Optimal Monetary Policy with Informational Frictions

Transcription

1 Opmal Moneary Polcy wh Informaonal Frcons George-Maros Angeleos MIT and NBER Jennfer La O Columba and NBER June, 208 Absrac Ths paper sudes opmal polcy n a busness-cycle seng n whch frms hold dspersed prvae nformaon abou he sae of he economy, or have a blurry undersandng abou due o raonal naenon. The nformaonal frcon s no only he source of nomnal rgdy bu also an mpedmen o he coordnaon of producon. The man lesson s ha he opmal moneary polcy does no arge prce sably; nsead, leans agans he wnd n he sense of argeng a negave relaon beween he nomnal prce level and real economc acvy. Ths polcy serves he goal of nducng he frms o ulze her nformaon and o coordnae her decsons n he bes possble manner. An addonal conrbuon s he adapaon of he prmal approach of he Ramsey leraure o a seng wh a rch form of nformaonal frcon. JEL codes: E32, E52, D6, D83. Keywords: busness cycles, ncomplee nformaon, raonal naenon, bounded raonaly, opmal polcy, prce sably. Ths paper exends, and subsumes, earler drafs ha concerned he same opc bu conaned a narrower mehodologcal conrbuon Angeleos and La O, 2008, 20. We are parcularly graeful o he edor, Harald Uhlg, and hree anonymous referees for dealed and consrucve feedback on he laes verson. We hank Rober Kng and Phlppe Bacchea for dscussng early versons of our paper, and Karhk Sasry for research asssance. We also benefed from commens receved n numerous conferences and semnars. Fnally, Angeleos acknowledges he fnancal suppor of he Naonal Scence Foundaon Award Emal: angele@m.edu, jenlao@columba.edu.

2 Inroducon In he las few years, a growng leraure explores he macroeconomc mplcaons of raonal naenon Sms, 2003, relaed forms of nformaonal frcon Woodford, 2003; Mankw and Res, 2002, and hgher-order uncerany Morrs and Shn, 998, In hs paper we sudy how such frcons affec he naure of he opmal moneary polcy and n parcular he desrably of prce sably. To hs goal, we consder a general-equlbrum macroeconomc seng n whch frms fx boh her prces and her producon decsons on he bass of ncomplee nformaon abou he sae of he economy. Ths amouns o nroducng no only a nomnal rgdy ha has been he focus of pror work, bu also an mpedmen o he coordnaon of producon. The man lesson s a novel raonale for leanng agans he wnd, ha s, for a polcy ha arges a negave relaon beween he nomnal prce level and real economc acvy. Such a polcy s opmal because provdes frms wh he rgh ncenves for how o ac on her nformaon abou he sae of he economy, as well as for how o collec such nformaon n he frs place. Ths raonale s dfferen from he one famlar from he exbook New Keynesan model. In ha conex, polces ha lean agans he wnd are jusfed by assumng ha he flexble-prce allocaons are subopmal and by leng moneary polcy subsue for mssng ax nsrumens. Furhermore, such polces nvolve a rade off beween mnmzng relave-prce dsorons and sablzng he oupu gap. By conras, none of hese properes apply n our conex. Undersandng hese suble pons and he precse naure of he opmal polcy requres a revson of he effcency benchmark relave o whch he oupu gap and he relave-prce dsorons ough o be measured once he nformaonal frcon s accommodaed. Ths brngs us o he mehodologcal conrbuon of our paper, whch s o exend he prmal approach of he Ramsey leraure, and more specfcally he mehods of Correa, Ncoln, and Teles 2008, o a framework n whch frms are nformaonally consraned. Framework. Our seng feaures a represenave household, cenralzed markes, and a connuum of monopolsc frms. Each such frm produces a dfferenaed commody, whch serves as an npu no he producon of a sngle fnal good, whch n urn can be used for consumpon and nvesmen. A benevolen Ramsey planner ses jonly he moneary and fscal polces, under full commmen. Lump-sum axaon s ruled ou, bu he ax sysem s oherwse rch enough o guaranee ha moneary polcy does no have o subsue for mssng ax nsrumens. These feaures make our framework comparable o, and ndeed nes, hose consdered n Lucas and Sokey 983, Char, Chrsano, and Kehoe 994 and Correa e al We depar from hese benchmarks by leng boh he frm s prce and s npu choces be measurable n a nosy, prvae sgnal of he aggregae sae of he economy.

3 In he man ex, he nformaonal frcon s reaed as exogenous. In Appendx C, s recas as he produc of a generalzed form of raonal naenon or of cosly nformaon acquson. Ths llusraes he flexbly of our mehods and he robusness of our nsghs. Nomnal vs Real Rgdy. The nformaonal consran on he frm s prcng-seng choce represens a nomnal rgdy. The consran on s npu choces nroduces a real rgdy. Alhough he leraure has proposed he frs feaure as an appealng subsue o scky prces and menu coss Mankw and Res, 2002; Woodford, 2003; Mackowak and Wederhol, 2009, hs feaure does no alone upse he key normave lessons of he New Keynesan paradgm. Indeed, a corollary of our analyss s ha when only prces are subjec o an nformaonal consran, he resuls of Correa, Ncoln, and Teles 2008 connue o apply: prce sably remans opmal nsofar as moneary polcy need no subsue for mssng ax nsrumens. The second feaure s herefore crucal. Because each frm condons s npu choce on a nosy and dosyncrac undersandng of he sae of he economy, producon can no longer be perfecly coordnaed across frms. As a resul, he effcency benchmark suded n Correa e al. 203 and he exsng leraure more generally s napproprae for gaugng opmal polcy n he envronmen under consderaon. Insead, he relevan benchmark embeds he real nformaon rgdy whn he feasbly consrans of he planner. I s hs mperfecon whch s responsble for he novel lessons delvered n our paper. A Prmal Approach. We sar by characerzng he enre se of he allocaons ha can be mplemened as equlbra wh he avalable polcy nsrumens. To shed lgh on he role of moneary polcy, we conduc hs exercse under wo scenaros. The one swches off he nomnal rgdy by droppng he nformaonal consran on he frms prcng decsons, he oher manans. Ths adaps he conceps of flexble-prce and scky-prce allocaons o our conex. We nex solve a relaxed problem n whch he planner faces only hree consrans: resource feasbly, he absence of lump sum axaon, and he real nformaonal rgdy dscussed above. As a resul of he laer frcon, he soluon o hs problem can dsplay posve cross-seconal dsperson n margnal producs as well as exoc busness-cycle properes such as nose or senmendrven flucuaons. These devaons could be msaken as obvous reasons for sablzaon polcy and ye hey are nheren n he planner s soluon whn hs envronmen hereby revsng he effcency benchmark relave o whch he concep of he oupu gap and ha of relave-prce dsorons should be defned. The mehodologcal par of our paper s compleed by showng ha he relaxed opmum s conaned whn he se of flexble-prce allocaons and by denfyng he combnaon of axes and For nsance, he busness cycle can be drven by forces akn o anmal sprs as formalzed n Angeleos and La O 203 and Benhabb, Wang, and Wen

