3 Optimal Informational Interest Rate Rule 3.1. Introduction

Transcription

1 3 Opimal Informaional Ineres Rae Rule 3.1. Inroducion Any public policy may be undersood as a public signal of he curren sae of he economy as i informs he views of he governmenal auhoriy o all agens. This paper consider he case where he ineres rae is informaive abou aggregae nominal demand - he fundamenal of our economy - o se ligh o wo main issues: (i) how his informaional effec changes price seing and, by consequence, inflaion dynamics and (ii) wha would be an opimal informaional ineres rae rule. We consider a model where he cenral bank has a noisy signal of aggregae nominal demand growh. Based on his informaion, i ses he ineres rae, which affecs he dynamics of he fundamenal. As firms needs o infer aggregae nominal demand o se heir prices and hey do no observe he informaion cenral bank has, when hey observe an ineres rae rise, hey consider wo opposie movemens: (i) aggregae nominal demand is expeced o rise and (ii) aggregae nominal demand will depress by he influence of he policy insrumen. We name he firs effec as he informaional power of ineres rae as i reveals he views of he moneary auhoriy. To evaluae his effec on prices, and by consequence, on inflaion, we sudy price seing in a model where firms have heerogeneous informaion and face sraegic complemenariy on heir acions. In his environmen, firms choose prices knowing ha heir payoffs depend no only on heir own acions, bu also on he prices he oher firms choose. Firms ry o predic one anoher's acions using he informaion hey have. We assume ha informaion is boh sicky (as in Mankiw and Reis (2002), each period only a fracion of he firms updaes heir informaion ses) and dispersed (as in Morris and Shin (2002), firms receive privae signals of he fundamenal when hey updae). Furhermore, he policy insrumen becomes a public signal since i is observable o all firms, including

2 Informaional Fricions and Inflaion Dynamics 49 hose ha have no been seleced o updae he informaion se. We isolae he informaional power of ineres rae by sudying how (equilibrium) price seing changes when firms ignore he fac ha he moneary auhoriy reveals informaion when i chooses he policy insrumen. This sraegy allows us o measure he influence of informaional power of ineres rae on welfare, evaluaed by hree differen measures - inflaion variance, ex-ane oal profi and cross-secional dispersion. Besides, i is possible o find an opimal informaional ineres rae rule, which is characerized by he parameers of he policy insrumen ha maximizes welfare, considering ha cenral bank knows ha firms will ake informaion from is acions. 22 The main consequence o inflaion dynamics regards he persisence of moneary shocks. Wihou he informaional effec, firms would undersand changes in he policy insrumen as isolaed movemens, insead of a response o changes in he fundamenal. Under his shor sigh, ineres rae affecs inflaion only insananeously. In conras, when ineres rae is undersood as a public signal of he fundamenal, he whole realizaion of ineres rae affecs inflaion, making he moneary shock persisen. We also show ha here is an opimal informaional ineres rule ha simulaneously maximizes all hree welfare crieria. To implemen his rule, cenral bank should avoid adding any moneary shock o he policy insrumen, while should use he level of informaional response ha minimizes inflaion variance, one of our welfare measures. We inroduce our basic model in he nex secion. In Secion 3.3 we give he basic seps we use o compue he equilibrium, considering ha is derivaion is analogous o he one described in chaper 2. In Secion 3.4, we define he informaional effec and build a counerfacual in oher o isolae i. We also presen some analyses based on how price seing changes when he informaional effec is considered. The core of he paper is Secion 3.5, where we use hree differen welfare crieria o measure he informaional effec and analyze is 22 Our erminology opimal informaional ineres rae rule should no be misundersood. By his expression, we do no mean finding which variables he moneary auhoriy should look o when i wans o minimize a welfare crierion. For his approach, see Woodford (2003).

