Technical Appendix for Central Bank Communication and Expectations Stabilization

Transcription

1 Technical Appendix for Cenral Bank Communicaion and Expecaions Sabilizaion Sefano Eusepi Bruce Preson y March 5, 00 Absrac This echnical appendix provides he microfoundaions and some calculaions underlying he model in he paper. Federal Reserve Bank of New York. sefano.eusepi@ny.frb.org. y Deparmen of Economics, Columbia Universiy and CAMA, Ausralian Naional Universiy. bp@columbia.edu

2 A Simple Model The following deails a simple model of oupu gap and in aion deerminaion ha is similar in spiri o Goodfriend and King (997), Roemberg and Woodford (999) and Svensson and Woodford (005). A coninuum of households face a canonical consumpion allocaion problem and decide how much o consume of available di ereniaed goods and how much labor o supply o rms for he producion of such goods. A coninuum of monopolisically compeiive rms produce di ereniaed goods using labor as he only inpu and face a price seing problem of he kind proposed by Roemberg (98). The major di erence is he incorporaion of non-raional beliefs, delivering an anicipaed uiliy model. The analysis follows Marce and Sargen (989) and Preson (005), solving for opimal decisions condiional on curren beliefs. Various mechanisms of persisence, such as habi formaion, price indexaion and inerial moneary policy are absraced from. This provides sharp, perspicuous analyical resuls. An earlier version of his paper, Eusepi and Preson (008a), demonsraes ha our conclusions regarding he value of communicaion in policy design remain perinen in models wih such modi caions.. Microfoundaions Households. and leisure Households maximize heir ineremporal uiliy derived from consumpion subjec o he ow budge consrain ^E i X T ln CT i h i T B i R B i + W h i + P P C i T i where B i denoes holdings of he one period riskless bond, R denoes he gross ineres paid on he bond, W he nominal wage, h i labor supplied by household i and T i lump-sum axes An analysis of price seing of he kind proposed by Calvo (983), as implemened by Yun (996), would lead o similar conclusions. I is also moivaed by Milani (007) and Eusepi and Preson (008b) which sugges ha purely forward looking business cycle models wih learning dynamics provide a superior characerizaion of various U.S. macroeconomic ime series han do raional expecaions models wih various persisence mechanisms.

3 and ransfers for household i. Financial markes are assumed o be incomplee and denoes pro s from holding shares in an equal par of each rm. Nominal income in any period is P Y i = W h i + P and P is he aggregae price level de ned below. ^Ei denoe he beliefs a ime held by each household i; which saisfy sandard probabiliy laws. Secion 3 describes he precise form of hese beliefs and he informaion se available o agens in forming expecaions. However, wo poins are worh noing. Firs, in forming expecaions, households and rms observe only heir own objecives, consrains and realizaions of aggregae variables ha are exogenous o heir decision problems and beyond heir conrol. They have no knowledge of he beliefs, consrains and objecives of oher agens in he economy: in consequence agens are heerogeneous in heir informaion ses in he sense ha even hough heir decision problems are idenical, hey do no know his o be rue. Second, given he assumed condiioning informaion for expecaions formaion, consumpion plans are made one period in advance and herefore predeermined. 3 Labor supply decisions are no predeermined and are condiioned on period informaion. 4 Each household consumes a composie good Z C i = 0 c i (j) dj which is made of a coninuum of di ereniaed goods, c i (j), each produced by a monopolisically compeiive rm j. The elasiciy of subsiuion among di ereniaed goods,, is ime-varying, wih E [ ] = >. This is a simple way of modeling ime-varying mark-ups, inroducing a rade-o beween in aion and oupu sabilizaion relevan o opimal policy design. A log-linear approximaion o he rs order condiions of he household problem provides he household Euler equaion ^C i = ^E h i ^Ci + (^{ + ) 3 We consider a model wih pricing and spending decisions deermined one period in advance so as o pu households, rms and policymakers on an idenical informaional fooing. This could similarly be achieved by he alernaive assumpion ha he cenral bank has a policy reacion funcion ha responds o one period ahead expecaions of in aion and agens condiion decisions on period informaion. All resuls coninue o hold. 4 This assumpion ensures markes clear in equilibrium. ()