4 moneary polcy ha mplemen as a scky-prce allocaon. As urns ou, he opmal axes are smlar o hose found n Lucas and Sokey 983 and Char, Chrsano, and Kehoe 994; hs s despe he fac ha axes play a novel role n our seng, namely hey may manpulae he decenralzed use of prvae nformaon. The opmal moneary polcy s dscussed nex. On Prce Sably. We now urn o he man appled conrbuon of our paper. In he New Keynesan framework, he opmal flexble-prce allocaon s ypcally replcaed wh a moneary polcy ha arges prce sably Correa, Ncoln, and Teles, We nsead show ha prce sably s nconssen wh replcaon of he opmal flexble-prce allocaon. Raher, opmaly requres a negave relaon beween he nomnal prce level and real economc acvy. The nuon for hs resul s suble, ye robus. Consder wo frms wh dfferen belefs, or dfferen degrees of opmsm, abou he sae of he economy. Because frms are raonal, such belef dfferences reflec dfferenal prvae nformaon. Furhermore, s socally opmal o le each frm condon s producon on such prvae nformaon hs s a key propery of he effcency benchmark denfed above. As a resul, s opmal for he more opmsc frm o produce more han he less opmsc one. In shor, effcency requres ha relave quanes vary wh relave belefs. In equlbrum, hs means ha relave prces mus also vary wh relave belefs; hs s smply a consequence of downward-slopng demand. When nomnal prces are flexble, he requse comovemen beween relave prces and relave belefs s rvally mplemenable. Bu now consder a world n whch frms face nformaonal consrans on her prce-seng decsons. For he more opmsc frm o produce more and charge a lower relave prce han he less opmsc frm, has o be ha he nomnal prce se by each frm s a decreasng funcon of her belef: more opmsc frms ough no only o produce more bu also o fx lower prces. Consder hen a shock ha causes a posve mass of frms o receve favorable prvae nformaon abou he underlyng economc fundamenals and he lkely level of aggregae demand. As explaned above, effcency requres ha hese frms produce more and se lower nomnal prces. Bu snce here s a posve mass of hem, he aggregae level of real economc acvy and he nomnal prce level have o move n he oppose drecon, whch explans he resul. To sum up, he documened form of leanng agans he wnd derves from hree basc properes: frms have dfferen nformaon and dfferen belefs abou he sae of he economy; here s socal value n leng relave producon vary wh relave belefs; and nomnal prces mus move n he drecon oppose o real quanes n order o mplemen he requse movemens n relave quanes and relave prces. Relaon o he Leraure. The frcons we are concerned wh are no only a pror plausble bu also conssen wh survey evdence Cobon and Gorodnchenko, 202, 205. They also help 3

5 accoun for varous salen feaures of he macroeconomc daa Angeleos and Huo, 208; Carroll e al., 208; Lorenzon, 2009; Mackowak and Wederhol, 205; Mankw and Res, Alhough a few oher works have also ouched on he queson of how such frcons affec opmal moneary polcy Ball, Mankw, and Res, 2005; Adam, 2007; Lorenzon, 200; Pacello and Wederhol, 204, our paper remans he frs o sudy hs queson n a seng n whch such frcons s he source no only of nomnal rgdy bu also of real rgdy n he sense defned earler. As already explaned, hs feaure s responsble for he novel lessons delvered n hs paper. Barrng ha feaure, he resuls of Correa, Ncoln, and Teles 2008 would have appled: he relevan effcency benchmark would no have o be modfed, and prce sably would have remaned opmal. Anoher noable aspec of our conrbuon s he flexbly of our prmal approach. Macroeconomc models wh nformaonal frcons can be hard o analyze due o he complexy n he dynamcs of hgher-order belefs. 2 As a resul, he leraure ypcally akes one of wo roues: eher mposes srong assumpons, ncludng he absence of capal accumulaon and specfc sgnal srucures, so as o solve for he equlbrum n closed form; or resors o numercal smulaons. In conras, our approach bypasses hese obsacles and delvers sharp heorecal resuls despe a flexble specfcaon of he nformaon srucure. Ths approach bulds a brdge beween he Ramsey leraure and he work of Angeleos and Pavan 2007, 2009, whch sudes effcency n a class of absrac ncomplee-nformaon games. Close, alhough no as flexble, varans of such an approach appear n Angeleos and La O 2008 and Lorenzon 200. Layou. The res of he paper s organzed as follows. Secon 2 ses up our framework. Secons 3 and 4 defne he approprae conceps of scky-prce and flexble-prce allocaons. Secon 5 defnes and characerzes he opmal allocaon. Secon 6 presens our key resuls on opmal moneary polcy. Secon 7 conans a smple, racable example ha helps llusrae he lessons of our paper more sharply. Secon 8 concludes. Appendces A and B conan he proofs. Appendx C conans an exenson wh endogenous nformaon acquson or raonal naenon. 2 The Framework In hs secon, we nroduce our framework. We frs descrbe he componens of he envronmen ha are nvaran o he nformaon srucure. We nex formalze he nformaonal frcon and s wo roles he nomnal and he real. Prelmnares. Perods are ndexed by {0,, 2,...}. There s a represenave household, whch pools all he ncome n he economy and makes consumpon, capal accumulaon, and 2 See he dscussons n Townsend 983, Huo and Takayama 205b and Nmark

6 labor supply decsons. There s a connuum of monopolscally compeve frms, ndexed by I = [0, ]. These frms produce dfferenaed goods, whch are used by a compeve real secor as nermedae npus no he producon of a fnal good. The laer n urn can be used for hree purposes: as consumpon; as nvesmen no capal; or as maerals, ha s, as nermedae npu n he producon of he dfferenaed goods. Fnally, here s a governmen, whch lacks lump-sum axaon bu can levy a varey of dsoronary axes and can ssue boh a conngen and non-conngen deb. Saes of Naure. In each perod, Naure draws a random varable s from a fne se S. Ths varable may conan no only nnovaons n he curren fundamenals, namely aggregae TFP, governmen spendng, and household preferences, bu also news abou fuure fundamenals Beaudry and Porer, 2006; Jamovch and Rebelo, 2009 or nose and senmen shocks Lorenzon, 2009; Angeleos and La O, 203. The aggregae sae of he economy, or he sae of Naure, n perod s comprsed by he hsory of draws of s τ for all τ {0,..., }. The sae s herefore an elemen of S S 0... S and s henceforh denoed by s s 0,..., s. Is uncondonal probably s denoed by µs. Tax and deb nsrumens. The governmen lacks access o boh lump-sum axaon and frmspecfc axes. I can noneheless mpose four knds of economy-wde axes: a proporonal ax on consumpon a rae τ c ; a proporonal ax on labor ncome a rae τ l ; a proporonal ax on capal ncome, ne of deprecaon, a rae τ k ; and a 00% ax on dsrbued profs. In addon, he governmen can ssue wo knds of deb nsrumens. The frs s a one-perod, non-conngen, deb nsrumen ha coss dollar n perod and pays ou + R n perod, where R denoes he nomnal neres rae beween and +. The second s a complee se of sae-conngen asses or Arrow secures. These are ndexed by s S +, hey cos Q,s dollars n perod, and hey pay ou dollar n perod + f sae s s realzed and 0 oherwse. Ther correspondng quanes are denoed by D,s. The quany of he non-conngen deb, on he oher hand, s denoed by B. The household. Le K denoe he capal sock accumulaed by he end of perod ; L he labor supply n perod ; r and w he pre-ax real values of he renal rae of capal and he wage rae n perod, respecvely; C and X he perod- real levels of consumpon and nvesmen, respecvely; and P he perod- prce level.e., he nomnal prce of he fnal good. The household s perod- budge consran can hen be expressed, n nomnal erms, as follows: +τ c P C +P X +B + Q,s D,s = τ l P w L + τ k P r K ++R B +D,s s Ss 5