3 Informaional Fricions and Inflaion Dynamics 50 quaniaive implicaions. We provide concluding remarks in Secion 3.6 and proofs omied in he main ex in he Appendix The Model Pricing Decisions There is a coninuum of firms indexed by z 0,1. Every period 1,2,..., each firm z chooses a price p z. We can derive from a model of monopolisic compeiion à la Blanchard and Kiyoaki (1987) ha he (log-linear) price decision ha solves a firm's profi maximizaion problem, p, is given by p rp 1 r, (3.1) where r is he degree of sraegic complemenariy, P 1 p ( z)dz is he 0 aggregae price level of he economy, and is a relevan fundamenal. In our framework, we can inerpre as being he curren sae of aggregae nominal demand of our economy. Fundamenal Dynamics In his paper, we consider he case where he dynamics of θ is parially driven by a policy insrumen, he ineres rae i, ha respond o he noisy informaion cenral bank has abou changes in he sae, y. Tha is θ = θ 1 σ i + ε, (3.2) y i = φ y + v, (3.3) ( θ θ ) + η, = (3.4) 1 where he errors are independen of one anoher -, (, k m) η + k v + m, ε - and 1 1 disribued according o ε N( 0, α ), ( 0, 1 η N γ ), and v N ( 0, µ ) noise erm. The ε may represen demand shock affecing he curren sae of he economy, while he noise v is a policy disurbance as, for example, a moneary policy shock. Finally, he shock η reflecs ha cenral banks' informaion abou he sae is also imprecise. Informaion As in Mankiw and Reis (2002), informaion is sicky. Every period only a

4 Informaional Fricions and Inflaion Dynamics 51 fracion ( 0,1) λ of firms receives new informaion abou he fundamenal. The probabiliy of being seleced o adus informaion is he same across firms and independen of hisory. However, as in chapers 1 and 2, we depar from his sandard sicky-informaion srucure, by allowing informaion o be also dispersed. As before, following Morris and Shin (2002) and Angeleos and Pavan (2007), we assume ha, when a firm updaes is informaion se, i receives no only informaion regarding he pas saes of he economy, k k 1, bu also a privae signal abou he curren sae, x z z. 1 As before, we assume ha ξ ( z) N( 0, β ) is independen of all oher errors. Furhermore, for each firm z, ξ ( z) is a idiosyncraic shock. Tha is, ξ ( z ) ξ ( ), (, k, z z ) 1 + k z 2 1, 2. We can combine equaions (3.3) and (3.4) o show ha ineres rae is a public signal of he fundamenal's change, which is available o all firms, including hose who have no been seleced o updae heir informaion ses. As a resul, he informaion se of a firm z ha was seleced o updae is informaion periods ago is I = 0 where { } i k k= z x z, 1,I,. The inroducion of a public signal on a sicky-dispersed informaion model has already been sudied in chaper 2. Neverheless, now he public signal is a policy insrumen ha also inerferes on he dynamics of he fundamenal. This fac changes how firms compue he equilibrium Equilibrium chooses In equilibrium, a firm z ha updaed is informaion se a period p z E p z. (3.5) From (3.1), i is clear ha firm z will have o predic no only he curren

5 Informaional Fricions and Inflaion Dynamics 52 sae θ, bu also he aggregae price level P. As P encompasses he prices se by oher firms, firm z mus also predic he behavior of he oher firms in he economy by forecasing oher firms' forecass abou he sae, forecasing he forecass of oher firms' forecass abou he sae, and so on and so forh. This explanaion highlighs he imporance of compuing order beliefs o find he equilibrium. However, because of he lineariy of he bes-response condiion (3.1) and he Gaussian specificaion of he informaion srucure, he equilibrium prices are linear combinaions of he observed signals. As a resul, i is possible o obain he unique linear equilibrium of his game using he much simpler approach of maching coefficien. In chapers 1 and 2, we use boh mehods. Reduced Form In order o derive he equilibrium exacly in he same way as we did in chaper 2, we rewrie his model as θ θ + u, (3.6) = 1 where i ( θ θ ) φ e, = φ + (3.7) 1 1 ε σ φ e e η + φ v and u. (3.8) 1 + σ φ The erm u aggregaes he policy shocks, while he noise e encompasses all he shocks affecing he sae. As 1 1 e N( 0, ω ), and N ( 0, ϕ ) u where ε is independen of e, we obain ha ω = γ + ( φ µ ) and ϕ [ + ( ) ]. 1 α σ φ ω (3.9) + σ φ Alhough rewriing our informaion srucure as (3.6) and (3.7) makes our model similar o he one sudied a chaper 2, here are imporan differences o consider. Firs, we can analyze he impac of he policy parameer, while in chaper 2 we fix φ = 1. However, he main difference beween hem lies on he fac ha u is no independen of e. Alhough his fac is no surprising, since i capures he endogeneiy of he variables, i affecs he way firms compue heir beliefs abou he fundamenal and, consequenly, alers he equilibrium.

