Inflation Persistence, Backward-Looking Firms, and Monetary Policy in an Input-Output Economy

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Inflation Persistence, Backward-Looking Firms, and Monetary Policy in an Input-Output Economy Brad E. Strum 2010-55 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

In(cid:135)ation Persistence, Backward-Looking Firms, and Monetary Policy in an Input-Output Economy (cid:3) Brad E. Strum Federal Reserve Board y November 4, 2010 Abstract Thispaperstudiestheimplicationsofin(cid:135)ationpersistence(generatedbybackward-looking price setters) for monetary policy in a New Keynesian "input-output" model(cid:151)a model with sticky prices in both intermediate and (cid:133)nal goods sectors. Optimal policy under commitment depends on the degree of in(cid:135)ation persistence in both sectors. Under discretion, speed-limit targeting(cid:151)targeting the change in the output gap(cid:151)outperforms price-level and in(cid:135)ation targetinginthepresenceofin(cid:135)ationpersistence. Ifin(cid:135)ationpersistenceislowintheintermediate goods sector, price-leveltargeting outperforms in(cid:135)ation targeting despite high in(cid:135)ation persistence in the (cid:133)nal goods sector. Keywords: In(cid:135)ation persistence, intermediate goods, monetary policy JELcodes: E50,E52,E58 This paper is a revised version of the second chapter of my doctoral dissertation. I would like to thank my (cid:3) mainadvisor,LarsSvensson,forhelpfulguidanceandcomments. IwouldalsoliketoespeciallythankRicardoReis, whoseguidanceandinsighthavebeenveryhelpful. AlanBlinderhasalsoprovidedmanyusefulcomments. Finally, IwouldliketothankparticipantsinthePrincetonStudentMacro/InternationalWorkshop. Theviewsexpressedin this paper are my own and do not necessarily represent those of the Board of Governors or the sta⁄of the Federal Reserve System. Iam solely responsible forany remaining errors. BoardofGovernorsoftheFederalReserveSystem,20thStreetandConstitutionAvenue,NW,Washington,DC, y 20551. E-mail: brad.e.strum@frb.gov. 1

1 Introduction How should monetary policy be conducted in a New Keynesian model in which prices are sticky at multiple stages of production? Huang and Liu (2005) and Strum (2009) examine this question using a forward-looking New Keynesian "input-output" model(cid:151)a model that has sticky prices in both the intermediate and (cid:133)nal goods sectors. They (cid:133)nd that a central bank paying attention to pricemovementsinboththe(cid:133)nalandintermediategoodssectorscanmoreablyminimizehousehold utilitylossesthanifitconsidersonlyonesector. Furthermore,Strum(2009)(cid:133)ndsthatifthecentral bank acts under discretion, it performs better if it targets price levels rather than in(cid:135)ation rates. One feature of the standard forward-looking New Keynesian framework is that it generates Phillips curves that do not relate current in(cid:135)ation to lagged in(cid:135)ation. Yet studies such as Fuhrer (1997), Rudebusch (2002), and Roberts (2005) (cid:133)nd an important empirical role for lagged in(cid:135)ation in the Phillips curve. Besides a⁄ecting the speci(cid:133)cation of the Phillips curve, in(cid:135)ation persistence can a⁄ect the evaluation of monetary policy in New Keynesian models. For example, when examining discretionary monetary policy regimes in one-sector New Keynesian models, Walsh (2003) and NessØn and Vestin (2005) show that the type of regime that performs best depends on the degree of in(cid:135)ation persistence in the Phillips curve. These(cid:133)ndingssuggestthequestion: HowismonetarypolicydesigninaNewKeynesianinputoutput model a⁄ected by in(cid:135)ation persistence? To answer this question, the standard model must be extended so that the Phillips curve exhibits in(cid:135)ation persistence. As recent scholarship has found, in(cid:135)ation persistence can arise in New Keynesian models in a number of di⁄erent ways.1 This paper builds in(cid:135)ation persistence into the input-output model developed by Huang and Liu (2005) and extended in Strum (2009) by following the approach pioneered in Gal(cid:237) and Gertler (1999), Amato and Laubach (2003), and Steinsson (2003): When resetting prices, some (cid:133)rms are modeled as employing a simple rule of thumb that uses information about past states of the world to set new prices. These (cid:133)rms may behave this way if, from time to time, they (cid:133)nd it too costly to gather new information and calculate the optimal forward-looking price. Firms using this rule 1See Woodford (2007) for a discussion on recent scholarship that explores other methods used to account for in(cid:135)ation persistence. 2

of thumb generate in(cid:135)ation persistence in the sectoral Phillips curves. Furthermore, compared to modelswithstickypricesinonesector,thismodelcanyielddi⁄erentdegreesofin(cid:135)ationpersistence in di⁄erent sectors. This paper examines four questions about monetary policy in New Keynesian models: How doesin(cid:135)ationpersistenceatmultiplestagesofproductiona⁄ecttheconductofoptimalpolicyunder commitment? Howdoesin(cid:135)ationpersistenceatmultiplestagesofproductiona⁄ectthetypeofloss function that should be assigned to a central bank acting under discretion? How well do di⁄erent regimesperformwhenpoliciesaresetusingincorrectassumptionsaboutin(cid:135)ationpersistenceorthe sourcesofshocks? Howwelldopoliciesderivedfromone-sectormodelsperformwhenimplemented in an input-output model? I(cid:133)ndthatthetimingandmagnitudeofthecentralbank(cid:146)sresponsestoshockswhenimplementing optimal monetary policy under commitment are a⁄ected not only by the presence of in(cid:135)ation persistence,butalsobytherelativedegreesofsectoralin(cid:135)ationpersistence. Ontheotherhand,the natureofthecentralbank(cid:146)sresponses(cid:151)whetherexpansionaryorcontractionary(cid:151)isnota⁄ectedby the degrees of in(cid:135)ation persistence. Whenstudyingmonetarypolicyunderdiscretion,I(cid:133)ndthatin(cid:135)ationpersistencea⁄ectsthetype of loss function that best minimizes household losses. As in Strum (2009), price-level targeting performs best in a forward-looking model. However, when in(cid:135)ation persistence is introduced, speed-limit targeting (a regime targeting the change in the output gap) performs best. Price-level targeting outperforms in(cid:135)ation targeting unless in(cid:135)ation persistence is high in both sectors. Whenaregimeischosenandthelossfunctioniscrafted,thegovernmentmaymakeanincorrect assessment of in(cid:135)ation persistence in the two sectors, the sources of shocks, or the need to use the input-output model. I (cid:133)nd that, given the degrees of in(cid:135)ation persistence in the two sectors, the type of regime that performs best is not a⁄ected by the government(cid:146)s assumptions regarding in(cid:135)ation persistence or the sources of shocks (when crafting the loss function). However, the type of regime that performs best is a⁄ected by the incorrect use of a one-sector model. Theremainderofthepaperisorganizedasfollows:Section2setsupthemodelandpresentsthe linearized version used for later analysis. Section 3 discusses the calibration. Section 4 examines 3

the qualitative characteristics of optimal policy under commitment for di⁄erent degrees of in(cid:135)ation persistence in the two sectors. Section 5 compares the performances of simple loss functions under themorerealisticcaseofdiscretionaryoptimization. Section6examinessomerobustnessproperties of the discretionary regimes. Section 7 concludes. 2 Model of an Input-Output Economy Huang and Liu (2005) develop a New Keynesian model with a vertical production chain consisting of two sectors. Firms in the (cid:133)rst sector produce (cid:133)nal (nondurable) goods using intermediate goods and labor. Final goods are consumed by households. In the second sector, intermediate (nondurable)goodsareproducedusingonlylabor. Intermediategoodsareusedonlyby(cid:133)nalgoods (cid:133)rmsinproduction. Each(cid:133)rmineachsectorproducesauniquedi⁄erentiatedgoodandengagesin monopolisticcompetitionwithinitssector. Pricesinbothsectorsaresticky,and(cid:133)rmsadjusttheir prices in a staggered manner in the spirit of Calvo (1983). There is one competitive market for homogenous labor that can be used by all (cid:133)rms. All (cid:133)rms are price takers in their input markets. Strum (2009) extends the model by introducing cost-push shocks and characterizing monetary policy as the minimization of an assigned loss function. This paper extends the model in Strum (2009) by assuming that in each sector and in each period, a fraction of (cid:133)rms (cid:133)nd that solving for the optimal forward-looking price is too costly. Following Gal(cid:237) and Gertler (1999), Amato and Laubach (2003), and Steinsson (2003), I assume that backward-looking (cid:133)rms in each sector employ a rule of thumb that uses past information to set a new price. This section presents the basic elements of the model and its linearized version used for subsequent analysis.2 2A detailed account ofthe modelis given in a technicalappendix that is available upon request. 4