7 The law of moon of he capal sock s gven by K = δk + X, where δ [0, ] s he deprecaon rae of capal. Fnally, he household s preferences are gven by her expecaon of U = β UC, L, s =0 where β 0, and U s srcly ncreasng and srcly concave n C, L. The frms. Consder monopols, ha s, he frm producng varey. Is oupu n perod s denoed by y and s gven by y = As F k, h, l, where As s an aggregae producvy shock, k s he capal npu, h s he fnal-good npu or maerals, l s he labor npu, and F s a Cobb-Douglas producon funcon. 3 The frm faces a proporonal revenue ax, a rae τ r. Is nomnal prof ne of axes s herefore gven by Π = τ r p y P r k P h P w l, where p denoes he nomnal prce of he nermedae good, P denoes he nomnal prce of he fnal good also, he prce level, and r and w denoe, respecvely, he real renal rae of capal and he real wage rae. The fnal good, n urn, s produced by a compeve real secor, whose oupu, Y, s a CES aggregaor of all he nermedae varees: [ Y = I y ] d, where >. The prof of he real secor s herefore gven by P Y I p y d and s maxmzaon yelds he demand curves faced by he monopolss. The governmen. The governmen s perod- budge consran, n nomnal erms, s gven by + R B + D,s + P G = B + s S + Q,s D,s + T where G = Gs s he exogenous real level of governmen spendng and T s he nomnal level 3 The Cobb-Douglas resrcon s wh some, bu no serous, loss of generaly. For deals, see Secon 4 n he Sepember 207 verson of hs paper hps://economcs.m.edu/fles/

8 of ax revenue, gven by T = τ r P Y + τ c P C + τ l P w L + τ k P r k + Π where Π are he aggregae frm profs. Wh some abuse of noaon, we le D = D,s s S + and Q = Q,s s S +. The planner conrols he vecor τ r, τ l, τ k, τ c, B, D along wh R, he nomnal neres rae. Fnally, o smplfy he exposon and keep he analyss comparable o ha of Correa, Ncoln, and Teles 2008, we absrac from he zero lower bound on he nomnal neres rae. Marke Clearng. Marke clearng n he goods marke s gven by C + H + X + G = Y, where X I x d s aggregae nvesmen and H I h d s he aggregae nermedaenpu use of he fnal good. Marke clearng n he labor marke, on he oher hand, s gven by I l d = L. The nformaonal frcon. The scenaro mos ofen suded n he leraure allows he frmspecfc varables p, k, h, l, y o be measurable n s, for all and all. We depar from hs benchmark by requrng ha each frm mus ac on he bass of an nosy, and dosyncrac, sgnal of s. As n he relaed leraure, he nose can be nerpreed eher as he produc of mperfec observably of he sae or as he produc of raonal naenon. More specfcally, he frcon akes he followng form. For every, s, and, Naure draws a random varable from a fne se Ω accordng o a probably dsrbuon ϕ. Ths varable represens he enre nformaon sgnal ha frm has n perod abou he underlyng sae of Naure. We denoe wh ϕ, s he jon probably of, s, wh ϕ s he probably of condonal on s, and wh ϕs he probably of s condonal on. Condonal on s, he draws are..d. across frms and a law of large number apples so ha ϕ s s also he fracon of he populaon ha receves he sgnal. 4 measurably resrcons. Fnally, we mpose he followng wo Propery. There exss funcons {h, k, l, y } such ha frm-level quanes sasfy h = h, k = k, l = l, s, y = y, s, for all, all, and all realzaons of uncerany. 4 See Uhlg 996 for an applcable law of large numbers wh a connuum of draws. 7

9 Propery 2. There exs funcons {p } such ha prces sasfy p = p for all, all, and all realzaons of uncerany. These properes consue, n effec, a defnon of nformaonally feasbly. Propery 2, whch requres p o be measurable n raher han s, nroduces he same knd of nomnal rgdy as he one feaured n Mankw and Res 2002, Woodford 2003, Mackowak and Wederhol 2009, and a growng leraure ha replaces Calvo-lke scky prces wh an nformaonal frcon. Relave o hs leraure, he key novely here s Propery. Ths adds a real frcon by requrng ha k, h be also measurable n. Fnally, leng l and hereby also y adjus o s guaranees ha supply can mee demand and markes clear for all realzaons of uncerany. 5 Inerpreaon and a few specal cases. Our formulaon allows for grea generaly. For nsance, because no resrcon s mposed on he dynamc srucure of he sgnals, we can accommodae arbrary learnng dynamcs or even he possbly of memory loss over me. Furhermore, may conan an arbrary nformaon, no only abou he fundamenals, bu also abou he belefs of oher frms. Ths allows us o accommodae rch hgher-order uncerany. Ths level of generaly hghlghs he flexbly of our prmal approach and he robusness of our lessons. I also perms us o nes a varey of specfc cases found n he leraure. To sar wh, consder models wh nosy Gaussan sgnals, as n Morrs and Shn 2002, Woodford 2003, and Angeleos and La O 200. These may be nesed by specfyng he underlyng aggregae TFP shock as a Gaussan random varable and leng each frm observe a par of sgnals abou, one prvae and one publc; see Secon 7 for an example along hese lnes. Alernavely, consder models wh scky nformaon as n Mankw and Res 2002 and Chung, Herbs, and Kley 205. Ths specfcaon s nesed n our framework by leng ϕ assgn probably µ o =, s and probably µ o =, where µ 0, s he probably wh whch a frm updaes s nformaon se wh perfec observaon of he underlyng sae and µ s he probably wh whch he frm s suck wh her old nformaon se. Fnally, consder he dfferen forms of raonal naenon found n Sms 2003, Mackowak and Wederhol 2009, Mya and Wallace 202 and Pavan 206, or he model of fxed observaon coss found n Alvarez, Lpp, and Pacello 20. For our purposes, hese approaches bol down o 5 Alhough we make a specfc modelng choce regardng whch npu choces are resrced o be conngen on, wha s essenal for our resuls s ha some npus are chosen on he bass of ncomplee nformaon, no he precse nerpreaon of hese npus. Moreover, he assumpon ha a leas one npu can adjus o he realzed s s sandard n boh he New Keynesan leraure and he recen leraure on he nformaonal foundaons of nomnal rgdy Mankw and Res, 2002; Woodford, 2003; Mackowak and Wederhol, Whou hs assumpon marke clearng would no be possble, and some form of raonng would have o be nroduced. 8