4 and he ineremporal budge consrain where X ^E T ^Ci T =! i + ^E X T ^Y i T () ^Y ln(y = Y ); ^C ln(c = C); ^{ ln(r = R); = ln (P =P ) ;! i = B i = Y ; and z denoes he seady sae value of any variable z. Solving he Euler equaion recursively backwards, aking expecaions a ime subsiuing ino he ineremporal budge consrain gives ^C i = ( )! i + ^E X T and h ( ) ^Y i T i (i T T + ) : (3) Opimal consumpion decisions depend on curren wealh a he beginning of he period,! i, and on he expeced fuure pah of income and he real ineres rae. 5 The opimal allocaion rule is analogous o permanen income heory, wih di erences emerging from allowing variaions in he real rae of ineres, which can occur eiher due o variaions in he nominal ineres rae or in aion. Nominal ineres raes a ec consumpion demand only hrough expecaions. Moreover, consumpion decisions depend on he enire expeced fuure pah of he nominal ineres rae, in conras wih Bullard Mira (00) and Orphanides and Williams (005), among ohers, where only he curren ineres rae maers for oupu deerminaion. This propery underscores he role of managing expecaions in policy design. Noe also, ha as households become more paien, curren consumpion demand is more sensiive o expecaions abou fuure macroeconomic condiions. Firms. There is a coninuum of monopolisically compeiive rms. Each di ereniaed consumpion good is produced according o he linear producion funcion Y j; = A h j; 5 Using he fac ha oal household income is he sum of dividend and wage income, combined wih he rs order condiions for labor supply and consumpion, delivers a decision rule for consumpion ha depends only on forecass of prices: ha is, goods prices, nominal ineres raes, wages and dividends. However, we make he simplifying assumpion ha households forecas oal income, he sum of dividend paymens and wages received. 3

5 where A denoes an aggregae echnology shock. Each rm chooses a price P j; in order o maximize is expeced discouned 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 period pro s and he quadraic erm he cos of adjusing prices as in Roemberg (98). 6 The ax,, on revenues is chosen o eliminae he seady sae disorion arising from monopolisic compeiion. Given he incomplee markes assumpion i is assumed ha rms value fuure pro s according o he marginal rae of subsiuion evaluaed a aggregae income Q ;T = T P Y P T Y T for T. 7 The inraemporal consumer problem implies aggregae demand for each di ereniaed good is Y j = Pj; P Y where Y denoes aggregae oupu and Z P = 0 (P j; ) dj is he associaed price index. Summing up, he rm chooses a sequence for P j; o maximize pro s, given he consrain ha demand should be sais ed a he posed price, aking as given P, Y ; and W. Again, given he informaion upon which expecaions are condiioned, prices are deermined one period in advance. 6 The resuls are similar o he case of a Calvo pricing model. 7 The precise deails of his assumpion are no imporan o he ensuing analysis so long as in he log linear approximaion fuure pro s are discouned a he rae T. 4

6 The rs-order condiion o he rm s problem is 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-linear approximaion provides ^P j; ^Pj; = ^E h ^Pj;+ ^Pj; i + ^E h ^s + ^ + ^P ^Pj; i where ^P = log P ; ^Pj; = log P j; ; ( ) Y = ; = ( ) denoes he mark-up and sais es ^ = ln( =); and ^s ln S = S is marginal coss (de ned below) in deviaions from seady sae. Collecing erms in he price of rm j provides + + L + L ^E ^Pj;+ = L ^E h ^s + + ^ + + ^P i + where L denoes he lag operaor. Facoring he polynomial and solving he unsable roo forward deermines he opimal price of he rm as where he roos and saisfy X ^P h^s (j) = ^P (j) + ^E ( ) T T + ^ T + ^P i T 0 < < ; > ; = and + = ( + + ) : The laer wo properies combined imply = ( ) ( ). Noing ha X T ^E T = ^P X T + ^E ^PT permis he opimal price decision o be wrien in erms of aggregae in aion as ( ^P j; = ^Pj; + P + ^E X T ) (^s T + ^ T ) + T : This condiion saes ha each rm s curren price depends on he expeced fuure pah of real marginal coss, he aggregae price level and cos-push shocks. 8 8 In an earlier version of his paper, Eusepi and Preson (008a), he rm s decision problem was simpli ed by making cerain assumpions abou he informaion available o rms when seing prices. Mike Woodford and an anonymous referee are hanked for encouraging he auhors o characerize he more general case presened here. The general enor of resuls is unchanged. 5 (4)