6 Informaional Fricions and Inflaion Dynamics 53 Expecaions I is imporan o undersand how a firm z ha updaed is informaion se a compues is beliefs abou a fundamenal θ m. Since a he momen i aduss is informaion se he firm observes all previous saes, 1, he firm will know for sure he value of θ m when Θ [ m ] = θ m θ. For m >. Therefore, E I ( z) m, θ m is no in he informaion se of firm z. However, i knows ha m 1 i m Since θ I ( z), i compues E I ( z) 1 [ ] m u i. θ as E m z 1 i m E u i z. As he process is Markovian, pas values of does no help o predic u i. Therefore, he only signals of u i he firm can build from z are w k 1 i k u k e k and z x z 1 u z. Bu, u is independen of i w, k i k, and of ( z), if i. Therefore, 23 E m where z 1 E u w, z 1 i m E u i w i 1 x 1 i 1 i m i i, (3.10) ˆ α + ω 1 and ˆ ω ασφ δ = κ = φ. (3.11) α + ω + β α + ω The firs hree componens presen in he expecaion represen [ I ( z) ] E θ and can be expressed as a convex combinaion of privae and public informaion. Tha is, x z,r,s N 1 x z r s, 1, where r 1 θ 1 σi = θ ε, and s φ i + θ 1 = θ + e are wo signals of he fundamenal θ. As he errors associaed o each of hese signals are independen of one anoher, he weighs represen he relaive precision 23 See Appendix A for deails.

7 Informaional Fricions and Inflaion Dynamics 54 associaed o each of hese signals. This sandard resul is presen in he models of Morris and Shin (2002) and Angeleos and Pavan (2007). The las erm shows how o build expecaions for imporance of θ, when m <. The weigh κ capures he m u k on he signal w k u k + e k =. I is worh noing ha κ is affeced by he public and policy precisions, α and ω, as well as he policy and srucural parameers, φ and. However, as ( z) u, when i x is no informaive abou i <, κ does no depend on he precision of privae informaion, β. In summary, a firm z ha las updaed is informaion se periods ago has he following forecass abou he sae θ m of he economy E m z 1 x z 1 i 1 i i i m : m m : m, (3.12) which are used o compue he linear equilibrium of he model. Linear Equilibrium In chaper 2, we use an expression analogous o (3.12) o derive he unique linear equilibrium of he game. The expression for he equilibrium price index is ( ˆ, δ ˆ κ ) Pˆ where P P, k 0 c k k k 0 d k i k. (3.13) and he coefficiens ( c, d ) are funcions of ( δ, κ ) given by 24 k k 1 1, c k 1 r r 1 1 r 1 1 if k 0 1 r r 1 1 r 1 1 k 1 1 r 1 1 k 1 if k 1, (3.14) 24 The erminology ( c, d ) is a noaional abuse, since i does no refer o a pair of ( ) generic coefficiens, bu o a pair of sequences { c } { d } i hroughou he ex., = 0 = 0. However, we use

8 Informaional Fricions and Inflaion Dynamics 55 d k, 1 k 1 r r 1 k. (3.15) Equaion (3.13) shows ha he price index (and, by consequence, inflaion) depends no only on he realizaions of θ, bu also on he realizaion of i. The presence of he policy insrumens on prices is a no exacly new resul. For insance, Ravenna and Walsh (2006) sudied he cos channel. A cos channel is presen when firms have o finance heir producions, making marginal cos depending direcly on he nominal rae of ineres. Neverheless, no only his resul comes from a differen source, he informaional power of ineres rae, bu i also has a much more permanen effec, since he whole realizaion of i appears on (3.13). Form (3.15), when sickiness is very high ( λ is small), his persisence becomes more relevan Informaional Effec In oher o measure he informaional effec, i is imporan o obain he dynamics of P when firms observe he ineres rae bu hey do no see i as a public signal Counerfacual If firms ignored ha ineres rae is informaive abou he curren sae of he economy, he dynamics of P would change because firms would modify he [ ] way hey compue E I ( z) θ. For m, a firm z ha updaed is m [ ] informaion se a period will obain E θ I ( z) from m m 1 i m i i 1 i m i. Firm z has wo signals of θ : a privae signal x z z, and a public signal composed by pas informaion abou he fundamenal and he policy insrumen