10 allowng each frm o choose s own ϕ, he jon dsrbuon of s sgnal and of he underlyng sae, and makng dfferen assumpons on se of feasble ϕ and he assocaed cognve coss. These possbles are nesed n he exenson suded n Appendx C. Relaon o Ramsey and New Keynesan leraures. When boh he nomnal and he real rgdy are assumed away meanng ha all prces and npus can be measurable n s, our framework reduces o a prooypcal Ramsey economy such as hose found n Lucas and Sokey 983 and Char, Chrsano, and Kehoe 994. More mporanly, our framework ness he New Keynesan seng of Correa, Ncoln, and Teles 2008 by droppng he measurably consran on h, k, l, y, mananng he measurably consran on p, and leng = s wh probably λ and = s wh probably λ, whch means ha a fracon λ of he frms mus se her prces one perod n advance whle he res can adjus her prces freely. Ths nesng perms us o clarfy hree elemenary pons. Frs, he earler resuls of Correa, Ncoln, and Teles 2008 drecly exend o he alernave, nformaon-based forms of nomnal rgdy consdered n Mankw and Res 2002, Woodford 2003, Mackowak and Wederhol 2009, and Alvarez, Lpp, and Pacello 20. Second, hese earler resuls do no drecly apply o our seng because, and only because, of he real rgdy formalzed n Propery above and of he assocaed mperfecon n he coordnaon of producon. And hrd, hs mperfecon s he sole source of our resul regardng he opmaly of moneary polces ha lean agans he wnd. All hese pons wll be made clear n he sequel. 3 Scky vs Flexble Prces: Defnons The dual role of he nformaonal frcon as boh a source of nomnal and real rgdy s a defnng feaure of our framework. Accordngly, we are ulmaely neresed n he scenaro n whch boh rgdes are presen. To undersand he role of moneary polcy n hs scenaro, s neverheless nsrumenal o sudy he alernave scenaro n whch he nomnal rgdy s arfcally shu down by leng all prces be measurable n s. Borrowng, and paraphrasng, he ermnology of he New Keynesan leraure, we henceforh refer o he scenaro ha embeds he nomnal rgdy as scky prces and o he one ha assumes away as flexble prces. In hs secon, we defne he ses of allocaons, prces, and polces ha can be par of an equlbrum under each scenaro. To sar wh, we nroduce some useful noaon. We henceforh represen an allocaon by a sequence ξ {ξ.} =0, where ξ. {k., h., l., y.; K., H., L., Y., C.} s a vecor of funcons ha map he realzaons of uncerany o he quanes chosen by he 9

11 ypcal frm for he frs four componens of ξ and he aggregae quanes for he remanng fve componens. We smlarly represen a prce sysem by a sequence ϱ {ϱ.} =0, where ϱ. {p., P., r., w., Q.} s a vecor of funcons ha map he realzaons of uncerany o he prce se by he ypcal frm, he aggregae prce level, he real wage rae, he real renal rae of capal, and he prces of he Arrow secures. We fnally represen a polcy wh a sequence θ = {θ.} =0, where θ. { } τ r., τ l., τ k., τ c., B., D., R. s a vecor of funcons ha map he realzaons of uncerany o he varous polcy nsrumens. Throughou our analyss, we le he doman of K., H., L., Y., C., P., r., w., Q., τ r., τ l., τ k., τ c., B., D., and R. be S. Ths reflecs he fac ha our analyss absracs from nformaonal frcons on he sde of eher he represenave household or he governmen. In conras, he nformaonal frcon of he frms s embedded n Properes and 2. We fnally express he aggregae level of oupu and he aggregae prce level as follows: Y s = [ Ω y, s ϕ s ] and P s = [ Ω p ϕ s ]. We can hen defne our noon of scky-prce equlbra as follows. 6 Defnon. A scky-prce equlbrum s a rple ξ, ϱ, θ of allocaons, prces, and polces ha sasfy Properes and 2 and are such ha: {C, L, K, B, D } solves he household s problem; {p, k, h, l, y } solves he frm s problem; he quany and he prce of he fnal good are gven by ; he governmen s budge consran s sasfed; v all markes clear. We nex defne our noon of flexble-prce equlbra by droppng he measurably consran on prces. Formally, we replace Propery 2 wh he followng propery: Propery 2. The prces sasfy p = p, s for all, all, and all realzaons of uncerany. 6 The only essenally novely n he defnon s he par of measurably consrans mposed on he frm s problem. The precse formulaon of hs problem, as well as ha of he household s problem, can be found n Appendx A. 0

12 Accordngly, we adjus he formula for he prce level n condon as follows: P s = p, s ϕ s 2 We can hen sae he relevan defnon as follows. Defnon 2. A flexble-prce equlbrum s a rple ξ, ϱ, θ of allocaons, prces, and polces ha sasfy he same condons as hose saed n Defnon, excep ha Propery 2 s replaced by Propery 2 and, accordngly, he prce level s gven by condon 2. We le X f and X s denoe he ses of he allocaons ha are par of a flexble-prce and a sckyprce equlbrum, respecvely. We also le X denoe he superse of all feasble allocaons, by whch we mean allocaons ha sasfy he economy s resource consrans along wh Propery. 4 Scky vs Flexble Prces: Characerzaon and Replcaon In hs secon we characerze, and compare, he ses of he allocaons ha can be par of eher a flexble-prce or a scky-prce equlbrum. Flexble-Prce Allocaons. Consder any flexble-prce equlbrum. The characerzaon of he household s problem s sandard. The characerzaon of he monopols s problem s slghly more exoc due o he heerogeney n he sgnal upon whch he npu choces are based. To conserve on noaon, we henceforh le, for any z {l, h, k}, MP z, s y, s As Y s z F k, h, l, s. In he eyes of he planner, MP z represens he margnal produc of npu z n frm, expressed n erms of he fnal good; n he eyes of he frm, capures he correspondng margnal revenue produc once s mulpled by χ, he recprocal of one plus he monopoly markup. As shown n Appendx A, we can express he frs-order condons of he frm as follows: τ r s χ MP l, s ws = 0,, s 3 E [ Ms { τ r s χ MP h, s } ] E [ Ms { τ r s χ MP k, s rs } ] = 0, 4 = 0,. 5 where Ms Ucs +τ c s and U c s s a shor-cu for he margnal uly of consumpon.

