Technical Appendix for Central Bank Communication and Expectations Stabilization

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1 Techncal Appendx for Cenral Bank Communcaon and Expecaons Sablzaon Sefano Eusep Federal Reserve Bank of New York Bruce Preson Columba Unversy and NBER Augus 0, 008 Absrac Ths echncal appendx provdes some calculaons underlyng he model used n Eusep and Preson (008). Conens A Smple Model. Mcrofoundaons Marke clearng, e cen oupu and aggregae dynamcs

2 A Smple Model The followng deals a smple model of oupu gap and n aon deermnaon ha s smlar n spr o Goodfrend and Kng (997), Roemberg and Woodford (999) and Svensson and Woodford (005). A connuum of households face a canoncal consumpon allocaon problem and decde how much o consume of avalable d erenaed goods and how much labor o supply o rms for he producon of such goods. A connuum of monopolscally compeve rms produce d erenaed goods usng labor as he only npu and face a prce seng problem of he knd proposed by Roemberg (98). The major d erence s he ncorporaon of non-raonal belefs, delverng an ancpaed uly model. The analyss follows Marce and Sargen (989) and Preson (005), solvng for opmal decsons condonal on curren belefs. Varous mechansms of perssence, such as hab formaon, prce ndexaon and neral moneary polcy are absraced from. Ths provdes sharp, perspcuous analycal resuls. An earler verson of hs paper, Eusep and Preson (007a), demonsraes ha our conclusons regardng he value of communcaon n polcy desgn reman pernen n models wh such mod caons.. Mcrofoundaons Households. and lesure Households maxmze her neremporal uly derved from consumpon subjec o he ow budge consran ^E X T ln CT h T B R B + W h + P P C T where B denoes holdngs of he one perod rskless bond, R denoes he gross neres pad on he bond, W he nomnal wage, h labor suppled by household and T lump-sum axes An analyss of prce seng of he knd proposed by Calvo (983), as mplemened by Yun (996), would lead o smlar conclusons. I s also movaed by Mlan (006) and Eusep and Preson (008a) whch sugges ha purely forward lookng busness cycle models wh learnng dynamcs provde a superor characerzaon of varous U.S. macroeconomc me seres han do raonal expecaons models wh varous perssence mechansms.

3 and ransfers for household. Fnancal markes are assumed o be ncomplee and denoes pro s from holdng shares n an equal par of each rm. Nomnal ncome n any perod s P Y = W h + P and P s he aggregae prce level de ned below. ^E denoe he belefs a me held by each household ; whch sasfy sandard probably laws. Secon 3 descrbes he precse form of hese belefs and he nformaon se avalable o agens n formng expecaons. However, wo pons are worh nong. Frs, n formng expecaons, households and rms observe only her own objecves, consrans and realzaons of aggregae varables ha are exogenous o her decson problems and beyond her conrol. They have no knowledge of he belefs, consrans and objecves of oher agens n he economy: n consequence agens are heerogeneous n her nformaon ses n he sense ha even hough her decson problems are dencal, hey do no know hs o be rue. Second, gven he assumed condonng nformaon for expecaons formaon, consumpon plans are made one perod n advance and herefore predeermned. 3 Labor supply decsons are no predeermned and are condoned on perod nformaon. 4 Each household consumes a compose good Z C = 0 c (j) dj whch s made of a connuum of d erenaed goods, c (j), each produced by a monopolscally compeve rm j. The elascy of subsuon among d erenaed goods,, s me-varyng, wh E [ ] = >. Ths s a smple way of modelng me-varyng mark-ups, nroducng a rade-o beween n aon and oupu sablzaon relevan o opmal polcy desgn. A log-lnear approxmaon o he rs order condons of he household problem provdes he household Euler equaon ^C = ^E h ^C + (^{ + ) 3 We consder a model wh prcng and spendng decsons deermned one perod n advance so as o pu households, rms and polcymakers on an dencal nformaonal foong. Ths could smlarly be acheved by he alernave assumpon ha he cenral bank has a polcy reacon funcon ha responds o one perod ahead expecaons of n aon and agens condon decsons on perod nformaon. All resuls connue o hold. 4 Ths assumpon ensures markes clear n equlbrum. ()