9 Informaional Fricions and Inflaion Dynamics 56 q 1 i. If firms did no consider i k informaive abou k ε, k <, where E m z E z 1 i m i i 1 x z q 1 i m i i 1 x z 1 i 1 i m i i,, and. (3.16) I is clear ha expression (3.16) is he same of (3.10), when we have ω = 0 in (3.11). This resul has an easy inerpreaion: ignoring he informaional power of ineres rae is equivalen o saying ha ineres rae is no informaive abou he curren sae of he economy (i.e., he variance associaed o he ineres rae is infiniy). From (3.16), we obain he expression for (he equilibrium) price index, where P is he funcion specified in (3.13). P P,, (3.17) Two main facs come sraighforward from he comparison of ( ˆ δ, ˆ κ ) wih ~ ( δ, ~ κ ): (i) ˆ ~ δ > δ and (ii) ~ κ is negaive, while κˆ can be negaive. The firs observaion ells us ha he privae signal becomes less imporan when agens consider ha ineres rae is informaive abou he curren sae. Tha is, he informaional power of ineres rae makes public informaion more valuable (relaive o privae informaion) o agens. This resul resembles he one obained in Morris and Shin (2002). Using (3.15), he second observaion shows ha ineres rae have a posiive impac on prices when ˆ κ > 0 (or equivalenly, ω > ασφ ). Therefore, if ineres rae is very informaive abou he curren sae of he economy (i.e., he precision of i, ω, is sufficienly high), an ineres rae upurn is undersood as a rise on he aggregae nominal demand, inducing firms o raise heir prices. In his conex, he informaional power of ineres rae is more imporan han is capabiliy of reducing aggregae nominal demand. This is clear he case when σ is small, since he policy insrumen will have a small impac on he aggregae

10 Informaional Fricions and Inflaion Dynamics 57 demand. The analysis for φ is no so obvious. We can used (3.8) o rewrie he condiion ˆ κ > 0 as 0, where 2. (3.18) Because is convex, when here is no real roo for (condiion µγ < ( 2ασ ) 2 ), for no value of φ ineres rae will have a posiive impac over prices. However, when has wo real roos ( ( 2ασ ) 2 when ( ) γµ > ), we have ˆ κ > 0 φ r 1, r 2, where r 1, r2 are he roos of. As boh roos are posiive, should be small enough ( φ ( 0, r )) or high enough ( φ (, ) 1 r 2 ), if he moneary auhoriy does no wans he informaional power of ineres rae o be dominan over is capabiliy of reducing aggregae nominal demand. When φ is high, he policy insrumen becomes srong enough o overcome he difficuly imposed by he informaional power of ineres rae. When φ is small, he policy insrumen is us no informaive abou movemens on aggregae nominal demand (a he limi case, φ = 0, he policy insrumen becomes a whie noise, i = v consequence, firms pay no aenion on he public signal Inflaion Dynamics ). As a In order o analyze he impac of he informaional power of ineres rae on inflaion, we use he expressions for Pˆ and P ~ o derive inflaion dynamics and sudy how is response o shocks is sensiive o he parameers of he model. Inflaion As Pˆ and P ~ differ only by he coefficiens ( c, d ˆ π π ( δ ˆ = ˆ, κ ) and ~ ~ π π ( δ, ~ κ ) independen shocks given by k k ), we can express inflaion as =, where π is wrien as a combinaion of

11 Informaional Fricions and Inflaion Dynamics 58, P P 1 0 c 1 0 d i i 1 0 c u 0 d u e u 1 e c l 0 l c e (3.19) and he coefficiens (c k,d k ) are given by (3.14) and (3.15), while l k, d 0 1 c 0, if k 0 d k d k 1 c k, if k 1. Using his expression, we can analyze how inflaion dynamics changes by he informaional power of he ineres rae. Calibraion The model's srucural parameers are r, λ, α, β, γ, µ, φ, and σ. Following Mankiw and Reis (2002), we use λ = and r = 0. 9 as our baseline values (see Table 3.1). Table 3.1: Baseline calibraion Parameer Descripion Range Benchmark Value r Degree of sraegic complemenariy 0, λ Degree of informaional sickiness 0, α Precision of he aggregae demand shock ε 1.00 β Precision of he privae informaion 1.00 γ Precision of of he informaion available o cenral bank 1.00 µ Precision of moneary shock 1.00 Cenral bank's informaional response coefficien 1.00 Elasiciy of he fundamenal wih respec o ineres rae 0.67 The value λ = implies ha firms adus heir privae informaion once a year, which is compaible wih he mos recen microeconomic evidence on price-seing. 25 For he remaining parameers, we se α = β = γ = µ = 1 as our 25 See, for example, Klenow and Malin (2010).