2.1 Households The economy is populated by a large number of identical, in(cid:133)nitely lived households. Households derive utility from consumption and leisure. Given a (cid:133)xed amount of time that households divide fully between leisure and labor each period, the household utility function can be written in terms of labor instead of leisure. Accordingly, households maximize expected lifetime utility, given by E 1 (cid:12)t[u(C ) v(N )] ; (1) 0 t t ( (cid:0) ) t=0 X whereE isthemathematicalexpectationsoperatorgiveninformationavailableattime0,(cid:12) (0;1) 0 2 is the subjective time discount factor, C is consumption, and N is labor hours. t t The period utility function for consumption is u(C )=log(C ). The consumption good, C , is t t t a composite of a continuum of di⁄erentiated (cid:133)nal goods in the spirit of Dixit and Stiglitz (1977), given by C t 1 y ft (i)((cid:18)ft(cid:0) 1)=(cid:18)ft di (cid:18)ft=((cid:18)ft(cid:0) 1) , (2) (cid:17) (cid:20)Z0 (cid:21) where(cid:18) isthetime-varyingelasticityofsubstitutionbetweenthedi⁄erentiated(cid:133)nalgoods,y (i) ft ft fori [0;1],andisassumedtoalwaysbegreaterthan1. Movementsin(cid:18) canrepresentchangesin ft 2 householdpreferencesorthebusinessenvironment. Theperioddisutilityfunctionforlabor,v(N ), t is linear and increasing in labor hours. Without loss of generality, labor hours are normalized so that N [0;1].3 t 2 Households have equal ownership in all (cid:133)rms and divide all pro(cid:133)ts equally among themselves. Labor is homogeneous and supplied equally by households to all (cid:133)rms through one market with one wage rate, which households take as given. I assume complete (cid:133)nancial markets. Finally, I assume standard budget-set and transversality conditions hold. 3In a setup like Hansen (1985),linear disutility arises if labor is assumed to be indivisible, which is the interpretation given in Huang and Liu (2005). 5

2.2 Final Goods Each (cid:133)nal goods (cid:133)rm i has access to a constant returns to scale (CRS) Cobb-Douglas production function y ft (i)=Y mt (i)’(A ft N ft (i))1 (cid:0) ’, (3) where A is a sectoral labor-augmenting technology factor, N (i) is the amount homogeneous ft ft labor used by (cid:133)rm i, and Y (i) is the amount of the composite intermediate good used by (cid:133)rm mt i. Thecompositeintermediategoodisacombinationofdi⁄erentiatedintermediategoodsgivenby the Dixit-Stiglitz aggregator Y mt (i) 1 y mt (i;j)((cid:18)mt(cid:0) 1)=(cid:18)mt dj (cid:18)mt=((cid:18)mt(cid:0) 1) , (4) (cid:17) (cid:20)Z0 (cid:21) where (cid:18) > 1 is the time-varying elasticity of substitution between di⁄erentiated intermediate mt goods, and y (i;j) is the amount of di⁄erentiated intermediate good j demanded by (cid:133)rm i. mt Variationsin(cid:18) canbeseenastechnologyshocks(theeasewithwhich(cid:133)rmsareabletosubstitute mt di⁄erentiated intermediate goods for one another) or as shocks to the business environment (such as changes to the monopoly power enjoyed by individual (cid:133)rms). Each (cid:133)rm minimizes costs to meet the demand for its good given its stated price. Final goods (cid:133)rmsadjusttheirpriceswithprobability1 (cid:11) eachperiod,where(cid:11) (0;1). Arandomfraction f f (cid:0) 2 1 (cid:19) of (cid:133)rms resetting prices (cid:133)nd it worthwhile to determine the price that maximizes discounted f (cid:0) expectedpro(cid:133)tsoverthetimethepriceisexpectedtopersist,where(cid:19) [0;1). Thismaximization f 2 problem is given by maxE 1 (cid:11)s tD [P (i)(1+(cid:28) ) V (i)]yd (i) , (5) Pft(i) t ( s=t f(cid:0) t;s ft f (cid:0) fs fs ) X where (cid:28) is a subsidy to (cid:133)nal goods producers, V (i) is the nominal marginal cost of production f fs in period s, D is the stochastic discount factor for nominal payments between periods t and s t;s (determined in the household maximization problem), and yd (i) is the total demand for (cid:133)rm i(cid:146)s fs output in period s. The price that solves the forward-looking maximization problem at time t is 6

denoted by Pf. ft The remaining fraction (cid:19) of (cid:133)rms setting a new price use a rule of thumb to determine a new f backward-looking price, Pb , instead of solving an explicit optimization problem. Steinsson (2003) ft suggests a generalization of the Gal(cid:237) and Gertler (1999) approach by allowing (cid:133)rms to also react to an indicator of the previous period(cid:146)s output gap. Generalizing the rule of thumb in Steinsson (2003), I assume that the backward-looking (cid:133)rms set prices according to (cid:14)f V =P P f b t =P f(cid:3);t (cid:0) 1 (cid:5) f;t (cid:0) 1 V f f n ; ; t t (cid:0) 1 1 =P f f n ; ; t t (cid:0) 1 1! , (6) (cid:0) (cid:0) where(cid:5) =P =P ;V denotesthenominalmarginalcostof(cid:133)nalgoodsattimet 1; f;t 1 f;t 1 f;t 2 f;t 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) thesuperscriptndenotesthevaluesinane¢ cientequilibrium;(cid:14) 0;andP denotesanindex f (cid:21) f(cid:3);t (cid:0) 1 of prices newly set at time t 1, given by (cid:0) logP =(1 (cid:19) )logPf +(cid:19) logPb . (7) f(cid:3);t (cid:0) 1 (cid:0) f f;t (cid:0) 1 f f;t (cid:0) 1 Gal(cid:237)andGertler(1999)formulatearulesimilartotheoneabove,excepttheye⁄ectivelyset(cid:14) =0. f Taking the log of (6) yields the rule logPb =logP +(cid:25) +(cid:14) v~ ; (8) ft f(cid:3);t 1 f;t 1 f f;t 1 (cid:0) (cid:0) (cid:0) where(cid:25) =log(cid:5) andv~ =log (V =P )= Vn =Pn . Inthesteadystate, f;t 1 f;t 1 f;t 1 f;t 1 f;t 1 f;t 1 f;t 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) h (cid:16) (cid:17)i this pricing rule is consistent with the steady-state price of all (cid:133)rms in the sector. The aggregate price level can be written as P ft = (cid:11) f P f 1 ; (cid:0) t (cid:18)f 1 t +(1 (cid:0) (cid:11) f )(1 (cid:0) (cid:19) f ) P f f t 1 (cid:0) (cid:18)ft +(1 (cid:0) (cid:11) f )(cid:19) f P f b t 1 (cid:0) (cid:18)ft 1=(1 (cid:0) (cid:18)ft) . (9) (cid:20) (cid:0) (cid:16) (cid:17) (cid:0) (cid:1) (cid:21) 7

2.3 Intermediate Goods Each intermediate goods (cid:133)rm j has access to a CRS production function given by y (j)=A N (j), (10) mt mt mt where A is a sectoral labor-augmenting technology factor, and N (j) is the amount of homomt mt geneous labor used by (cid:133)rm j. Otherwise, their cost minimization and price-setting problems are similar to the ones faced by (cid:133)nal goods (cid:133)rms. Intermediate goods (cid:133)rms adjust their prices each period with a probability of 1 (cid:11) , where (cid:11) (0;1). A fraction 1 (cid:19) of (cid:133)rms resetting their m m m (cid:0) 2 (cid:0) prices choose a price, denoted by Pf , that maximizes expected discounted pro(cid:133)ts over the time mt when the price is expected to remain (cid:133)xed. Similar to the case of (cid:133)nal goods (cid:133)rms, I assume (cid:19) [0;1). The remaining (cid:19) (cid:133)rms engage in backward-looking behavior, setting a new price m m 2 denoted by Pb . The functional forms that describe the price-setting behavior of intermediate mt (cid:133)rms are similar to those of (cid:133)nal goods (cid:133)rms and are obtained by replacing "f" with "m" in (5), (6), (7), (8), and (9). 2.4 Government The government serves two purposes in this model. First, it assigns a loss function to an independent central bank. The central bank acts to minimize its assigned loss function. I assume that the central bank can react to and a⁄ect state variables in the current period. Second, the governmentcollectslump-sumtaxesfromhouseholdstoprovidesubsidiesto(cid:133)rmssothatthesteady-state equilibrium is not distorted from ine¢ ciencies arising from monopolistic competition. Finally, I ignorethepossibleinteractionsbetweenmonetaryand(cid:133)scalpolicythatwouldbepresentinaricher model. 8