4 and he neremporal budge consran where X ^E T ^C T =! + ^E X T ^Y T () ^Y ln(y = Y ); ^C ln(c = C); ^{ ln(r = R); = ln (P =P ) ;! = B = Y ; and z denoes he seady sae value of any varable z. Solvng he Euler equaon recursvely backwards, akng expecaons a me subsung no he neremporal budge consran gves ^C = ( )! + ^E X T and h ( ) ^Y T ( T T + ) : (3) Opmal consumpon decsons depend on curren wealh a he begnnng of he perod,!, and on he expeced fuure pah of ncome and he real neres rae. 5 The opmal allocaon rule s analogous o permanen ncome heory, wh d erences emergng from allowng varaons n he real rae of neres, whch can occur eher due o varaons n he nomnal neres rae or n aon. Nomnal neres raes a ec consumpon demand only hrough expecaons. Moreover, consumpon decsons depend on he enre expeced fuure pah of he nomnal neres rae, n conras wh Bullard Mra (00) and Orphandes and Wllams (005), among ohers, where only he curren neres rae maers for oupu deermnaon. Ths propery underscores he role of managng expecaons n polcy desgn. Noe also, ha as households become more paen, curren consumpon demand s more sensve o expecaons abou fuure macroeconomc condons. Frms. There s a connuum of monopolscally compeve rms. Each d erenaed consumpon good s produced accordng o he lnear producon funcon Y j; = A h j; 5 Usng he fac ha oal household ncome s he sum of dvdend and wage ncome, combned wh he rs order condons for labor supply and consumpon, delvers a decson rule for consumpon ha depends only on forecass of prces: ha s, goods prces, nomnal neres raes, wages and dvdends. However, we make he smplfyng assumpon ha households forecas oal ncome, he sum of dvdend paymens and wages receved. 3

5 where A denoes an aggregae echnology shock. Each rm chooses a prce P j; n order o maxmze s expeced dscouned value of pro s X where j; = ( ^E j ) P j; P Y j; Q ;T P T j;t W Pj; h j P P j; denoes perod pro s and he quadrac erm he cos of adjusng prces as n Roemberg (98). 6 The ax,, on revenues s chosen o elmnae he seady sae dsoron arsng from monopolsc compeon. Gven he ncomplee markes assumpon s assumed ha rms value fuure pro s accordng o he margnal rae of subsuon evaluaed a aggregae ncome Q ;T = T P Y P T Y T for T. 7 The nraemporal consumer problem mples aggregae demand for each d erenaed good s Y j = Pj; P Y where Y denoes aggregae oupu and Z P = 0 (P j; ) dj s he assocaed prce ndex. Summng up, he rm chooses a sequence for P j; o maxmze pro s, gven he consran ha demand should be sas ed a he posed prce, akng as gven P, Y ; and W. Agan, gven he nformaon upon whch expecaons are condoned, prces are deermned one perod n advance. 6 The resuls are smlar o he case of a Calvo prcng model. 7 The precse deals of hs assumpon are no mporan o he ensung analyss so long as n he log lnear approxmaon fuure pro s are dscouned a he rae T. 4

6 The rs-order condon o he rm s problem s Pj; P ^E = P j; P ^E Pj;+ Q ;+ j; P j; " + ^E Pj; Y P Pj;+ Pj; P P j;!# ( ) S : P A log-lnear approxmaon provdes ^P j; ^Pj; = ^E h ^Pj;+ ^Pj; + ^E h ^s + ^ + ^P ^Pj; where ^P = log P ; ^Pj; = log P j; ; ( ) Y = ; = ( ) denoes he mark-up and sas es ^ = ln( =); and ^s ln S = S s margnal coss (de ned below) n devaons from seady sae. Collecng erms n he prce of rm j provdes + + L + ^E ^Pj;+ = L ^E h ^s + + ^ + + ^P + where L denoes he lag operaor. Facorng he polynomal and solvng he unsable roo forward deermnes he opmal prce of he rm as where he roos and sasfy X ^P h^s (j) = ^P (j) + ^E ( ) T T + ^ T + ^P T 0 < < ; > ; = and + = ( + + ) : The laer wo properes combned mply = ( ) ( ). Nong ha X T ^E T = ^P X T + ^E ^PT perms he opmal prce decson o be wren n erms of aggregae n aon as ( ^P j; = ^Pj; + P + ^E X T ) (^s T + ^ T ) + T : Ths condon saes ha each rm s curren prce depends on he expeced fuure pah of real margnal coss, he aggregae prce level and cos-push shocks. 8 8 In an earler verson of hs paper, Eusep and Preson (007a), he rm s decson problem was smpl ed by makng ceran assumpons abou he nformaon avalable o rms when seng prces. Mke Woodford and an anonymous referee are hanked for encouragng he auhors o characerze he more general case presened here. The general enor of resuls s unchanged. 5 (4)