12 Informaional Fricions and Inflaion Dynamics 59 benchmark value o keep he baseline calibraion as neural as possible regarding he relaive imporance of each ype of informaion. To highligh how he informaional power of ineres rae changes inflaion dynamics, we vary four parameers of he model. More specifically, we chose he parameers associaed o he precision ω (i.e., γ, µ, and φ ) and he primiive parameer σ. The imporance of ω o he informaional effec is clear, since he parameers (κ ~, ~ δ ) and (κˆ, δˆ ) differ solely by wheher ω = 0 or no. Besides, precision ω encompasses all policy dimensions, including he precision of policy informaion γ, policy insrumen shocks µ, and he degree o which policy responds sysemaically o informaion φ. The imporance of he primiive parameer σ comes from he fac ha i represens he endogeneiy of he variables. When σ = 0, he ineres rae does no inerfere on he dynamics of he fundamenal, alhough i coninues o be a public signal of he fundamenal growh. Impulse response We sudy inflaion's impulse responses o wo ypes of shocks - ε and e. From (3.2), ε is a shock ha has a direc impac on he aggregae nominal demand, while, from (3.7), e is a composie policy shock ha affecs he fundamenal hrough i. Figure (F1a) shows how inflaion's responses o a demand shock changes wih he parameers of he model - φ, σ, γ, and µ. ε Figure 3.1: Inflaion's impulse responses o a (uni) shock ε

13 Informaional Fricions and Inflaion Dynamics 60 Some paerns appear in he four graphs. For all parameers, when firms consider ha ineres rae is informaive abou he sae of he economy, he response is aenuaed. This resul suggess ha, when firms are beer informed, price seing becomes more sensible, as firms esimae beer he magniude of he shock. Furhermore, all four graphs show he same paern for he counerfacual: inflaion drops hugely a = 0 and becomes posiive aferwards. The raionale behind his observaion is simple: if firms do no consider he ineres rae informaive abou he curren sae, hey infer ha he ineres rae rise hey observe will make inflaion drop. This behavior is sronger a = 0, when no firm has informaion abou he sae θ, bu gradually vanishes over ime as firms ge informed abou he sae and correcly idenify he occurrence of a posiive shock on aggregae demand. When firms consider he ineres rae informaive abou he curren sae, we do no observe a srong drop a = 0, since he informaional power of ineres rae makes firms correcly predic ha he rise hey observe a ineres rae can come from a posiive shock on aggregae nominal demand. For some se of parameers, inflaion can be even posiive a = 0. I is also imporan o analyze he influence of each parameer separaely. Considering ha firms ake informaion from ineres rae, inflaion increases less when: (i) cenral bank responds more aggressively o he fundamenal growh (higher values of φ ), (ii) he precision of he informaion cenral bank has (γ ) is higher or (iii) he moneary shock has smaller variance. However, changes in he elasiciy of he fundamenal wih respec o ineres rae (σ ) modify inflaion's impulse response only a = 0. We can derive his resul analyically: using he definiion of l and he consan κˆ in equaion (3.19), we see ha boh c + φ l and l σ c do no depend on σ, 1. When we analyze he counerfacual, we see ha i does no move wih any of hese parameers when 1. Alhough no ploed in Figure (F1a), inflaion drops more inensively a poin = 0 when φ and σ increases, while i remains unchanged for variaions of γ and µ. As firms ignore he informaional power of ineres rae, parameers sricly associaed o he precision of ineres rae, ω, do no move inflaion on he counerfacual case. As parameers φ and σ are associaed o he ransmission of he policy insrumen o inflaion, hey move

14 Informaional Fricions and Inflaion Dynamics 61 inflaion when no firm has informaion abou he shock. Aferwards, however, firms do no consider ha ineres rae will coninue o rise, since firms do no see ineres rae as a response o he informaion cenral bank has abou he fundamenal growh. Figure (F2a) shows how inflaion's responses o a demand shock e changes wih he parameers of he model - φ, σ, γ, and µ. Figure 3.2: Inflaion's impulse responses o a (uni) shock e. The firs imporan observaion we ake from Figure (F2a) is ha shock e is no persisen when firms ignore he fac ha ineres rae is informaive abou he sae of he economy. This observaion is easy o undersand analyically: ~ l ~ δ, ~ κ ~ ~ δ, ~ ~ c κ, we have l σ c~ = 0, 1 in equaion defining l ( ) and ( ) l (3.19). As ineres rae is no persisen by iself, persisence on ineres rae comes from he fac ha cenral bank is reacing o changes in he fundamenal, which evolves according o a Markovian process. If firms did no see he ineres rae as a public signal abou he fundamenal, hey would never incorporae his inerial behavior in heir forecass. This shor-sigh behavior generaes he same paern in all four graphs when we analyze he counerfacual cases: inflaion hugely drops a = 0, and becomes null aferwards. I is imporan o sress ha wha creaes persisence on inflaion is no how cenral bank is reacing, bu raher how firms change price seing when hey undersand ha changes in ineres rae are persisen. When we analyze he influence of each parameer separaely, we observe