7 The real marginal cos funcion is S = w A = C A where he second equaliy comes from he household s labor supply decision. Log-linearizing we obain ^s = ^C ^a ; so ha curren prices depend on expeced fuure demand and echnology. The responsiveness of curren prices o changes in expeced demand depends on he degree of nominal rigidiy. A low degree of nominal rigidiy implies a high value of (corresponding o a low value of he cos ): in his case rms respond aggressively o changes in perceived demand because price changes are less cosly. The opposie occurs in he case of higher coss of price adjusmen. The degree of price rigidiy plays a key role in he sabiliy analysis.. Marke clearing, e cien oupu and aggregae dynamics The model is closed wih assumpions on moneary and scal policy. The scal auhoriy, aside from levying axes o eliminae he seady sae disorion from monopolisic compeiion, is assumed o follow a zero deb policy in every period and his is undersood o be rue by agens. 9 Moneary policy is discussed in deail in he subsequen secion. For now i su ces o noe ha a nominal ineres rae rule is implemened. For a more general reamen of he ineracions of scal and moneary policy under learning dynamics see Eusepi and Preson (008c) and Evans and Honkapohja (007). General equilibrium requires ha he goods marke clears, so ha Z A h ( ) = C dj = C : (5) This condiion saes ha oupu ne of adjusmen coss is equal o aggregae consumpion, deermining he equilibrium demand for labor h a he wage w = C. This relaion sais es he log-linear approximaion ^h + ^a = ^C = ^Y. 9 This implies agens do no need o forecas fuure ax obligaions as in he analyses of Eusepi and Preson (007b, 007d). 6

8 I is useful o characerize he e cien level of oupu ha would occur absen nominal rigidiies and disorionary shocks under raional expecaions. Under hese assumpions, opimal price seing implies he log-linear approximaion E ^Y e = E ^a : Hence predicable movemens in he e cien rae of oupu are enirely deermined by he aggregae echnology shock. Nominal bonds are also in zero ne supply requiring Z 0 B i di = 0: Aggregaing rm and household decisions, using (3) and (4), provides x = ^E X T [( )x T (i T T + ) + ^r e ] (6) and = ( ) ^E X ( ) T [( ) (x T + ^ T ) + T ] (7) where Z 0 ^E i di = ^E gives average expecaions; x = ^Y E ^Y e of oupu from is expeced e cien level; and ^r e = ^Y e + denoes he log-deviaion ^Y e he corresponding e cien rae of ineres. The average expecaions operaor does no saisfy he law of ieraed expecaions due o he assumpion of compleely imperfec common knowledge on he par of all households and rms. Because agens do no know he beliefs, objecives and consrains of ohers in he economy, hey canno infer aggregae probabiliy laws..3 The Moneary Auhoriy The moneary auhoriy minimizes a sandard quadraic loss funcion under he assumpion ha agens have raional expecaions. This approach follows a now subsanial lieraure on learning dynamics and moneary policy see Howi (99) for he seminal conribuion and Bullard and Mira (00), Evans and Honkapohja (003) and Preson (006, 008), iner alia, for subsequen conribuions moivaed by he quesion of robusness of sandard policy advice o small deviaions from he raional expecaions assumpion. For alernaive 7

9 reamens of policy design ha exploi knowledge of privae agen learning see Gaspar, Smes, and Vesin (005), Molnar and Sanoro (005) and Preson (006, 008). The opimal policy problem is min E X T + x x T subjec o he consrains x = E x + E (i + r e ) (8) = E x + E + + E ^ (9) which are he model implied aggregae demand and supply equaions under raional expecaions. 0 The weigh x > 0 deermines he relaive prioriy given o oupu gap sabilizaion. A second-order accurae approximaion o household welfare in his model can be shown o imply a speci c value for x : Because his is no cenral o our conclusions, and because his more general noaion permis indexing a broader class of policy rules, we adop his objecive funcion. The rs-order condiion under discreion is E = x E x : (0) Hence opimal policy dicaes ineres raes o be adjused so ha predicable movemens in in aion are negaively relaed o hose in he oupu gap. This argeing rule combined wih he srucural relaions (8) and (9) can be shown o deermine he raional expecaions equilibrium pahs fi ; ; x g as linear funcions of he exogenous sae variables r e ; ^. Wihou loss of generaliy, and o make he analysis as simple and ransparen as possible, 0 These expressions follow direcly from (6) and (7) on noing ha ^E sais es he law of ieraed expecaions under he assumpion of raional expecaions households and rms know he objecives, beliefs and consrains of oher agens and can herefore deermine aggregae probabiliy laws in equilibrium. Also, a he raional expecaions equilibrium x = E x and = E. These equivalences are no rue under learning. Policies under opimal commimen could similarly be analyzed wihou subsanial di erences in he conclusions of his paper. However, because such policies inroduce hisory dependence, analyical condiions are somewha edious and we herefore ake he case of discreion for convenience. 8