13 These condons have a smple nerpreaon. The frm seeks o equae he cos of each npu wh s afer-ax margnal revenue produc. The only dfference among he hree condons s he exen o whch hs goal s acheved. Because labor s conngen on he realzed sae s, s margnal revenue produc s equaed wh he real wage sae-by-sae. By conras, he oher wo condons hold only on average, ha s, n expecaon condonal on he frm s sgnal. Combnng he opmaly condons of he frm wh hose of he household, mposng marke clearng, and solvng ou for he prces and he polcy nsrumens, we reach he followng resul. Proposon. A feasble allocaon, ξ X, s par of a flexble-prce equlbrum f and only f he followng wo properes hold. The allocaon sasfes β µ s [ U c s C s + U l s L s ] = 0. 6,s For every, here exs funcons ψ r, ψ l, ψ c, ψ k : S R + such ha ψ r s χ MP l, s ψ l s = 0, s 7 E [ ψ r s χ MP h, s ψ c s ] = 0 8 E [ψ r s χ MP k, s ψ k s ] = 0 9 Condon 6 s famlar from he Ramsey leraure. I encapsulaes he absence of lump-sum axaon and follows drecly from he neremporal budge consran of he governmen, afer replacng he equlbrum prces and he polcy nsrumens n erms of he allocaon. Consder nex condons 7, 8 and 9. Were he nformaonal frcon absen, each frm would know s and hese condons would reduce o, respecvely, MP l, s = ψl s ψ r s, MP h, s = ψc s ψ r s, and MP k, s = ψk s ψ r s,,, s. Snce he ψ s are free varables, hese condons would requre ha he margnal produc of each npu s equaed across all frms, for all and s. Ths defnes wha we call perfec coordnaon n producon. I also means ha he sole role of he ax nsrumens n ha benchmark s o conrol he wedges beween he common MRTs of he frms and he correspondng MRSs of he household. When nsead he nformaonal frcon s presen, each frm condons her choces on an dosyncrac sgnal of s. As a resul, margnal producs are ypcally no equaed across frms. Ths manfess as an aggregae TFP loss lke ha quanfed n Davd, Hopenhayn, and Venkaeswaran 206. I also means ha he avalable ax nsrumens may play a new role: her conngency 2

14 on s nfluences how frms ulze her prvae nformaon. Ths enables he planner o conrol no only he response of aggregae oupu o aggregae TFP and oher shocks, bu also he crossseconal dsperson n produced quanes and margnal producs. I s hs new role of he axes ha s encoded no condons 7-9. We llusrae hs pon wh an example n Secon 7. Scky-Prce Allocaons. We now add back he nomnal rgdy Propery 2. As n he New Keynesan model, hs allows he realzed monopoly markup o flucuae around he deal one. Formally, here now exss a random varable χ, s, represenng he recprocal of he realzed markup, such ha he followng properes are rue. Frs, he opmaly condons 3-5 are modfed by replacng χ wh χ, s. And second, he followng opmaly condon s added: E [Ms Y s / y, s / τ r s { χ, s χ } ] = 0 0 Ths condon capures he opmal prce-seng behavor of he frm. I requres, n essence, ha he rsk-adjused expecaon of he realzed markup concdes wh he deal one. Adapng Proposon o hese modfcaons, we reach he followng resul. Proposon 2. A feasble allocaon, ξ X, s par of a scky-prce equlbrum f and only f he followng hree properes hold. The allocaon sasfes 6. For every, here exss funcons, ψ r, ψ l, ψ k, ψ c : S R + and χ : Ω S R + such ha he followng condons hold: χ, s ψ r s MP l, s ψ l s = 0, s E [ χ, s ψ r s MP h, s ψ c s ] = 0 2 [ E χ, s ψ r s MP k, s ] ψ k s = 0 3 E [Y s / y, s / ψ r s { χ, s χ } ] = 0 4 The funcon χ : Ω S R + s log-separable n he sense ha here exs posve-valued funcons χ and χ s such ha log χ, s = log χ + log χ s s, s. 5 Clearly, he only dfferences from Proposon are he emergence of he wedge χ, s n condons -3 and he addon of condons 4 and 5. As already explaned, condon 4 follows from he opmal prce-seng behavor of he frm. Condon 5, on he oher hand, follows from he so-elasc demand srucure; see he Appendx for deals. 3

15 Replcaon. Through he lens of Proposon 2, χ, s represens an addonal conrol varable for he planner, one ha encapsulaes he power of moneary polcy over real allocaons. Ths power s non-rval, bu s also resraned by condons 4 and 5. Snce boh condons are auomacally sasfed by leng χ, s = χ, he followng s mmedae. Corollary. Every flexble-prce allocaon can be replcaed as a scky-prce allocaon: X f X s. Ths proves ha an approprae moneary polcy can undo he nomnal rgdy, bu does no ell us wheher such a polcy s opmal or how looks lke. We address hese quesons nex. 5 The Ramsey Opmum In hs secon we defne and characerze he effcency benchmark ha s relevan for our purposes. Ths leads o our man resuls regardng he opmal moneary polcy. An approprae effcency benchmark. Our ulmae goal s o solve he problem of a Ramsey planner who maxmzes welfare over X s, he se of scky-prce allocaons. To hs goal, we frs characerze he allocaon ξ ha maxmzes welfare over an enlarged se, denoed by X R and conssng of all echnologcally and nformaonally feasble allocaons ha sasfy only condon 6. Tha s, from he sx mplemenably consrans seen n Proposon 2, we manan he frs one, whch encapsulaes he absence of lump-sum axaon, bu drop he remanng ones. Ths s akn o allowng he planner o mpose a compleely flexble se of npu- and sgnal-specfc axes. Proposon 3. There exss a consan Γ 0, capurng he shadow value of governmen revenue, such ha ξ, he opmal allocaon over he enlarged se X R, s gven by he feasble allocaon ha sasfes he followng condons: Ũ c s MP l, s + Ũl s = 0, s 6 E [Ũc s { MP h, s } ] = 0 7 E [Ũc s { MP k, s κ s } ] = 0 8 for some funcon κ : S R + ha capures ha ne-of-ax renal rae of capal and ha sasfes Ũ c s = βe [Ũc s + { + κ s + δ } s ] s, 9 where Ũcs and Ũl s are shorcus for C Ũ C s, L s, s ; Γ and LŨ C s, L s, s ; Γ, respecvely, and where Ũ C, L, s; Γ U C, L, s + Γ [ C C U C, L, s + L L U C, L, s]. 4