15 Informaional Fricions and Inflaion Dynamics 62 ha inflaion becomes less negaive for smaller values of φ. The impulse response does no move wih σ, when we have 1. We define he informaional effec over a variable as he difference ha occurs on his variable when we replace (3.13) wih (3.17). We are going o analyze hree differen crieria o measure he informaional effec: (i) inflaion variance; (ii) cross-secional price dispersion, and (iii) ex ane aggregae profi of he firms Efficiency Crieria The firs crierion we analyze is inflaion variance. Woodford (2003) derives a welfare based loss funcion as he second order approximaion of he uiliy funcion of a represenaive household. On a sandard sicky prices model à la Calvo (1983), his loss funcion is a weighed average of squared oupu gap and inflaion. This means ha any cenral bank ha wans o minimize he expeced value of his welfare based loss funcion should care abou inflaion variance. Alhough oupu gap is also presen in his loss funcion, Woodford (2003) shows ha, under sandar parameers values, is weigh is very small compared o inflaion's. As a resul, he variance of inflaion is as a proxy of welfare for a vas lieraure of moneary models. Neverheless, as we do no consider he exisence of price rigidiy in our model, i would no be opimal o pursue he minimizaion of inflaion variance as he policy obecive. Ball e al. (2005) show ha he welfare based policy obecive when here is informaional rigidiy is o minimize a cross-secional dispersion. Therefore we consider his our second efficiency crierion. Finally, following he efficiency benchmark for dispersed informaion models proposed by Angeleos and Pavan (2007), we consider ex-ane aggregae profis as our hird opimal crierion. Under his crierion, welfare is evaluaed from he perspecive of firms. To highligh he informaional effec, we show how hese hree crieria evolve wih he four parameers φ, σ, γ, and µ. We have already discussed he imporance of hese parameers for he model. We also obain he parameers of he ineres rule ha minimizes hose crieria. We assume ha cenral bank canno change γ, he precision of he policy

16 Informaional Fricions and Inflaion Dynamics 63 informaion y. Neverheless, we consider ha he cenral bank can choose no only φ, he cenral bank's informaional response coefficien, bu also µ, he precision of he moneary shock Inflaion Variance Considering expression (3.19), he variance of inflaion can be wrien as var c, d c l 2 2 E 2 0 l c 2 2 E e c l l c 2, where ( c, d ) is eiher ( cˆ, dˆ ) or ( c~ ~, d ). This expression shows ha inflaion variance depends on ω, even when firms ignore he fac ha he ineres rae is informaive abou he curren sae of he economy. The raionale behind his observaion is: firms may discard par of he informaion hey have, bu his behavior does no change he way cenral bank reacs o informaion on aggregae nominal demand growh. Therefore, we coninue o have i = φ u + φe. As he ineres rae inerferes on he dynamics of he fundamenal, and by exension on prices, he variance of e coninue o influence he variance of inflaion. Under his framework, we define he informaional effec over he inflaion variance as var var. Fugure 3.3: Evoluion of π for differen parameers.