2.5 Linearized Model Ilog-linearizethemodelusinglog-deviationsfromahypotheticalnon-distortede¢ cientequilibrium (theequilibriumthatwouldobtainifpriceswere(cid:135)exibleandtherewerenoshockstotheelasticities of substitution).4 The natural rate of interest is the real interest rate that would obtain in the e¢ cient equilibrium. I list the key variables and symbols from the model in Table 1. Symbol Meaning (cid:25) in(cid:135)ation in sector k, k f;m ((cid:133)nal, intermediate) kt 2f g Q relative price of intermediate goods in terms of (cid:133)nal goods: Q P =P t t mt ft (cid:17) q^ log-deviation of sticky-equilibrium relative price from steady state t q^ log-deviation of e¢ cient-equilibrium relative price from steady state t(cid:3) q~ relative price gap: q~ q^ q^ t t (cid:17) t (cid:0) t(cid:3) c~ output gap (sticky relative to e¢ cient level) t ^{ log-deviation of gross nominal interest rate from steady state (log[(1+i )=(1+(cid:22){)]) t t r^ log-deviation of gross natural real interest rate from steady state t(cid:3) a^ log-deviation of the technology factor from steady state in sector k kt (cid:26) autocorrelation of the technology factor in sector k k ’ measure of importance of intermediate goods in the production of (cid:133)nal goods (cid:27) 1 intertemporal elasticity of substitution in consumption (cid:0) (cid:11) probability that a (cid:133)rm in sector k keeps its previously set price k (cid:18) stochastic elasticity of substitution between di⁄erentiated goods in sector k kt (cid:22)(cid:18) steady-state value of (cid:18) k kt E x mathematical expected value of x at time t t t+1 t+1 (cid:19) fraction of backward-looking price setters in sector k k (cid:14) exponent for rule-of-thumb reaction to lagged real indicator in sector k k Table 1: List of Symbols Thehouseholdintertemporalconsumptionequationisobtainedfromthehousehold(cid:146)s(cid:133)rst-order conditions. Its log-linearized version is given by 1 c~ =E c~ (^{ E (cid:25) r^ ). (11) t t t+1 (cid:0) (cid:27) t (cid:0) t f;t+1 (cid:0) t(cid:3) I interpret c~ as the output gap since only households purchase (cid:133)nal goods. t The log-linearized pricing equations for backward-looking (cid:133)rms can be combined with the loglinearized (cid:133)rst-order equations for forward-looking (cid:133)rms to obtain Phillips curves for each sector, 4The model is log-linearized initially without using the assumption that u(Ct) = log(Ct). For the numerical analysis,thisassumption isimposed by setting (cid:27)=1in the log-linearized equations. A fullderivation isgiven in a technicalappendix that is available upon request. 9

namely, (cid:25) = (cid:31)f(cid:12)E (cid:25) +(cid:31)f(cid:25) +(cid:31)f(’q~ +(1 ’)(cid:27)c~) (12) ft 1 t f;t+1 2 f;t (cid:0) 1 3 t (cid:0) t +(cid:31)f(’q~ +(1 ’)(cid:27)c~ )+u , 4 t (cid:0) 1 (cid:0) t (cid:0) 1 ft (cid:25) = (cid:31)m(cid:12)E (cid:25) +(cid:31)m(cid:25) +(cid:31)m((cid:27)c~ q~) (13) mt 1 t m;t+1 2 m;t (cid:0) 1 3 t (cid:0) t +(cid:31)m((cid:27)c~ q~ )+u , 4 t (cid:0) 1 (cid:0) t (cid:0) 1 mt where, for k f;m , 2f g (cid:11) (cid:31)k = k ; 1 (cid:19) (1 (cid:11) +(cid:11) (cid:12))+(cid:11) k k k k (cid:0) (cid:19) (cid:31)k = k ; 2 (cid:19) (1 (cid:11) +(cid:11) (cid:12))+(cid:11) k k k k (cid:0) (1 (cid:11) )[(1 (cid:11) (cid:12))(1 (cid:19) ) (cid:11) (cid:12)(cid:19) (cid:14) ] (cid:31)k = (cid:0) k (cid:0) k (cid:0) k (cid:0) k k k ; 3 (cid:19) (1 (cid:11) +(cid:11) (cid:12))+(cid:11) k k k k (cid:0) (cid:19) (cid:14) (1 (cid:11) ) (cid:31)k = k k (cid:0) k ; 4 (cid:19) (1 (cid:11) +(cid:11) (cid:12))+(cid:11) k k k k (cid:0) (1 (cid:11) (cid:12))(1 (cid:19) )(1 (cid:11) ) u = (cid:0) k (cid:0) k (cid:0) k ^(cid:18) . kt 1 (cid:22)(cid:18) [(cid:19) (1 (cid:11) +(cid:11) (cid:12))+(cid:11) ] kt k k k k k (cid:0) (cid:0) (cid:0) (cid:1) As noted in Section 2, though (cid:19) is allowed to be zero, (cid:11) > 0 and (cid:22)(cid:18) > 1; therefore, none of the k k k denominators can be zero. The term ’q~ +(1 ’)(cid:27)c~ is equivalent to the real marginal cost gap t t (cid:0) for (cid:133)nal goods producers, while (cid:27)c~ q~ is equivalent to the real marginalcost gap forintermediate t t (cid:0) goods producers. The sectoral Phillips curves reveal how the percentage of backward-looking (cid:133)rms a⁄ects the sectoralPhillipscurves. Anincreasethepercentageofbackward-looking(cid:133)rmsincreasestheweight of lagged in(cid:135)ation relative to the forward-looking component ((cid:31)k is decreasing and (cid:31)k is increasing 1 2 in (cid:19) ). The coe¢ cient of the sectoral Phillips curves with respect to the sector(cid:146)s current real k marginalcostgap(and,therefore,theoutputgap)decreasesasthepercentageofbackward-looking 10

(cid:133)rms increases ((cid:31)k is decreasing in (cid:19) ).5 On the other hand, the coe¢ cient of the sectoral Phillips 3 k curves with respect to the sector(cid:146)s lagged real marginal cost gap increases as the percentage of backward-looking (cid:133)rms increases ((cid:31)k is increasing in (cid:19) ). 4 k The weight on intermediate goods in the production function of (cid:133)nal goods, represented by ’, a⁄ectsthecoe¢ cientsofthecurrentandlaggedoutputgapsinthe(cid:133)nalgoodsPhillipscurve, given by (cid:31)f(1 ’)(cid:27) and (cid:31)f(1 ’)(cid:27), respectively. As in the forward-looking model in Strum (2009), 3 (cid:0) 4 (cid:0) highervaluesof’correspondtolowervaluesofthecoe¢ cientofthecurrentoutputgapinthe(cid:133)nal goodsPhillipscurve. Moreover,highervaluesof’alsocorrespondtolowervaluesofthecoe¢ cient of the lagged output gap in the (cid:133)nal goods Phillips curve.6 InstandardhybridNewKeynesianPhillipscurves,thecost-pushshocksarerepresentedbyterms like u . In this model, the term u arises from ^(cid:18) , variations in the elasticities of substitution kt kt kt of di⁄erentiated goods. The magnitude of a one-standard-deviation shock to (cid:18) does not depend kt on the percentage of backward-looking (cid:133)rms. However, as the expression for u shows, the e⁄ects kt of (cid:135)uctuations in the elasticities of substitution are attenuated by higher percentages of backwardlooking (cid:133)rms. For the state-space representation of the model, I rewrite this term as u =(cid:27) ((cid:19) )(cid:17) , k f;m , (14) kt uk k kt 2f g where (cid:17) an i.i.d. white noise process that is uncorrelated with all other stochastic variables (with kt variance normalized to 1), and (cid:27) ((cid:19) ) is decreasing in (cid:19) . In this setup, I refer to (cid:17) as the uk k k kt cost-push shock in sector k. As noted in Table 1, the "relative price" refers to the ratio of the price index for intermediate 5Givenvaluesfor(cid:11)k,(cid:12),and(cid:19)k,asu¢ cientlylargevaluefor(cid:14)k canleadto(cid:31)k 3 beingnegative. Thisneveroccurs forany ofthe calibrations considered in this paper. 6However, an increase in ’has the opposite e⁄ect on the coe¢ cients of q~t in the (cid:133)nal goods Phillips curve. As Huang and Liu (2005) point out, the real marginal cost gap for the (cid:133)nal goods sector can be written as v~ft = ’q~t+(1 ’)(cid:27)c~t=(cid:27)c~t ’v~mt,wherev~mt=(cid:27)c~t q~t istherealmarginalcostgapintheintermediategoodssector. (cid:0) (cid:0) (cid:0) If the (cid:133)nal goods sector Phillips curve is written in this way, the value of ’ does not a⁄ect the slope of the (cid:133)nal goods Phillips curve with respect to the output gaps. Instead, the value of ’ determines how (cid:135)uctuations in the realmarginalcostgapintheintermediatesectora⁄ectthe(cid:133)nalgoodsPhillipscurve. HuangandLiu(2005)suggest that ’v~mt could be viewed as a cost-push shock in the (cid:133)nalgoods Phillips curve. 11

