregate tehnology shoks is F t (L) ε t = a a t. (64) At the optimal monetary poliy, the prie level does not respond to aggregate tehnology shoks. Proof. See Appendix B. The reason for this result about optimal poliy is the following. The response of the eonomy to aggregate tehnology shoks under perfet information is effiient. Furthermore, by offsetting the effet of aggregate tehnology shoks on the profit-maximizing prie the entral bank an repliate the perfet-information response of real variables to aggregate tehnology shoks. To see this, note that the profit-maximizing prie (47) an be written as µ p i,t =(1 ) p t + m t a a t + λ λ t. By setting F t (L) ε t = a a t, the entral bank an offset the effet of aggregate tehnology shoks on the profit-maximizing prie. Prie setters in firms then put no weight on their noisy signals onerning aggregate tehnology and thus there is no ineffiient prie dispersion due to the noise in the signal onerning aggregate tehnology. In addition, the prie level then does not respond to aggregate tehnology shoks and therefore the response of the onsumption level to aggregate tehnology shoks equals a a t, whih equals the effiient response of the onsumption level to aggregate tehnology shoks. See equations (44) and (43). 22

23 6.2 Endogenous information struture Proposition 3 (Endogenous signal preision) Consider the Ramsey problem (42)-(60), where the signal preisions 1/σ 2 ³ η and 1/σ 2 ζ are given by the solution to problem (57)-(60). Consider equilibria with the property that the prie level p t is a linear funtion of the shoks. If >0, the unique optimal monetary poliy response to aggregate tehnology shoks is F t (L) ε t = a a t. (65) At the optimal monetary poliy, the prie level does not respond to aggregate tehnology shoks. Proof. See Appendix C. The reason for this result about optimal poliy is the same as in the previous subsetion: The response of the eonomy to aggregate tehnology shoks under perfet information is effiient and by offsetting the effet of aggregate tehnology shoks on the profit-maximizing prie the entral bank an repliate the perfet-information response of real variables to aggregate tehnology shoks. There is one differene to the previous subsetion. At the optimal monetary poliy, prie setters in firms now devote no attention to aggregate tehnology beause the profit-maximizing prie does not respond to aggregate tehnology shoks. 7 Optimal monetary poliy response to markup shoks In this setion, we derive the optimal monetary poliy response to markup shoks. Our main result is the following. Complete 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. For ease of exposition, we assume in this setion that there are no aggregate tehnology shoks. This assumption simplifies the notation in Propositions 4, 5 and 6, and has no impat on the optimal monetary poliy response to markup shoks. 7.1 Exogenous information struture In the model with an exogenous information struture, the optimal monetary poliy response to markup shoks in the ase of an i.i.d. desired markup is given by the following proposition. 23

24 Proposition 4 (Exogenous signal preision) Consider the Ramsey problem (42)-(56), where the varianes of noise σ 2 η and σ 2 ζ are exogenous. Suppose σ2 ν > 0, ρ λ =0and σ 2 ε = a 1 =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, (66) + σ2 ζ σ 2 λ t = σ 2 ζ g σ 2 0 λ λ λ t, (67) + σ2 ζ σ 2 λ p i,t p t = g 0 + λ ζ i,t. (68) + σ2 ζ σ 2 λ Furthermore, if σ 2 ζ > 0, the unique optimal monetary poliy g 0 R is g0 = (1 δ ) λ. (69) σ 2 ζ + δ 2 σ 2 λ At the optimal monetary 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 D. The main result in Proposition 4 is that in the model with an exogenous information struture and an i.i.d. desired markup, omplete prie stabilization in response to markup shoks is never optimal. To understand this result, note first what happens when the entral bank does not hange the monetary poliy instrument in response to markup shoks (i.e., g 0 =0). In this ase, a positive markup shok (i.e., a shok that raises the desired markup) inreases the profit-maximizing prie. Prie setters in firms therefore put a positive weight on their signals onerning the desired markup whih auses ineffiient prie dispersion due to noise in the signal ( ross-setional ineffiieny ). Furthermore, the prie level inreases whih - given the onstant money supply - auses a fall in onsumption ( aggregate ineffiieny ). To redue ineffiient prie dispersion, the entral bank an ounterat the effet of a positive markup shok on the profit-maximizing prie with a ontrationary monetary poliy (i.e., by lowering the money supply). The profit-maximizing prie then inreases by less in response to a positive markup shok, implying that prie setters in firms put less 24