17 Informaional Fricions and Inflaion Dynamics 64 Figure (F3a) shows how he informaional effec over variance evolves when we modify he policy parameers φ, γ, and µ and he primiive parameer σ regarding wo siuaions: (i) firms consider he policy insrumen informaive, var ( πˆ ), and (ii) hey do no, ( ~ π ) var. Equaion (3.7) decomposes ineres rae in wo pars: he response o he fundamenal ( φ ( θ θ ) 1 ) and a noise ( e ). As he variance of he noisy par diminishes (γ or µ grows), ineres rae becomes more sysemaic. As hese parameers are exclusively associaed o he qualiy of public informaion, he informaional effec over he inflaion variance grows wih hem. Wheher π becomes posiive when γ or µ grows, i depends on he calibraion. For small values of σ, we can obain ( ˆ π ) var( ~ π ) var >, if γ and µ are sufficienly high. When σ is small, he policy insrumen does no inerfere much on he fundamenal dynamics, being us a public signal. As we have seen in chaper 2, he precision of he public signal increases inflaion variance. The explanaion is idenical o he one presened in Angeleos and Pavan (2007): as he public signal increases coordinaion in price seing, he variance of inflaion increases. However, for higher values of, as shown in figure (F3a), inflaion variance diminishes when γ or µ grows, even when we consider ha firms ake informaion from ineres rae. Therefore, in order o decrease inflaion variance, cenral bank should pursue a coninuous improvemen on qualiy of he informaion used o ake policy decisions, as well as no adding any moneary shock. When we analyze he influence of σ, we have o consider wo effecs: (i) he influence of he policy insrumen on he fundamenal dynamics increases and (ii) he shocks incorporaed in he policy insrumen are amplified. The firs effec helps o lower inflaion variance, as firms will change he way hey compue expecaions when hey perceive he policy insrumen as an effecive mean of driving he fundamenal (i.e., cenral bank will need smaller variaions on he policy insrumen o move he fundamenal). The second effec increases inflaion variance as he fundamenal becomes more volaile. Firms do no compue he firs effec when hey disregard he fac ha he policy insrumen is a response o changes in he fundamenal. Therefore, we observe increasing inflaion volailiy

18 Informaional Fricions and Inflaion Dynamics 65 when we analyze he counerfacual case. When firms ake informaion from he ineres rae, boh effecs are considered. For small values of σ, he firs effec is dominan, while he second effec pushes inflaion variance up when σ increases. The ne effec, measured by small values of σ. π, is a decreasing funcion of σ ha is posiive for The mos complex analysis is relaed o he cenral bank's informaional response coefficien, φ. Depending on he calibraion, i can eiher produce he smooh behavior shown in Figure (F3a) or i can produce an overshooing on var ( πˆ ), and by consequence on π, for small values of φ. This overshooing shows ha here is a region where inflaion variance increases wih φ when firms consider he policy insrumen informaive abou he sae. Alhough no proven analyically, we believe ha his overshooing is produced when we have wo posiive real roos in (3.18), in which case here is a region where ineres rae increases prices. Besides he exisence of an overshooing, we observe he same wo effecs we sudied for he case of σ. As before he firs effec is dominan for small values of φ, while he second effec makes inflaion variance increases. Again, π is a decreasing funcion of φ, when his parameer is sufficienly high. I is no a coincidence ha σ and φ have similar influence on he inflaion variance, once he facor σφ measures how inensively cenral bank's informaion his he fundamenal dynamics. Considering our baseline calibraion, minimizaion of inflaion variance recommends ha he cenral bank should no add any noise o is policy rule ( µ ), making he ineres rae as much informaive abou he fundamenal as possible. Besides, he cenral bank should posiively reac o he informaion received ( φ 1.5) Ex ane Toal Profi As in chaper 2, we use a modified version of he efficiency crierion proposed by Angeleos and Pavan (2007) ha represens ex-ane oal profis.

19 Informaional Fricions and Inflaion Dynamics 66 E,I 0 1 n x, 1,I 2 df x,i df,i x,i,i h,i df,i, [ ] where n ( x Θ, I ) p ( x, Θ, I ) ( 1 r) + rp (, I ), 1 1 θ Θ is he obecive funcion ha guaranees profi maximizaion and ( Θ, ) muliplier associaed o he consrain η is he Lagrange I h, I P, I 0 1 p x, 1,I df x, I. x Using (3.13), he generic expression for he equilibrium aggregae price index, we can wrie n ( x, I ) of he independen shocks, Θ 1 as a funcion of he parameers (κ, δ ) and n x, 1,I z k 0 k k k 0 k e k, e where 1 r 1 r 1 1. (3.20) Using his expression, we obain ex ane oal profi as a funcion of ( κ, δ ) and of he variances of he shocks. However, we can come wih a much simpler expression when we use (3.11) o wrie E Π as a funcion of ω, E (3.21) For he case where firms ignore he informaional power of ineres rae (he counerfacual), we have o evaluae his expression considering ω = 0. I is imporan o highligh ha Ω is also a funcion of ω, since i depends on ρ. Under his framework, we define he informaional effec over ex ane oal profi as E E E 0. Figure (F4a) shows how E changes wih some parameers of he model.