goods to the price index for (cid:133)nal goods. The relative price gap evolves, by de(cid:133)nition, according to q~ =q~ +(cid:25) (cid:25) +(1 ’)((cid:1)a^ (cid:1)a^ ). (15) t t 1 mt ft mt ft (cid:0) (cid:0) (cid:0) (cid:0) I assume that the technology factors are stationary(cid:151)that is, (cid:26) < 1 for k f;m (cid:151)and evolve j kj 2 f g according to a^ =(cid:26) a^ +(cid:27) (cid:15) , (16) k;t+1 k kt ak k;t+1 where, for sector k, a^ is the log-deviation of the technology factor from steady state; (cid:15) , the kt k;t+1 productivity shock, is an i.i.d. white noise process uncorrelated with all other stochastic variables (withavarianceof1);(cid:27) isaconstantusedtocalibratethevarianceoftheshocktothetechnology ak factor; and (cid:26) is the autoregressive coe¢ cient.7 k In this model, the nominal interest rate does not appear in the objective function of the central bankandisnotaconstraintinthecentralbank(cid:146)smaximizationproblem. Therefore,Isimplifythe setup by treating c~ as the instrument of the central bank. Once the model is solved, I use (11) t to determine the interest rates that are consistent with the desired equilibrium. I compute price levels using the identity p p +(cid:25) for k f;m , where p can be interpreted as the k;t+1 kt k;t+1 kt (cid:17) 2 f g log-deviation in the price level from its initial value. I represent the structural equations of the economy as X X C t+1 t =A +Bc~ + " , (17) 2 3 2 3 t 2 3 t+1 HE y y 0 t t+1 t 6 7 6 7 6 7 4 5 4 5 4 5 7Strum (2009) makes a similar assumption, whereas Huang and Liu (2005) assume a stationary log-di⁄erence AR(1) process. 12

where u ft 2 3 u mt 6 7 6 a^ 7 6 ft 7 6 7 6 7 6 a^ mt 7 (cid:17) f;t+1 6 7 q~ 6 7 t 2 3 6a^ 7 (cid:17) X t =6 6 f;t (cid:0) 17 7 , y t = 2 (cid:25) ft 3 , " t+1 = 6 m;t+1 7 . 6 6 6 6 6 6 a^ q m ~ t ; (cid:0) t (cid:0) 1 1 7 7 7 7 7 7 6 6 6 6 4 (cid:25) mt 7 7 7 7 5 6 6 6 6 6 6 (cid:15) (cid:15) m f; ; t t + + 1 1 7 7 7 7 7 7 6 7 4 5 6(cid:25) 7 6 f;t (cid:0) 17 6 7 6(cid:25) 7 6 m;t 17 6 (cid:0) 7 6 7 6 c~ t 1 7 6 (cid:0) 7 4 5 The matrices A;B;C; and H are given in the appendix. 2.6 Household Loss Function The second-order Taylor approximation to the household(cid:146)s utility function is given by = 1 u C(cid:22) C(cid:22) E 1 (cid:12)tLh+t.i.p.+O (cid:24) 3 , (18) L 0 (cid:0)2 c 0 t k k (cid:18) (cid:0) (cid:1) (cid:19) X t=0 (cid:16) (cid:17) where C(cid:22) is the steady-state level of consumption, "t.i.p." is a collection of terms independent of policy,andO (cid:24) 3 isacollectionofthird-orderandhigher-orderterms. Theperiodlossfunction k k (cid:16) (cid:17) is given by Lh = A c~2+A (cid:25)2 (19) t 0 t 1 ft +A (cid:25)2 +A ((cid:27)c~ q~)2 2 mt 3 t (cid:0) t +A ((cid:1)(cid:25) )2+A ((cid:1)(cid:25) )2 4 ft 5 mt +A (cid:1)(cid:25) (’q~ +(1 ’)(cid:27)c~ )+A (cid:1)(cid:25) ((cid:27)c~ q~ ) 6 ft t 1 t 1 7 mt t 1 t 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) +A (’q~ +(1 ’)(cid:27)c~ )2+A ((cid:27)c~ q~ )2, 8 t 1 t 1 9 t 1 t 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) 13

with coe¢ cients A = (cid:27) 0 (cid:22)(cid:18) (cid:11) A = f f ; 1 1 (cid:11) (cid:12) 1 (cid:11) (cid:0) f (cid:18) (cid:0) f(cid:19) ’(cid:22)(cid:18) (cid:11) A = m m ; 2 1 (cid:11) (cid:12) 1 (cid:11) (cid:0) m (cid:18) (cid:0) m(cid:19) A = ’(1 ’); 3 (cid:0) (cid:22)(cid:18) (cid:19) A = f f ; 4 1 (cid:11) (cid:12) (1 (cid:11) )(1 (cid:19) ) (cid:0) f (cid:18) (cid:0) f (cid:0) f (cid:19) ’(cid:22)(cid:18) (cid:19) A = m m ; 5 1 (cid:11) (cid:12) (1 (cid:11) )(1 (cid:19) ) (cid:0) m (cid:18) (cid:0) m (cid:0) m (cid:19) (cid:22)(cid:18) 2(cid:19) (cid:14) A = (cid:0) f f f ; 6 1 (cid:11) (cid:12) 1 (cid:19) (cid:0) f (cid:18) (cid:0) f(cid:19) ’(cid:22)(cid:18) 2(cid:19) (cid:14) A = (cid:0) m m m ; 7 1 (cid:11) (cid:12) 1 (cid:19) (cid:0) m (cid:18) (cid:0) m(cid:19) A = (cid:22)(cid:18) f (1 (cid:0) (cid:11) f )(cid:19) f (cid:14)2 f ; 8 1 (cid:11) (cid:12) 1 (cid:19) f f ! (cid:0) (cid:0) ’(cid:22)(cid:18) (1 (cid:11) )(cid:19) (cid:14)2 A = m (cid:0) m m m , 9 1 (cid:11) (cid:12) 1 (cid:19) (cid:0) m (cid:18) (cid:0) m (cid:19) where (cid:1)(cid:25) (cid:25) (cid:25) . This derivation is technically correct as long as (cid:11) , (cid:11) , (cid:19) , and (cid:19) kt kt k;t 1 f m f m (cid:17) (cid:0) (cid:0) are not equal to 1, which was assumed in Section 2.8 The period loss function is a natural extension of period loss functions in other models. Under theassumptionofonesectorwithstickyprices, the(cid:133)rstlineoftheperiodlossfunctionisobtained. The assumption that both intermediate and (cid:133)nal goods sectors have sticky prices adds the second line. The third line is obtained if backward-looking price setters are included and (cid:14) = 0 (the k formulation in Gal(cid:237) and Gertler, 1999). The last two lines of the loss function arise if backwardlooking (cid:133)rms react to past marginal cost gaps in their rule of thumb. In(cid:135)ation in each sector corresponds to lower household utility since the interaction of sticky prices and in(cid:135)ation produces a set of suboptimal relative prices of di⁄erentiated goods, which then 8The derivation is given in a technical appendix that is available upon request. A value of 1 for (cid:11)f, (cid:11)m, (cid:19)f, or (cid:19)m would mean dividing by 0. These parameters equal 1 only if price setters never update their prices or are all backward-looking;these cases are not analyzed in this paper. 14

leadstoine¢ cientmixesofgoodsineachsector. Inthelossfunction,therealmarginalcostgapin the intermediate goods sector is connected to the relative price(cid:146)s role in the allocation of resources across sectors. Finally, as is standard in other models, deviations of consumption ((cid:133)nal output) from the e¢ cient level correspond to higher utility losses. In order to understand the connection between the household loss function and the proportion of backward-looking price setters, I present the loss function weights for a number of combinations of forward-looking and backward-looking price setters in Table 2. The calibration of the model determining these weights is explained in Section 3. The main e⁄ect of backward-looking price setters is a dramatic increase in the importance of smoothing the change of in(cid:135)ation, represented by A and A : 4 5 A A A A A A A A A A 0 1 2 3 4 5 6 7 8 9 (cid:19) =(cid:19) =0 0:01 1 0:6 0:002 0 0 0 0 0 0 f m (cid:19) =0:7;(cid:19) =0 0:01 1 0:6 0:002 3:11 0 0:13 0 0:001 0 f m (cid:0) (cid:19) =0:7;(cid:19) =0:2 0:01 1 0:6 0:002 3:11 0:20 0:13 0:009 0:001 0:000 f m (cid:0) (cid:0) (cid:19) =0:7;(cid:19) =0:7 0:01 1 0:6 0:002 3:11 1:87 0:13 0:080 0:001 0:001 f m (cid:0) (cid:0) Table 2: Household Loss Function Weights 3 Calibration of the Model Thissectiondiscussesthecalibrationusedforthebenchmarkmodel. Theassumptionthatu(C )= t log(C )impliesthat(cid:27) =1. Isetthesubjectivetimediscountfactorto(cid:12) =0:99,implyingthatthe t annual real interest rate in the steady state is about 4 percent, given that I interpret a time period asaquarter. Thesteady-statevaluesoftheelasticitiesofsubstitutionforthedi⁄erentiatedgoods, (cid:22)(cid:18) and(cid:22)(cid:18) ,aresetto10,whichimpliesasteady-statemarkupofabout11percent. Consistentwith f m earlier empirical work (e.g., Carlton, 1986; and Blinder et al., 1998) and following Huang and Liu (2005) and Strum (2009), I set the average price contract equal to one year, which means setting (cid:11) =(cid:11) =0:75. I also follow Huang and Liu (2005) and Strum (2009) in setting ’=0:6. f m Technologyshocksaretypicallyrepresentedassmallbutpersistent(see,forexample,Cooleyand 15