10 we assume ha he exogenous processes are deermined by r e = r r e + " r ^ = ^ + " where 0 < r ; < and (" r ; " ) are independenly and idenically disribued random variables, wih auoregressive coe ciens known o households and rms. Under hese assumpions where i = r r e + ^ = x + ( ) + x ( ) delineaes he desired sae coningen evoluion of nominal ineres raes required o implemen he opimal equilibrium. Following Svensson and Woodford (005), raher han adoping he argeing rule (0) direcly as he policy rule, we insead assume he cenral bank implemens policy according o he nominal ineres rae rule i = i + ^E + x ^E x () = i + ^E + x ^E x () where > 0 and x ( x =) > 0. This serves o limi he informaion ha he cenral bank requires o implemen moneary policy. Moreover, an explici policy rule can be learned by marke paricipans wihou hem specifying a complee model of he economy. The cenral bank is assumed o observe privae forecass hrough survey daa or o have an idenical inernal forecasing model. This rule has he propery ha if beliefs converge o he underlying raional expecaions equilibrium hen i is consisen wih implemening opimal policy under a raional expecaions equilibrium. This follows immediaely from observing in his case ha ^E + x ^E x = 0 This assumpion can be dispensed wih wihou alering resuls. Because hese shocks are exogenous and assumed o be observed by agens, i is immediae ha esimaing a rs order process for each shock will recover he rue auoregressive coe cien wih probabiliy going o one as he sample size goes o in niy. 9

11 which in urn implies i = i as required for opimaliy under raional expecaions. Noe also ha i ness an expecaions-based Taylor rule as a special case, albei wih a sochasic consan. 3 References Bullard, J., and K. Mira (00): Learning Abou Moneary Policy Rules, Journal of Moneary Economics, 49(6), Calvo, G. (983): Saggered Prices in a Uiliy-Maximizing Framework, Journal of Moneary Economics,, Eusepi, S., and B. Preson (007): Sabilizaion Policy wih Near-Ricardian Households, unpublished, Columbia Universiy. (008a): Cenral Bank Communicaion and Macroeconomic Sabilizaion, American Economic Journal: Macroeconomics, Forhcoming. (008b): Expecaions, Learning and Business Cycle Flucuaions, NBER Working Paper 48. (008c): Expecaions Sabilizaion under Moneary and Fiscal Policy Coordinaion, NBER Working Paper 439. Evans, G. W., and S. Honkapohja (003): Expecaions and he Sabiliy Problem for Opimal Moneary Policies, Review of Economic Sudies, 70(4), (007): Policy Ineracion, Learning and he Fiscal Theory of Prices, Macroeconomic Dynamics, forhcoming. Gaspar, V., F. Smes, and D. Vesin (005): Opimal Moneary Policy under Adapive Learning, unpublished working paper. 3 The sochasic consan is largely irrelevan o he sabiliy analysis under learning dynamics. Also, if he assumpion of discreionary opimizaion is unappealing, hen a rule of his form wih appropriaely de ned sochasic consan can implemen he opimal equilibrium under commimen see Preson (006). 0

12 Goodfriend, M., and R. G. King (997): The New Neoclassical Synhesis and he Role of Moneary Policy, in 997 NBER Macroeconomics Annual. NBER. Howi, P. (99): Ineres Rae Conrol and Nonconvergence o Raional Expecaions, Journal of Poliical Economy, 00(4), Marce, A., and T. J. Sargen (989): Convergence of Leas-Squares Learning in Environmens wih Hidden Sae Variables and Privae Informaion, Journal of Poliical Economy, pp Milani, F. (007): Expecaions, Learning and Macroeconomic Persisence, Journal of Moneary Economics, 54, Molnar, K., and S. Sanoro (005): Opimal Moneary Policy When Agens Are Learning, unpublished, Iniversia Pomeu Fabra. Preson, B. (005): Learning Abou Moneary Policy Rules when Long-Horizon Expecaions Maer, Inernaional Journal of Cenral Banking, (), 8 6. (006): Adapive Learning, Forecas-Based Insrumen Rules and Moneary Policy, Journal of Moneary Economics, 53. (008): Adapive Learning and he Use of Forecass in Moneary Policy, Journal of Economic Dynamics and Conrol, 3(4), Roemberg, J. J. (98): Monopolisic Price Adjusmen and Aggregae Oupu, Review of Economic Sudies, 64, Roemberg, J. J., and M. Woodford (999): Ineres-Rae Rules and an Esimaed Sicky-Price Model, in Moneary Policy Rules, ed. by J. Taylor, pp Universiy of Chicago Press, Chicago. Svensson, L. E., and M. Woodford (005): Implemening Opimal Moneary Policy Through In aion Forecas Targeing, in The In aion Targeing Debae, ed. by B. S. Bernanke, and M. Woodford. Universiy of Chicago Press.

13 Yun, T. (996): Nominal Price Rigidiy, Money Supply Endogeneiy, and Business Cycles, Journal of Moneary Economics, 37,