16 To undersand hs resul, momenarly shu down he nformaonal frcon. condons 6-8 reduce o he followng: MP l s = Ũl s Ũ c s, In hs case, MP h s =, and Ũ c s = βe [Ũc s + δ + MP k s + ] s, where MP z s now denoes he common margnal produc of npu z n all frms. The frs condon s dencal o he one found n Lucas and Sokey 983 and denfes he opmal ax on labor. The second condon mples ha he ax on he nermedae npu s zero, an example of he resul n Damond and Mrrlees 97: axes should no nerfere wh producve effcency. The las condon s dencal o ha found n Char, Chrsano, and Kehoe 994 and relaes o he celebraed Chamley-Judd resul abou he opmaly of zero axes on capal ncome. Now add back n he nformaonal frcon. In general, opmaly requres ha each frms condon her choces on her prvae nformaon abou he underlyng sae. Because such nformaon conans dosyncrac nose, he margnal producs are no more equaed across he frms. In comparson o he prevous leraure, hs propery may be msnerpreed as a sympom of producve neffcency and relave-prce dsorons; bu hrough he lens of Proposon 3, s undersood as he by-produc of he socally opmal decenralzed use of nformaon. Ths explans how our analyss revss he concep of relave-prce dsorons. Proposon 3 also revses he concep of he oupu gap. Because he CES srucure mples ha he socal value of producng an exra un of any gven good ncreases wh he quanes of oher goods, he planner fnds opmal o le he frms coordnae her npu choces. 7 means ha he hrd bes characerzed here allows a frm s producon o vary, no only wh her nformaon abou he underlyng fundamenals, bu also wh her belefs abou he belefs of oher frms. The busness cycle can hus be drven by seemngly exoc senmens, of he knd formulaed n Angeleos and La O 203 and Benhabb, Wang, and Wen 205 and quanfed n Angeleos, Collard, and Dellas 207 and Huo and Takayama 205a. Under a radonal polcy perspecve, such flucuaons can be msnerpreed as flucuaons n he oupu gap; bu hrough our analyss, hey are recas as flucuaons n poenal oupu. To sum up, no only do he observable properes of he opmum have o be modfed, bu also he famlar goals of mnmzng relave-prce dsorons and sablzng he oupu gap mus be revsed before we may undersand he role of moneary polcy. Implemenaon. We now show how he opmum characerzed n Proposon 3 can be mplemened wh he avalable polcy nsrumens. 7 Formally, he opmal allocaon can be undersood as he Perfec Bayesan Equlbrum of a game of sraegc complemenary, n lne wh he more absrac analyss n Angeleos and Pavan Ths 5

17 Recall ha X R s a superse of boh X f and X s because allows he planner o make he producon choces of each frm an arbrary funcon of her prvae nformaon, whereas X f and X s resran ha conrol n he manner descrbed n par of, respecvely, Proposons and 2. Ye, he addonal conrol afforded by X R s mmaeral for opmaly. In parcular, as shown n Appendx A, we have ha ξ X f, meanng ha ξ can be mplemened as a flexble-prce equlbrum. By Corollary, we have X f X s. I follows ha ξ can be mplemened as a scky-prce allocaon wh a moneary polcy ha replcaes flexble-prces. And because X s X R, we have ξ maxmzes welfare over all scky-prce allocaons. Combnng hese fndngs, and denfyng he axes ha suppor ξ as an equlbrum, we reach he followng resul. Theorem. The allocaon ξ obaned n Proposon 3 denfes he opmal allocaon and s mplemened wh: a moneary polcy ha replcaes flexble prces; and he followng se of axes: τ l s + τ c s = U l s /U c s Ũ l s /Ũc s, τ k s =, τ r s =, + τ c s = δ U c s Ũ c s 20 where U c, U l, Ũc, and Ũl are evaluaed a ξ and where δ > 0 s any sae-nvaran scalar. Par exends he relaed resul of Correa, Ncoln, and Teles 2008 o he class of economes under consderaon. Par generalzes he opmal axaon resuls of Lucas and Sokey 983 and Char, Chrsano, and Kehoe 994. There are, however, hree suble dfferences. Frs and foremos, replcang flexble prces s no more synonymous o argeng prce sably. We expand on hs pon n he nex secon. Second, he relevan wedges are evaluaed a an allocaon whose observable properes may dffer from hose characerzed n he aforemenoned works, for he reasons already explaned. Ths opens he door o he possbly ha he cyclcal properes of he opmal axes are dfferen even hough he ax formulas obaned are essenally he same. Thrd, he consumpon ax may play a novel role. In he exsng leraure, τ c s ypcally resrced o be zero and hs resrcon s whou loss of opmaly nsofar as publc deb s sae-conngen and he he zero lower bound on he nomnal neres rae s non-bndng. Here, nsead, s generally necessary o le τ c vary wh he sae of Naure so as o make sure ha he frms face he rgh prce of rsk when seng her prces. The las wo sublees can be sdesepped by mposng he followng, homohec specfcaon for preferences: U C, L = C γ γ η L+ɛ + ɛ, 2 6

18 for γ, ɛ, η > 0. In hs case, he opmal allocaon s mplemened wh a sae-nvaran ax on labor, a zero ax on capal, and a zero ax on consumpon; See Lemma 5 and s proof n Appendx A. From an appled perspecve, he mos mporan lesson herefore seems o be he ncompably of replcang flexble prces wh argeng prce sably, o whch we urn nex. 6 On he Opmal Cyclcaly of he Prce Level Whn he New Keynesan framework, he logc n favor of prce sably s ha mnmzes relave-prce dsorons or, equvalenly, maxmzes producve effcency. We now explan why hs logc s upse once he nformaonal frcon s aken no consderaon and he effcency benchmark s revsed along he lnes we descrbed n he prevous secon. We sar by nong ha, along he opmal allocaon, he oupu of each frm can be expressed as he logarhmc sum of wo componens: one measurable n he frm s prvae nformaon and he oher measurable n he realzed sae. Lemma. There exs posve-valued funcons Ψ and Ψ s such ha, along he opmal allocaon, he oupu of a frm can be expressed as log y, s = log Ψ + log Ψ s s, s. 22 The precse values of hese componens follow from he soluon o he opmaly condons n Proposon 3. In Appendx A see, n parcular, he proof of Lemma 2, we show how Ψ may be expressed as a funcon of he npu choces ha frms make on he bass of her mperfec observaon of he sae of he economy, whereas Ψ s capures he adjusmen n he labor npu ha akes place n order for supply o mee he realzed demand, and markes o clear, a he se prces. In he example suded n he nex secon, all hese objecs can be solved n closed form as smple funcons of he avalable sgnals. I hen becomes evden how he opmal npu choces and he aforemenoned oupu componens covary wh he sae of he economy. For he presen purposes, however, suffces o noe he followng general pons. Whereas Ψ s capures he componen of he oupu ha s common across all frms, Ψ capures he componen ha s drven by each frm s prvae nformaon. The laer componen can be hough of as a proxy of he frm s dosyncrac belef abou he sae of he economy. Along he opmal allocaon, hs ypcally means ha an opmsc frm s assocaed wh a hgher Ψ, and produces more, han a pessmsc one. Furhermore, Ψ s he only source of varaon n relave quanes and, hereby, n relave 7