Prescott, 1995, and Gomme and Rupert, 2007). I set the AR(1) coe¢ cients for process governing the evolution of the technology factors to (cid:26) = (cid:26) = 0:95. I set the standard deviation of the f m innovations to the technology factor process in each sector, (cid:27) , to 0:02. ak Asnotedearlier,Iassumethatthecost-pushshocksarewhitenoiseprocessesthatdonotdepend on (cid:19) . I set (cid:27) ((cid:19) ) so that the standard deviation of u is 0:004 in the purely forward-looking k uk k kt case. I chose this value so that a negative two-standard-deviation shock does not cause the central bank to hit the zero bound on the nominal interest rate when implementing optimal policy with commitment.9 When (cid:19) >0, I use the expressions for u derived earlier to adjust (cid:27) ((cid:19) ). k kt uk k As I am not aware of any models with estimates of (cid:14) , I set these values so that the coe¢ cient k on lagged marginal cost in the backward-looking rule of thumb equals that of the coe¢ cient on current marginal cost in the Phillips curves in which there are no backward-looking price setters (similar to the approach in Steinsson, 2003). This leads me to set (cid:14) =(cid:14) =0:086. f m Anumberofauthorshave(cid:133)tempiricalestimatestohybridPhillipscurves. Fuhrer(1997)(cid:133)nds that setting more relative weight on lagged in(cid:135)ation does better, whereas Gal(cid:237) and Gertler (1999) (cid:133)nd that more weight should be put on the forward-looking term. I take a middle-of-the-road approach and set (cid:19) = 0:7, implying (cid:31)f = 0:52 and (cid:31)f = 0:48 in (12), which is in line with the f 1 2 estimates in Roberts (2005). Since Clark (1999) (cid:133)nds that the prices of goods at earlier stages of processing are more responsive to monetary policy shocks, I assume that the intermediate goods sector does not have greater persistence than the (cid:133)nal goods sector. Consequently, I set (cid:19) equal m to or less than (cid:19) , with three possible values for (cid:19) : 0;0:2; and 0:7. f m 9AlthoughStrum (2009)sets(cid:27)uk sothatthestandarddeviationofukt is0:02when(cid:133)rmsareforward-looking(a valueinlinewithWalsh,2003),thisvaluewouldmeanthatnegativecost-pushshockswouldleadthenominalinterest rate to hit the zero bound too often when the central bank acts under commitment to minimize the household loss function. 16

4 Optimal Policy with Commitment Inthissection,Iassumethatthecentralbankisassignedthehouseholdlossfunction. Furthermore, I assume that the central bank can credibly commit to state-contingent future actions. The intertemporal loss function of the central bank is given by E 1 (1 (cid:12))(cid:12)tLs. (20) 0 (cid:0) t t=0 X Withoutlossofgenerality,Ihavemultipliedthehouseholdlossfunctionby(1 (cid:12)),dividedoutthe (cid:0) leading coe¢ cient u C(cid:22) C(cid:22), and brought the coe¢ cient 1=2 into the period loss function.10 In c (cid:0) state-space form, the pe(cid:0)rio(cid:1)d loss function is given by 1 Ls = Y (cid:3)sY ; (21) t 2 t0 t where c~ t 2 3 (cid:27)c~ q~ t t (cid:0) 6 7 6 (cid:25) 7 6 ft 7 X t 6 7 6 (cid:1)(cid:25) 7 2 3 6 ft 7 Y t =D y t =6 6 7 7 . (22) 6 7 6’q~ t 1 +(1 ’)(cid:27)c~ t 17 6c~ 7 6 (cid:0) (cid:0) (cid:0) 7 6 t7 6 7 6 7 6 (cid:25) 7 4 5 6 mt 7 6 7 6 (cid:1)(cid:25) 7 6 mt 7 6 7 6 7 6 (cid:27)c~ t 1 q~ t 1 7 6 (cid:0) (cid:0) (cid:0) 7 4 5 The matrices D and (cid:3)s are given in the appendix. To obtain its state-contingent policy plan, the 10I also dropped the terms independent of policy and of third-order or higher. Multiplying the loss function by (1 (cid:12)) converts the loss from the discounted sum of period losses to the period value that would produce the (cid:0) discounted sum ifit occurred in every period(cid:151)a "constant period loss." 17

central bank solves the Lagrangian X X t+1 t (cid:10) 0 = E 0 X t 1 =0 (1 (cid:0) (cid:12))(cid:12)t8 >>>>< Ls t + (cid:20) (cid:24) 0t+1 (cid:23) 0t (cid:21) 0 B B H(cid:22) 2 6 6 y c~ t+1 3 7 7 (cid:0) A(cid:22)2 6 6 y c~ t 3 7 7 (23) B 6 t+17 6 t7 >>>>: C B @ 6 4 7 5 6 4 7 5 (cid:0) 2 6 6 0 0 3 7 7 " t+1 1 C C 9 >>>>= + 1 (cid:0) (cid:12) (cid:12) (cid:24) 00 (cid:0) X 0 (cid:0) X(cid:22) 0 (cid:1) , 6 7 C 6 4 7 5 C A >>>>; where I 0 0 H(cid:22) = 2 3 and A(cid:22)= A B : 0 H 0 (cid:20) (cid:21) 6 7 4 5 I have used the law of iterated expectations to write the Lagrangian more compactly. The initial conditions of X are given and equal to X(cid:22) . I have written the vector of Lagrangian multipliers t 0 relating to the non-predetermined variables as (cid:23) to emphasize that these variables depend on 0t information available at time t.11 TheresultsinSection2showthatin(cid:135)ationpersistencea⁄ectsthemodelinthreewaysthatmatter for policymakers: First, higher in(cid:135)ation persistence in a particular sector causes the household utility function to put more weight on smoothing in(cid:135)ation in that sector. Second, for the calibrations considered in this paper, a higher degree of sectoral in(cid:135)ation persistence leads to a (cid:135)atter sectoralPhillipscurvewithrespecttotheoutputgap.12 Third,highersectoralin(cid:135)ationpersistence means that policymakers must work against a greater degree of momentum when attempting to a⁄ect sectoral in(cid:135)ation. How do these additional complications arising from in(cid:135)ation persistence a⁄ect optimal policy under commitment? Figure 1 presents the optimal commitment responses to positive shocks under di⁄erent combinations of sectoral in(cid:135)ation persistence. The standard forward-looking case 11The solution procedure isgiven in a technicalappendix thatisavailable on request. See S(cid:246)derlind (1999)fora good exposition on the techniques used forproblems ofthis kind. 12This is true as long as (cid:31)k 3 >0. As noted earlier, given values for (cid:11)k, (cid:12), and (cid:19)k, su¢ ciently large values of (cid:14)k cancause(cid:31)k tobenegative,inwhichcasethemagnitudeofthecoe¢ cientoftheoutputgapwouldbeincreasingin 3 (cid:19)k. Forallofthe calibrations considered in this paper,(cid:31)k 3 remains positive. 18

is obtained by setting (cid:19) = (cid:19) = 0. Two cases of high (cid:133)nal goods in(cid:135)ation persistence and low f m intermediategoodsin(cid:135)ationpersistenceareobtainedbysetting(cid:19) =0:7and(cid:19) =0or0:2. Finally, f m high in(cid:135)ation persistence in both sectors is obtained by setting (cid:19) =(cid:19) =0:7. f m Productivity in Final Productivity in Intermediate Cost Push in Final Cost Push in Intermediate )e la n im o N e ta R tse re tn ta R d e z ila u n3 3 4 . . . 4 8 9 1 3 3 4 . . . 4 8 9 1 4. 4 5 5 4 5 6 In A3.7 3.7 ( 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 p a )n o 0.1 0.05 0.4 la e R G e ta R tse re ita iv e D tn e c 0.0 0 5 0.05 0 0. 0 2 0 0 0 . . . 0 2 4 6 tn I re P ( 0.05 0 5 10 15 0.1 0 5 10 15 0 5 10 15 0 5 10 15 )e ta R 0.2 0.4 1.5 0.05 la n iF n o ita lfn I d e z ila u n n 0 0 . . 4 2 0 0 0 .2 . 0 2 0. 0 1 5 0.05 0 A 0.5 0.1 ( 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 )e e ta id e m re tn n o ita lfn I ta R d e z ila u n 0 0 0 . . . 0 2 4 6 0 0 0 0 . . . 6 4 2 . 0 2 0 0 0 . . 2 1 . 0 1 0. 0 1 5 I n A 0.2 0.5 ( 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 Quarters Purely Forward Looking i = 0 m i = 0.2 m i = 0.7 m Figure 1: Impulse Responses to Shocks under Optimal Commitment 19