20 Informaional Fricions and Inflaion Dynamics 67 For all cases, we have ha E Π is posiive, meaning ha welfare is higher when firms use ineres rae o ake informaion abou he sae. We have shown ha E Π is a funcion of he precision of ineres rae, ω. Therefore, he counerfacual does no change wih any parameer, since we consider ω = 0 when firms do no ake informaion from ineres rae. As he elasiciy of he fundamenal wih respec o ineres rae, σ, does no affec ω, no vary wih σ. Furhermore, he firs derivaive of always posiive, meaning ha γ, and µ. E Π, and by exension E Π, does E Π wih respec o ω is E Π is an increasing funcion of he parameers φ, Figure 3.4: Evoluion of EΠ for differen parameers The firs derivaive of E Π wih respec o ω is always posiive. Therefore, if he cenral bank is ineresed in his crierion, i should increase ω as much as possible. One way of aaining his obecive is o increase he precision of he policy insrumen, moneary shocks, should be avoided Cross-Secional Dispersion µ. This resul ells us again ha policy shocks, like Following similar seps o Woodford (2002), Ball e al. (2005) showed ha he second order approximaion of agens' uiliy funcion in a model wih sicky informaion is a weighed average of oupu gap variance and cross-secional dispersion plus erms ha are independen of policy. As we focus on he crosssecional dispersion, our crierion is a proxy of he crierion proposed in Ball e al.

21 Informaional Fricions and Inflaion Dynamics 68 (2005), where Var z is given by EV E Var z p z P, Var z p z P p z P 2 dz p z P dz 2. Wriing his crierion in a manner similar o Angeleos and Pavan (2007) we obain 26 EV E E p P 2. This expression shows ha he expeced cross-secion dispersion is relaed o he ex-ane oal profi. As before we wrie ( p ) independen shocks, P as a funcion of k 0 d k m k 0 d k e m. Taking he expeced value of his expression, we obain a funcion of ω, EV 1 E p P 2 2 k k, (3.22) where Ω k is defined in (3.20). As in he former crierion, we have o consider ω = 0 o analyze he counerfacual. Wih he expression for EV 1, we can wrie he informaional effec over cross-secional dispersion as EV E EV 1 E 0 EV 1 0. As in he former crierion, we have ha: (i) for all parameers, EV is always posiive, (ii) EV is a growing funcion of ω, (iii) he counerfacual does no vary wih φ, σ, γ, and µ, (iv) EV does no change wih σ. Observaions (ii) and (iii) leads o he resul: EV is a increasing funcion of φ, γ, and µ. Therefore, we conclude ha welfare, evaluaed eiher from firms' poin of view or as he second order approximaion of agens' uiliy funcion, presens he same characerisics. 26 See Appendix G for deails.

22 Informaional Fricions and Inflaion Dynamics 69 Figure 3.5: Evoluion of EV for differen parameers 3.7. Conclusions We use a sicky-dispersed informaion model o show how he informaional power of ineres rae, defined as he informaion firms ake from he policy insrumen, changes price seing, and by consequence, inflaion dynamics. Pricing decisions modifies due o he fac ha firms aler he way hey compue expecaion on he curren sae of he economy. The main consequence on inflaion dynamics regards persisence of he moneary shock. As ineres rae is no inerial, persisence on ineres rae comes from cenral bank's reacion o informaion on changes in he fundamenal, which evolves according o a Markovian process. If firms did no see ineres rae as cenral bank's reacion funcion, any change in he policy insrumen will be undersood as an isolaed movemen ha affecs inflaion only insananeously. In conras, when ineres rae is undersood as a public signal of he fundamenal, he whole realizaion of ineres rae affecs inflaion. I is imporan o sress ha wha creaes persisence on inflaion is no how cenral bank is reacing, bu raher how firms change price seing when hey undersand ha changes in ineres rae are persisen. We also analyze how he informaional power of ineres rae affecs hree differen welfare crieria: (i) inflaion variance, (ii) cross-secional dispersion, and (iii) ex-ane oal profi. We showed ha he las wo crieria depends exclusively on he precision of he policy insrumen, ω. Therefore, o implemen he opimal

23 Informaional Fricions and Inflaion Dynamics 70 informaional ineres rule cenral bank should eiher increase as much as possible he informaional response coefficien, φ, or he precision of he moneary shock, µ. When we focus on he inflaion variance, we observe ha, as before, he precision of he moneary shock should increase as much as possible. However, φ should assume a finie and posiive value depending on he baseline calibraion of he model. Following his recommendaion, cenral bank maximizes all hree crieria a he same ime. The lessons we ake from his opimal informaional ineres rule are: (i) cenral bank should avoid adding moneary shocks o he ineres rae and (ii) here is always an opimal informaional response coefficien, φ.