19 prces. Indeed, by he relave demand for he goods produced by frms and j, we have log p log p j = [ log ys, log ys, j ] Usng condon 22, we hen ge log p log p j = [ log Ψ log Ψ j ], whch verfes ha he relave prce of any wo frms s nversely relaed o her relave belef, as measured by he log-dfference beween Ψ and Ψ j. Inuvely, f opmsc frms are o produce more han pessmsc ones, hey mus also charge lower relave prces. Ths elemenary nsgh underles our resul regardng he subopmaly of prce sably. As long as frm does no know j and, symmercally, frm j does no know, her relave prce can be nversely relaed wh her relave quany only f he nomnal prce of frm s self negavely relaed o her belef, as capured by Ψ, and smlarly for j. Formally, has o be ha log p = z log Ψ, for some varable z ha s commonly known o he frms meanng ha he prces of all he frms can be conngen on z. Aggregang he above, we ge ha he aggregae prce level mus sasfy log P s = z log Bs, 23 where [ Bs Ψ ] dµ s. We hus reach he followng resul. Theorem 2. Along any scky-prce equlbrum ha mplemens he opmal allocaon, he prce level s negavely correlaed wh he average belef and real economc acvy, as proxed by Bs. Ths s our man resul regardng he sub-opmaly of prce sably. Is applcably hnges on relang he objec Bs o a more concree measure of economc acvy. Ths s done n Secon 7 whn an example ha allows for an explc soluon of he opmal allocaon and he opmal prce level. Tha example reles on assumng away capal and mposng a Gaussan nformaon srucure. Bu even whou hese resrcons, he followng resul can be shown. Lemma 2. Along any scky-prce equlbrum ha mplemens he opmal allocaon, Bs s, o a frs-order log-lnear approxmaon, a log-lnear combnaon of he aggregae quanes of frm npus; Bs s herefore procyclcal f npus are also procyclcal. 8

20 Ths corroboraes he nerpreaon of Bs as proxy for he aggregae level of economc acvy and he nerpreaon of Theorem 2 as a case for leanng agans he wnd. The logc for our resul follows drecly from our earler dscusson abou he relaon beween relave prces and relave belefs. Because opmaly requres ha he oupu of each frm vares wh s belef abou he sae of he economy, and because relave prces are nversely relaed o relave quanes, he nomnal prce of a frm has o move n he oppose drecon ha s belef and s oupu. A he aggregae level, hs ranslaes o negave co-movemen beween he prce level and real oupu propery ha resembles nomnal GDP argeng. I s worh nong, however, wo sublees. Frs, Theorem 2 allows for a ceran degree of nomnal ndeermnacy: as evden n condon 23, he prce level s ndeermnae vs-a-vs any varable z ha s common knowledge o he frms. Ths s because frms can perfecly coordnae her prce responses o any such shock, whch n urn guaranees ha varyng he response of moneary polcy o z affecs he varaon n he prce level whou affecng he real allocaons. 8 Second, Theorem 2 conans also a case for prce sably: f he opmal allocaon s nvaran wh a shock, hen s opmal o sablze he prce level vs-a-vs ha shock. Consder, for example, a pure sunspo, namely a shock ha s orhogonal no only o he underlyng fundamenals bu also o he enre herarchy of belefs abou hem. Alernavely, absrac from capal accumulaon and consder a shock o belefs of fuure TFP. In eher case, he opmal allocaon remans sable. If he moneary auhory fals o sablze he prce level wh respec o he shock under consderaon, he producon of a posve mass of frms wll vary wh, conradcng opmaly. We close hs secon by emphaszng ha our resul hnges on allowng he nformaonal frcon o be a real frcon n he sense of Propery. We formalze hs pon below. Proposon 4. Suppose we manan Propery 2 bu drop Propery ; ha s, we manan he nomnal role of he nformaonal frcon bu drop he real one. Then, he opmal allocaon s mplemened by argeng prce sably. Ths s essenally he man resul of Correa, Ncoln, and Teles Recall ha he seng n ha paper may be nesed n our framework when he real rgdy s assumed away and he nomnal rgdy s such ha a fracon λ of he frms se her prces one perod n advance n whch case = s whle he remanng are free o adjus her prces n whch case = s. Proposon 4 herefore replcaes he man resul of ha paper and also exends he alernave, nformaon-based foundaons of he nomnal rgdy proposed by Mankw and Res 2002, Woodford 2003, Mackowak and Wederhol 2009, and ohers. 8 The source of hs ndeermnacy s smlar o ha n he older leraure on nomnal confuson Lucas, 972; Barro, 976; s clearly welfare-rrelevan n our seng; and can refned away by mposng ha no shock s common knowledge. We suspec hs ndeermnacy dsappears also f we add a Calvo frcon, even a ny one, for hs helps anchor he opmal prce level a all 0 o P, he hsorcal prce level. 9

21 To sum up, wha drves he parcular knd of leanng agans he wnd documened n our paper, s precsely he real be of he nformaonal frcon, capured heren by Propery. 9 7 An Illusraon In hs secon we use a racable Gaussan example one smlar o hose suded n Woodford 2003 and Angeleos and La O 200 o llusrae he man lessons of our paper. In parcular, we frs demonsrae how he polcy nsrumens can manpulae he decenralzed use of nformaon and can possbly nsulae aggregae oupu from he effecs of nose, senmens, and he lke. We nex characerze he opmal allocaon, conras o s complee-nformaon counerpar, and show how he prce level moves n he oppose drecon han aggregae oupu. Se up. We absrac from capal, le governmen spendng be consan, specfy preferences as n condon 2, and add dosyncrac TFP shocks. The producon funcon s hus gven by y = A h η α l α, 24 where α 0, and η 0, and where A, he producvy of frm n perod, s comprsed of boh an aggregae and a frm-specfc componen. In parcular, a log A = a + v, where a log A s he aggregae componen and v s he dosyncrac one. The processes of a and v are Gaussan, saonary, and orhogonal o one anoher. The dosyncrac componen v s..d. across frms bu can be correlaed over me whn a frm. The aggregae componen a can also be correlaed over me. We fnally le each frm know s own producvy, a, bu no he underlyng aggregae componen, a. The resuls presened below mpose no furher resrcons on he process for a and he avalable sgnals abou. Ths perms us o accommodae rch learnng dynamcs as well as rch hgherorder uncerany. For nsance, by leng a follow an AR process and each frm observe a nosy prvae sgnal of a n each perod, we can accommodae he knd of neral belef dynamcs suded n Woodford 2003, Nmark 2008, Angeleos and Huo 208, and elsewhere. To fx deas, however, he reader may resrc aenon o he specal case n whch a s..d. over me and frm s nformaon n perod s gven by he par a, z, where z s a nosy 9 Ths also explans why Adam 2007 and Pacello and Wederhol 204, whch absrac from he real rgdy ha has been he focus of our paper, le moneary polcy subsue for mssng ax nsrumens. Ball, Mankw, and Res 2005 also absrac from he real rgdy, bu focus on a dfferen ssue, he ranson from a subopmal o an opmal polcy. 20