The (cid:133)rst and second columns of Figure 1 show the central bank(cid:146)s responses to positive onestandard-deviationproductivityshocks. Thetoprowshowsthe(annualized)valuesofthenominal interestratethatareconsistentwiththecentralbank(cid:146)spolicy. However,therealinterestrategap, shown in the second row, indicates the nature of monetary policy (whether policy is expansionary or contractionary). Initially, a positive productivity shock in the (cid:133)nal goods sector induces the central bank to enact contractionary policy (a positive real interest rate gap), whereas the central bank pursues expansionary policy (a negative real interest rate gap) in response to a positive productivity shock in the intermediate goods sector. The nature of the central bank(cid:146)s response does not depend on the degrees of in(cid:135)ation persistence in the two sectors. However, the timing and magnitude of the central bank(cid:146)s responses to productivity shocks depend on the degrees of in(cid:135)ation persistence in the two sectors. When in(cid:135)ation persistence is high in the (cid:133)nal goods sector but low in the intermediate goods sectors, the central bank delays its maximalresponseandincreasesthemagnitudeofitsmaximalresponse. Whenin(cid:135)ationpersistence is high in both sectors, the central bank also delays its maximal response relative to the forwardlooking case; however, unlike the case of unequal in(cid:135)ation persistence, the maximal response is of lesser magnitude than in the forward-looking case. Finally, the bottom two rows show that when a sector is populated by a large number of backward-looking price setters, in(cid:135)ation (or de(cid:135)ation) in that sector has a delayed and muted maximal response to a productivity shock in either sector. The third and fourth columns of Figure 1 show the dynamics following positive one-standarddeviation cost-push shocks. The simulations show that, unlike the case with productivity shocks, the central bank engages in contractionary policy when responding to positive cost-push shocks in either sector. On the other hand, just as is the case with productivity shocks, the nature of the response does not depend on the degrees of in(cid:135)ation persistence in the two sectors. The timing of the central bank(cid:146)s maximal response to a cost-push shock does not depend on the degrees of in(cid:135)ation persistence in the two sectors; however, the magnitude does. The maximal response comes in the initial period whether or not in(cid:135)ation is persistent. The magnitude of the central bank(cid:146)s response to a cost-push shock is strongly a⁄ected by the attenuation of the e⁄ect of 20

cost-push shocks on sectoral in(cid:135)ation that occurs as the percentage of backward-looking (cid:133)rms in a sectorincreases. Accordingly,themagnitudeofthecentralbank(cid:146)smaximalresponsetoacost-push shock decreases noticeably relative to the forward-looking case if in(cid:135)ation persistence is high in the sector hit by the shock. Finally, as the bottom two rows show, when a sector is characterized by highin(cid:135)ationpersistence,themagnitudeoftheinitialjumpinin(cid:135)ation(orde(cid:135)ation)inthatsector following a cost-push shock in either sector is lower than in the forward-looking case. Figures 2 and 3 show that the behavior of the price levels of (cid:133)nal and intermediate goods followingpositiveone-standard-deviationcost-pushshocksdependsonthepercentageofbackwardlookingpricesettersinthesectorhitbytheshock. So,forexample,theresponseofthepricelevelof (cid:133)nal goods to a cost-push shock in the intermediate goods sector depends critically on the number of backward-looking (cid:133)rms in the intermediate goods sector. Consistent with Steinsson (2003), the price levels of both sectors do not return to their pre-shock levels if there are backward-looking price setters. However, if the intermediate goods sector has few backward-looking price setters, the price levels of both sectors converge to levels very close to their pre-shock values following a cost-push shock in the intermediate goods sector. Furthermore, even when in(cid:135)ation persistence is high in the intermediate goods sector, the magnitudes of the permanent e⁄ect of an intermediate goodscost-pushshockonpricelevelsinbothsectorsarelowerthaniftheshockoccurredinthe(cid:133)nal goods sector (when persistence is high in the (cid:133)nal goods sector). Given that previous studies have found that price-level targeting under discretion works well in the forward-looking case, Figures 2 and 3 suggest that price-level targeting may perform well in an input-output model with in(cid:135)ation persistence, especially if the intermediate goods sector has fewer backward-looking (cid:133)rms, is more likelytoexperiencecost-pushshocksthanthe(cid:133)nalgoodssector,orhasbothofthesecharacteristics. 21

0.3 0.03 Purely Forward Looking i = 0 0.25 m i = 0.2 0.02 m leve i m = 0.7 L 0.2 lan 0.01 ig irO 0.15 m o rf n 0 o 0.1 itaive D 0.01 tnec 0.05 reP 0.02 0 0.05 0.03 0 10 20 30 0 10 20 30 Quarters Quarters Final Goods Shock Intermediate Goods Shock Figure 2: Final Goods Price-Level Responses to Cost-Push Shocks 0.08 0.3 Purely Forward Looking i = 0 0.06 0.25 i m = 0.2 m leve 0.04 i m = 0.7 L 0.2 lan ig irO 0.02 0.15 m o rf n 0 o 0.1 itaive 0.02 D tnec 0.04 0.05 reP 0.06 0 0.08 0.05 0 10 20 30 0 10 20 30 Quarters Quarters Final Goods Shock Intermediate Goods Shock Figure 3: Intermediate Goods Price-Level Responses to Cost-Push Shocks 5 Discretionary Policy Regimes Althoughananalysisoftheoptimalcommitmentpolicyisusefulforunderstandingthecharacteristics of desirable policy, Clarida, Gal(cid:237), and Gertler (1999) point out that central banks do not make 22

binding commitments. Furthermore, simple loss functions are easier for the public to understand andmonitor,facilitatingtransparencyandcommunication. Accordingly,Iexaminehouseholdutility losses that arise under alternative simple loss functions assigned to central banks acting under discretion. I consider the same combinations of in(cid:135)ation persistence as in the previous section andexaminethreeclassesoflossfunctions: in(cid:135)ationtargeting(IT),price-leveltargeting(PT),and speed-limit targeting (SL). I also report household losses under the optimal commitment policy (COM). Underin(cid:135)ationtargeting,thecentralbankactstokeepin(cid:135)ationandothervariables(suchasthe output gap) close to a set of target values.13 Price-level targeting di⁄ers from in(cid:135)ation targeting in that the central bank reacts to deviations of the price level from a target value. Speed-limit targetingstabilizes(cid:135)uctuationsinthechangeintheoutputgapinadditiontoothertargetvariables. Inaforward-lookingmodel,Strum(2009)(cid:133)ndsthattargetingin(cid:135)ationorpricelevelsinbothsectors dominatessingle-sectortargetingregimes; therefore,Iconsideronlyregimesthattargetin(cid:135)ationor price levels in both sectors. I represent these regimes by loss functions given by LIT = (cid:21) (cid:25)2 +(cid:21) (cid:25)2 +(cid:21) c~2; f ft m mt c t LPT = (cid:21) p2 +(cid:21) p2 +(cid:21) c~2; f ft m mt c t LSL = (cid:21) (cid:25)2 +(cid:21) (cid:25)2 +(cid:21) (c~ c~ )2: f ft m mt c t (cid:0) t (cid:0) 1 The government sets the weights (cid:21) , (cid:21) , and (cid:21) in each regime. The weights must be calculated f m c with care since arbitrarily chosen weights in loss functions may a⁄ect the ranking of the regimes. To address this problem, I use numerical methods to (cid:133)nd the set of weights for each policy regime that yields the minimum household loss for that regime type, subject to the constraint that each coe¢ cient be nonnegative.14 13In(cid:135)ation targeting is of particular interest since di⁄erent forms of it are pursued by central banks around the world. See Bernanke et al.,1999,fora good discussion. 14Thenumericalmethodusedto(cid:133)ndtheoptimalweightsstartswithaninitialguessofweightsandthenexamines how the household losses change as a function of the joint set of weights. The candidate optimal set of weights for eachregimewasfoundbyfollowingthedecreasinghouseholdlossesasafunctionofthesetofweightstoaminimum. To check the result,Iemployed a simulated annealing technique given in Yang et al. (2005). 23