7 The real margnal cos funcon s S = w A = C A where he second equaly comes from he household s labor supply decson. Log-lnearzng we oban ^s = ^C ^a ; so ha curren prces depend on expeced fuure demand and echnology. The responsveness of curren prces o changes n expeced demand depends on he degree of nomnal rgdy. A low degree of nomnal rgdy mples a hgh value of (correspondng o a low value of he cos ): n hs case rms respond aggressvely o changes n perceved demand because prce changes are less cosly. The oppose occurs n he case of hgher coss of prce adjusmen. The degree of prce rgdy plays a key role n he sably analyss.. Marke clearng, e cen oupu and aggregae dynamcs The model s closed wh assumpons on moneary and scal polcy. The scal auhory, asde from levyng axes o elmnae he seady sae dsoron from monopolsc compeon, s assumed o follow a zero deb polcy n every perod and hs s undersood o be rue by agens. 9 Moneary polcy s dscussed n deal n he subsequen secon. For now su ces o noe ha a nomnal neres rae rule s mplemened. For a more general reamen of he neracons of scal and moneary polcy under learnng dynamcs see Eusep and Preson (008b) and Evans and Honkapohja (007). General equlbrum requres ha he goods marke clears, so ha Z A h ( ) = C dj = C : (5) Ths condon saes ha oupu ne of adjusmen coss s equal o aggregae consumpon, deermnng he equlbrum demand for labor h a he wage w = C. Ths relaon sas es he log-lnear approxmaon ^h + ^a = ^C = ^Y. 9 Ths mples agens do no need o forecas fuure ax oblgaons as n he analyses of Eusep and Preson (007b, 007d). 6

8 I s useful o characerze he e cen level of oupu ha would occur absen nomnal rgdes and dsoronary shocks under raonal expecaons. opmal prce seng mples he log-lnear approxmaon E Under hese assumpons, ^Y e = E ^a : Hence predcable movemens n he e cen rae of oupu are enrely deermned by he aggregae echnology shock. Nomnal bonds are also n zero ne supply requrng Z B d = 0: and where Aggregang rm and household decsons, usng (3) and (4), provdes Z 0 = 0 x = ^E X T [( )x T ( T T + ) + ^r e ] (6) ( ) ^E X ( ) T [( ) (x T + ^ T ) + T ] (7) ^E d = ^E gves average expecaons; x = ^Y E ^Y e of oupu from s expeced e cen level; and ^r e = ^Y e + denoes he log-devaon ^Y e he correspondng e cen rae of neres. The average expecaons operaor does no sasfy he law of eraed expecaons due o he assumpon of compleely mperfec common knowledge on he par of all households and rms. Because agens do no know he belefs, objecves and consrans of ohers n he economy, hey canno nfer aggregae probably laws. References Calvo, G. (983): Saggered Prces n a Uly-Maxmzng Framework, Journal of Moneary Economcs,, Eusep, S., and B. Preson (007a): Cenral Bank Communcaon and Macroeconomc Sablzaon, NBER Workng Paper 359. (007b): Columba Unversy. Sablzaon Polcy wh Near-Rcardan Households, unpublshed, 7

9 (008a): Expecaons, Learnng and Busness Cycle Flucuaons, unpublshed, Columba Unversy and Federal Reserve Bank of New York. (008b): Expecaons Sablzaon under Moneary and Fscal Polcy Coordnaon, unpublshed, Columba Unversy and Federal Reserve Bank of New York. Evans, G. W., and S. Honkapohja (007): Polcy Ineracon, Learnng and he Fscal Theory of Prces, Macroeconomc Dynamcs, forhcomng. Goodfrend, M., and R. G. Kng (997): The New Neoclasscal Synhess and he Role of Moneary Polcy, n 997 NBER Macroeconomcs Annual. NBER. Marce, A., and T. J. Sargen (989): Convergence of Leas-Squares Learnng n Envronmens wh Hdden Sae Varables and Prvae Informaon, Journal of Polcal Economy, pp Mlan, F. (006): Expecaons, Learnng and Macroeconomc Perssence, Journal of Moneary Economcs, 54, Preson, B. (005): Learnng Abou Moneary Polcy Rules when Long-Horzon Expecaons Maer, Inernaonal Journal of Cenral Bankng, (), 8 6. Roemberg, J., and M. Woodford (999): Ineres-Rae Rules and an Esmaed Scky- Prce Model, n Moneary Polcy Rules, ed. by J. Taylor. Unversy of Chcago Press, Chcago. Roemberg, J. J. (98): Monopolsc Prce Adjusmen and Aggregae Oupu, Revew of Economc Sudes, 64, Svensson, L. E., and M. Woodford (005): Implemenng Opmal Moneary Polcy Through In aon Forecas Targeng, n The In aon Targeng Debae, ed. by B. S. Bernanke, and M. Woodford. Unversy of Chcago Press. Yun, T. (996): Nomnal Prce Rgdy, Money Supply Endogeney, and Busness Cycles, Journal of Moneary Economcs, 37,