22 sgnal gven by z = a + σ υ υ + σ ɛ ɛ 25 where ɛ and υ are, respecvely, dosyncrac and aggregae noses, ndependen of one anoher and of a. We le scalars σ ɛ > 0 and σ υ > 0 parameerze he level of he wo noses, respecvely. In hs example, he shock υ s a source of correlaed nose n frms frs- and hgher-order belefs. Also noe ha he case of a nosy publc sgnal s nesed by leng σ ɛ 0, whereas he case wh purely prvae nformaon and no aggregae nose s nesed by leng σ υ 0. Remark. As already noed, he example nroduced above can be hough of as a hybrd of Woodford 2003 and Angeleos and La O 200. Woodford 2003 assumes away he real rgdy and les moneary polcy nduce an exogenous Gaussan process for nomnal GDP. Angeleos and La O 200 shus down he nomnal rgdy and absracs from boh fscal and moneary polcy. Relave o hese earler works, we no only combne he wo forms of rgdy n he same example, bu also work ou he opmal polcy. Manpulang he Decenralzed Use of Informaon. Before characerzng he opmal polcy, we fnd useful o llusrae how he sae conngency of he polcy nsrumens can nfluence he decenralzed use of nformaon and hereby he sochasc process of aggregae oupu. To hs end, we specfy he ax sysem such ha he relevan wedges are log-lnear funcons of aggregae producvy and aggregae oupu only. In parcular, we mpose log τ r A, Y = ˆτ 0 + ˆτ A log A + ˆτ Y log Y for some scalars ˆτ 0, ˆτ A, ˆτ Y R. We hen le he remanng ax raes sasfy τ k s = τ c s = 0 and +τ l s = / τ r s. The scalars ˆτ 0, ˆτ A, ˆτ Y can hen be hough of as he polcy coeffcens. Fnally, here and for he res of hs secon, we consder he log-lnearzed approxmaon of he equlbrum allocaons around he seady sae n whch A akes s uncondonal mean value. Proposon 5. Consder he economy and he axes descrbed above. In any flexble-prce equlbrum, GDP sasfes, up o a log-lnear approxmaon, log GDP s = γ 0 + γ A log A + γ u u, 26 where he scalars γ 0, γ A, γ u are pnned down by he polcy coeffcens ˆτ 0, ˆτ A, ˆτ Y and where u s a Normally dsrbued random varable, orhogonal o log A, wh mean 0 and varance. Furhermore, by appropraely choosng he polcy coeffcens ˆτ A, ˆτ Y, he planner can mple- 2

23 men any par γ A, γ u nsde he se Υ, where Υ { γ A, γ u R 2 : eher γ u > 0 and γ A > ˆγ + γ u, or γ u < 0 and γ A < ˆγ + γ u } 27 and where ˆγ s a consan ha depends on he underlyng preference, echnology, and nformaon parameers bu s nvaran o polcy and he mplemened allocaon. To undersand hs resul, noe ha u s he sandardzed resdual of regressng aggregae oupu on he curren aggregae producvy. Ths resdual s zero n he absence of he nformaonal frcon bu no when s presen. For nsance, n he aforemenoned specal case n whch a s..d. over me and he sgnals are as n condon 25, u concdes wh υ, he aggregae nose n he avalable sgnals. More generally, u encapsulaes all aggregae varaon n he frms frs- and hgher-order belefs ha s orhogonal o he curren fundamenals TFP. Wh hese pons n mnd, Proposon 5 can be read as follows: by appropraely desgnng he coeffcens ˆτ A and ˆτ Y, he planner can affec boh he covaraon of aggregae oupu wh he curren fundamenal and s resdual varaon due o nose or hgher-order uncerany. Ths s because hese ax coeffcens conrol how sensve a frm s ne-of-axes revenue s o, respecvely, TFP and he acons of oher frms. As a resul, hese coeffcens ndrecly conrol he ncenves each frm has n reacng o dfferen peces of nformaon abou hese objecs. In sum, polcy coeffcens may be used o conrol he decenralzed use of nformaon. I can be shown ha a smlar resul apples o moneary polcy wh he analogues of ˆτ A and ˆτ Y beng he responsveness of he nomnal neres rae o aggregae producvy and aggregae oupu, respecvely. Ths llusraes our pon ha famlar polcy nsrumens, wheher fscal or moneary, play novel roles once he nformaonal frcon s accommodaed. The Ramsey Opmum. Consder, as a reference pon, he opmal allocaon n he absence of he nformaonal frcon; hs corresponds n effec o he Lucas-Sokey benchmark. In hs case, can be shown ha aggregae oupu s gven by log Y = γ LS 0 + γ LS A log A, for some scalars γ0 LS and γa LS > 0 ha depend on he preferences, echnology, and level of governmen spendng or he ax dsoron. Consder now he case n whch he nformaonal frcon s presen. By Proposon 5, here exs polces such ha aggregae oupu s gven by 26 wh γ A = γa LS and γ u = 0. Tha s, s feasble for he planner o boh nduce he same covaraon beween aggregae oupu and aggregae producvy as n he frconless benchmark and o nsulae aggregae oupu from nose, senmens, ec. 22

24 Ths s made possble by combnng a ˆτ Y hgh enough so ha he ne-of-axes reurns are nvaran o aggregae oupu and a ˆτ A low enough so ha he ne-of-axes reurns are suffcenly sensve o aggregae producvy. The former propery guaranees ha he frms dsregard nformaon ha s useful n predcng he choce of oher frms bu s no useful n predcng aggregae producvy; he laer ensures ha he frms respond wh enough srengh o varaon n aggregae producvy. Ths may sound lke a wn-wn suaon. Bu s no. When he planner nduces he frms o dsregard nformaon abou one anoher s choces over nformaon abou he fundamenals, she exacerbaes he ms-coordnaon of producon and mplemens an neffcenly hgh level of crossseconal dsperson n quanes. To economze on hs margn, he opmal allocaon allows he frms o ulze ha knd of nformaon, hus also allowng aggregae oupu o move wh nose, senmens, ec. Tha s, opmaly calls for γ u > 0. For essenally he same reason, he opmal allocaon also nduces a lower covaraon beween aggregae oupu and aggregae producvy han n he frconless benchmark: he alernave requres ha he frms respond oo srongly o her prvae nformaon and nduces oo much cross-seconal dsperson n quanes. Tha s, opmaly calls for γ A < γ LS A. These pons are esablshed formally n he nex proposon, whch characerzes he process of aggregae oupu along he opmal allocaon. Proposon 6. In any equlbrum ha mplemens he opmal allocaon, GDP s gven by log GDP = γ 0 + γ A log A + γ uu, 28 where u s a Normally dsrbued random varable, orhogonal o log A, wh mean 0 and varance, and where he scalars γ A and γ u are unquely deermned by he underlyng preference, echnology, and nformaon parameers. Furhermore, 0 < γ A < γ LS A and γ u > Ths resul llusraes how he effcency benchmark denfed n our paper dffers from ha found n he leraure. Frs, GDP feaures a lower sensvy o he underlyng fundamenal han n he Lucas-Sokey benchmark. And second, GDP vares wh nose, senmens, belefs, ec. Moneary Polcy. We now urn aenon o he opmal moneary polcy and he assocaed prce level. Theorems and 2, of course, apply. The goal s o llusrae he parcular form of leanng agans he wnd ha obans n he example under consderaon. 23