Onedisadvantageofthebackward-lookingruleisthatoptimizingagentsmightnotusethesame backward-looking rule in di⁄erent regimes. Nevertheless, two factors suggest that the approach in this paper is useful. First, although di⁄erent regimes lead to di⁄erent short-run dynamics of in(cid:135)ation,alloftheregimesareconsistentwiththesamesteady-statein(cid:135)ationrateofzero. Second, (cid:133)rms do not adopt the rule to optimize or as a long-term rule, but rather at random times when information costs are high. Therefore, the backward-looking rule can be seen as a simple rule employed by (cid:133)rms when they occasionally (cid:133)nd the costs of information gathering and processing prohibitively expensive. The central bank(cid:146)s problem can be put in a standard linear-quadratic setup.15 I calculate and report the expected household losses under each loss function as E regime % of C(cid:22) =100 L , u C(cid:22) C(cid:22) (cid:2)C (cid:3) (cid:0) (cid:1) where C(cid:22) is the quarterly steady-state value of consumption ((cid:133)nal goods output) and E regime is L the expected discounted sum of household losses minus the losses that would occur in(cid:2)the e¢ ci(cid:3)ent (cid:135)exible-price equilibrium.16 Table 3 reports the coe¢ cients and expected household losses for each regime according to the percentages of backward-looking (cid:133)rms (and, hence, degrees of in(cid:135)ation persistence) in the two sectors.17 Two observations jump out immediately. First, the absolute levels of losses are highest in the fully forward-looking model and lowest in the model with high persistence in both sectors. When the percentage of backward-looking price setters is high, the attenuation of the e⁄ects of cost-push shocks in the sectoral Phillips becomes important. Second, even though there is always a clear ranking of regime performance, the di⁄erence between the bestand worst regimes decreases as in(cid:135)ation persistence increases. 15The particular details of my solution are given in a technical appendix that is available upon request. See S(cid:246)derlind (1999) for a good exposition on the techniques used forthese types ofproblems. 16Strum (2009) reports the constant period loss, that is the loss that would produce the discounted sum if it occurred at that constant level each period, forever. The constant period loss is obtained by multiplying the discounted sum by (1 (cid:12)). 17Thecommitment (cid:0) policycontainsacoe¢ cientfortherealmarginalcostgapthatisnotreportedinTable3since the discretionary regimes do not include this variable. 24

(cid:19) =(cid:19) =0 (cid:19) =0:7; (cid:19) =0 f m f m (cid:21) (cid:21) (cid:21) % of C(cid:22) (cid:21) (cid:21) (cid:21) % of C(cid:22) f m c f m c IT 1 0:72 0:012 13:46 1 0:48 0:007 10:50 PT 1 0:56 0:010 12:30 1 0:94 0:019 10:01 SL 1 0:61 0:010 12:51 1 0:51 0:008 9:90 COM 1 0:60 0:009 12:18 1 0:60 0:009 9:81 (cid:19) =0:7; (cid:19) =0:2 (cid:19) =(cid:19) =0:7 f m f m (cid:21) (cid:21) (cid:21) % of C(cid:22) (cid:21) (cid:21) (cid:21) % of C(cid:22) f m c f m c IT 1 0:58 0:005 10:41 1 0:68 0:003 9:59 PT 1 0:97 0:019 10:30 1 0:74 0:038 9:66 SL 1 0:55 0:005 10:07 1 0:65 0:004 9:57 COM 1 0:60 0:009 10:01 1 0:60 0:009 9:44 Table 3: Benchmark Model Results The qualitative results are summarized in Table 4. Consistent with Strum (2009), price-level targetingproducesthelowesthouseholdlossesinthepurelyforward-lookingmodel. Whenin(cid:135)ation persistence is high in the (cid:133)nal goods sector but low in the intermediate goods sector, speed-limit targeting performs best. At the same time, price-level targeting outperforms in(cid:135)ation targeting. Whenin(cid:135)ationpersistenceishighinbothsectors,speed-limittargetingperformsbest. However,in this instance, in(cid:135)ation targeting performs better than price-level targeting (and only slightly worse thanspeed-limittargeting). TheseresultsarebroadlyconsistentwithWalsh(2003),who(cid:133)ndsthat speed-limit targeting outperforms price-level targeting once moderate levels of in(cid:135)ation persistence arereached,andthatin(cid:135)ationtargetingperformswellwhenin(cid:135)ationpersistenceishigh. However, these results show that the degrees of in(cid:135)ation persistence in both intermediate and (cid:133)nal goods sectors can be important when ranking regime performance. Regime Ranking In(cid:135)ation Persistence (Final/Intermediate) First Second Third Forward-Looking: (cid:19) =0; (cid:19) =0 PT SL IT f m High/Low: (cid:19) =0:7; (cid:19) 0;0:2 SL PT IT f m 2f g High/High: (cid:19) =0:7; (cid:19) =0:7 SL IT PT f m Table 4: Benchmark Model Qualitative Results 25

6 Robustness of Discretionary Regimes In the previous section, the government crafted loss functions based on an accurate assessment of in(cid:135)ation persistence, the sources of shocks to the economy, and the model of the economy to be used. However,monetarypolicymaynotbepracticedinsuchfavorableconditions. Inthissection, I examine how discretionary regimes perform when the central bank minimizes loss functions that were crafted based on assumptions that may not be true. 6.1 The Degrees of In(cid:135)ation Persistence I begin by examining how policies crafted under incorrect assumptions about in(cid:135)ation persistence perform. Speci(cid:133)cally, I follow a two-step process: First, I assume that the government chooses loss function coe¢ cients based on assumptions regarding in(cid:135)ation persistence in each sector that may not be accurate. Second, I determine the household losses that occur when the central bank implementsthesepoliciesinthetrueeconomy. Table5reportstheresultsfromthisexercise. For example, in the block of columns under "0=0," the table shows the household losses for policies crafted under four assumed combinations of degrees of in(cid:135)ation persistence, and implemented in a forward-looking world. Price-level targeting performs best in the forward-looking world in three of the four scenarios. Whenin(cid:135)ationispersistentinthe(cid:133)nalgoodssectorbutforward-lookingintheintermediategoods sector, price-level targeting performs best in two of the four scenarios, while speed-limit targeting performsbestintheothertwoscenarios. Whenin(cid:135)ationpersistenceishighinthe(cid:133)nalgoodssector andeitherlow(butpositive)orhighintheintermediategoodssector,speed-limittargetingperforms best. Althoughin(cid:135)ationtargetingneverperformsbest,itoutperformsprice-leveltargetinginthree of the four scenarios when in(cid:135)ation persistence is high in both sectors. Finally, as the true levels of in(cid:135)ation persistence rise, the losses from making the wrong assumptions about the degrees of in(cid:135)ation persistence decrease. 26

True (cid:19) =(cid:19) 0=0 0:7=0 f m Assumed (cid:19) =(cid:19) 0=0 0:7=0 0:7=0:2 0:7=0:7 0=0 0:7=0 0:7=0:2 0:7=0:7 f m IT 13:46 13:67 14:07 15:20 10:58 10:50 10:80 11:98 PT 12:30 12:60 12:63 12:69 10:46 10:01 10:01 10:16 SL 12:51 12:57 12:79 12:97 9:93 9:90 10:04 10:31 True (cid:19) =(cid:19) 0:7=0:2 0:7=0:7 f m Assumed (cid:19) =(cid:19) 0=0 0:7=0 0:7=0:2 0:7=0:7 0=0 0:7=0 0:7=0:2 0:7=0:7 f m IT 10:65 10:59 10:41 10:73 9:81 9:85 9:67 9:59 PT 10:81 10:31 10:30 10:47 9:90 9:78 9:80 9:63 SL 10:17 10:15 10:07 10:13 9:62 9:66 9:60 9:57 Table 5: Benchmark Household Losses under Alternative Assumptions about In(cid:135)ation Persistence 6.2 The Sources of Shocks Next,Iexamineregimeperformancewhenthelossfunctionisbasedontheassumptionthatshocks hiteitheronlythe(cid:133)nalgoodssectororonlytheintermediategoodssector,when,infact,shockshit both sectors. I assume that the government correctly perceives the degrees of in(cid:135)ation persistence in both sectors when it sets the loss functions. I follow a two-step process similar to the one above: I calibrate loss function coe¢ cients that are optimal under the assumption that shocks hit only one of the sectors, then I run the implied policies in economies that are subject to shocks in bothsectors. Table6reportstheresultsforthefourpossiblecombinationsofin(cid:135)ationpersistence. The columns labeled "F" show the results when the central bank is assigned a loss function based on the assumption that shocks arise only in the (cid:133)nal goods sector. The columns labeled "M" show the results when the loss function is based on the assumption of shocks coming only from the intermediate goods sector. For example, in the two-column block under "0:7=0," the column labeled"F"showsthehouseholdlossesunderregimesbasedontheassumptionthatshockshitonly the (cid:133)nal goods sector, when, in fact, shocks hit both sectors. If the government crafts the loss function based on the incorrect assumption that shocks hit only one sector, better results are usually obtained if the government assumes that shocks hit the intermediate goods sector. Regardless of which sector is assumed to be hit by shocks, price-level targeting performsbestin the forward-lookingcase. When in(cid:135)ation persistence is high in the (cid:133)nal goodssectorandeitherloworhighintheintermediategoodssector,speed-limittargetingperforms 27

best. Although in(cid:135)ation targeting ranks last both when the economy is fully forward-looking and whenin(cid:135)ationpersistenceishighinthe(cid:133)nalgoodssectorbutlowintheintermediategoodssector, it outperforms price-level targeting when in(cid:135)ation persistence is high in both sectors. True (cid:19) =(cid:19) 0=0 0:7=0 0:7=0:2 0:7=0:7 f m Assumed shocks F M F M F M F M IT 14:16 13:71 12:22 10:54 11:45 10:42 9:60 9:60 PT 12:32 12:31 10:09 10:02 10:39 10:32 9:64 9:64 SL 12:52 12:53 9:96 9:90 10:14 10:07 9:57 9:57 Table 6: Benchmark Household Losses under Incorrect Assumptions about the Sources of Shocks 6.3 Assumption of a One-sector Model Finally, suppose that the central bank is assigned a loss function based on a standard one-sector model in which in(cid:135)ation is assumed to be (cid:133)nal goods in(cid:135)ation. I consider two alternatives: a loss function based on a forward-looking one-sector model and a loss function based on a one-sector model with many backward-looking price setters. Table 7 reports the household losses when these policies are implemented in the four input-output economies examined in earlier sections. For example, in the two-column block under "0:7=0," the column labeled "0" shows the household losses under regimes based on the assumption of a one-sector model with no in(cid:135)ation persistence, when the true economy has an input-output structure with high in(cid:135)ation persistence in the (cid:133)nal goods sector and no in(cid:135)ation persistence in the intermediate goods sector. When the one-sector regimes are incorrectly implemented in input-output economies, regime rankingsfromearlierexercisesdonotholdastightly. Intheforward-lookinginput-outputeconomy, one-sector in(cid:135)ation targeting performs best if a forward-looking model is assumed, whereas onesector price-level targeting performs best if in(cid:135)ation persistence is assumed when crafting the loss functions.18 If in(cid:135)ation is persistent in at least one sector in the true input-output economy, one-sector in(cid:135)ation targeting always performs best. 18Thisresultdi⁄ersfrom Strum(2009)because,in thecalibrationin thispaper,thecost-push shocksareassumed tohavemuchsmallerstandarddeviations. Ifthestandarddeviationsofcost-pushshockswereassumedtobeslightly larger,thenprice-leveltargetingcraftedundertheassumptionofaforward-lookingone-sectormodelwouldperform best in the forward-looking input-output economy. 28

True (cid:19) =(cid:19) 0=0 0:7=0 0:7=0:2 0:7=0:7 f m Assumed (cid:19) 0 0:7 0 0:7 0 0:7 0 0:7 (f) IT 15:93 25:93 12:35 13:21 12:85 13:94 11:81 13:92 PT 16:90 16:29 14:53 14:08 15:51 14:99 16:86 16:07 SL 16:93 23:72 13:29 14:55 14:02 15:50 14:43 16:97 Table 7: Benchmark Household Losses under One-Sector Model Assumption 6.4 Summary of Robustness Results The three exercises in this section point to a few qualitative results about the robustness of loss functions that are formed under assumptions that may not be true of the economy in which they are implemented. Table 8 reports the best-performing regime for each characterization of the economy under the possibly incorrect assumptions studied in the previous three exercises and the best-performing regime when correct assumptions are used. The type of regime that performs best under di⁄erent combinations of in(cid:135)ation persistence is not a⁄ected by the government(cid:146)s assumptionsregardingin(cid:135)ationpersistenceorthesourcesofshocks(whencraftingthelossfunction). In particular, if the economy is fully forward-looking, price-level targeting usually performs best. When in(cid:135)ation is persistent in one or both sectors, speed-limit targeting usually performs best. However, the type of regime that performs best is a⁄ected by the incorrect use of a one-sector model. In this case, in(cid:135)ation targeting performs best in one of the two forward-looking cases and in every case when in(cid:135)ation in the true economy is persistent in one or both sectors. True In(cid:135)ation Persistence (Final/Intermediate): Forward-looking High/Low High/High Type of Possibly Incorrect Assumption Degrees of In(cid:135)ation Persistence PT SL SL Sources of Shocks PT SL SL One Sector Model IT or PT IT IT Correct Assumptions PT SL SL Table 8: Best Regimes according to Assumptions 29

7 Conclusion Adding in(cid:135)ation persistence (through backward-looking price setters) to a New Keynesian model in which prices are sticky in both intermediate and (cid:133)nal goods sectors alters the household loss functionandthesectoralPhillipscurves. Consequently,thedegreesofin(cid:135)ationpersistenceinboth sectors can a⁄ect the implementation and design of monetary policy in New Keynesian models. When conducting the optimal commitment policy, the nature of the central bank(cid:146)s responses to shocks(cid:151)whether expansionary or contractionary(cid:151)is not a⁄ected by the degrees of in(cid:135)ation persistence. However, the timing and magnitude of the central bank(cid:146)s responses shocks can be a⁄ected by the degrees of in(cid:135)ation persistence in the two sectors. When in(cid:135)ation is persistent, the maximal response to productivity shocks by the central bank is delayed relative to the forwardlookingcase. Whenin(cid:135)ationpersistenceishighinthe(cid:133)nalgoodssectorbutlowintheintermediate goods sector, the central bank(cid:146)s maximal response is greater in magnitude than in the forwardlooking case. On the other hand, the magnitude of the central bank(cid:146)s maximal response is lower than in the forward-looking case when in(cid:135)ation persistence is high in both sectors. The timing of thecentralbank(cid:146)sresponsetocost-pushshocksdoesnotdependonin(cid:135)ationpersistence. However, the magnitude of the central bank(cid:146)s response to a cost-push shock decreases as the percentage of backward-looking (cid:133)rms and the degree of in(cid:135)ation persistence in the sector hit by the shock increase. When the central bank acts under discretion, the type of regime that performs best depends on the degrees of in(cid:135)ation persistence in both sectors. As in Strum (2009), price-level targeting performs best when both sectors are fully forward-looking. Speed-limit targeting performs best whenin(cid:135)ationpersistenceishighinthe(cid:133)nalgoodssectorbutlowintheintermediategoodssector. In this case, both speed-limit targeting and price-level targeting outperform in(cid:135)ation targeting. Whenin(cid:135)ationpersistenceishighinbothsectors,speed-limittargetingstillperformsbest;however, in this case, in(cid:135)ation targeting outperforms price-level targeting. When crafting the loss function to assign to the central bank, incorrect assumptions can be madeaboutthedegreeofin(cid:135)ationpersistence,thesourcesofshocks,orwhethertouseaone-sector 30

model. Underthecalibrationconsideredhere,thetypeofregimethatperformsbestunderdi⁄erent combinations of in(cid:135)ation persistence is not a⁄ected by the government(cid:146)s assumptions regarding in(cid:135)ation persistence or the sources of shocks (when crafting the loss function). However, the type of regime that performs best is a⁄ected by the incorrect use of a one-sector model. Finally, in assessing these results, it is important to remember that the mechanism generating in(cid:135)ation persistence in the model may be important. Further research into the sources of in(cid:135)ation persistence would enable clearer connections between the design of policy in a model and in the real world. Nevertheless, this paper has shown that accounting for both sticky prices and in(cid:135)ation persistenceatdi⁄erentstagesofproductioncanbeimportantforsticky-pricemodelsusedtostudy monetary policy. In(cid:135)ation dynamics and policy trade-o⁄s can depend on the degrees of in(cid:135)ation persistence at multiple stages of production. 31

A Structural Matrices for the Economy I represent the structural equations of the economy as X X C t+1 t =A +Bc~ + " : 2 3 2 3 t 2 3 t+1 HE y y 0 t t+1 t 6 7 6 7 6 7 4 5 4 5 4 5 The matrices that describe the exact state-space structural relations for the economy are given by 0 0 0 2 3 H = 0 (cid:31)f(cid:12) 0 , 1 6 7 60 0 (cid:31)m(cid:12)7 6 1 7 6 7 4 5 (cid:26) 0 0 0 0 0 0 0 0 0 0 0 0 uf 2 3 0 (cid:26) 0 0 0 0 0 0 0 0 0 0 0 um 6 7 6 0 0 (cid:26) 0 0 0 0 0 0 0 0 0 07 6 f 7 6 7 6 7 6 0 0 0 (cid:26) m 0 0 0 0 0 0 0 0 07 6 7 6 7 6 0 0 1 0 0 0 0 0 0 0 0 0 07 6 7 6 7 6 0 0 0 1 0 0 0 0 0 0 0 0 07 6 7 6 7 A=6 6 0 0 0 0 0 0 0 0 0 0 1 0 0 7 7 ; 6 7 6 7 6 0 0 0 0 0 0 0 0 0 0 0 1 07 6 7 6 7 6 0 0 0 0 0 0 0 0 0 0 0 0 17 6 7 6 7 6 7 6 0 0 0 0 0 0 0 0 0 0 0 0 07 6 7 6 7 6 0 0 (cid:9) (cid:9) (cid:9) (cid:9) 1 0 0 0 1 1 17 6 (cid:0) (cid:0) (cid:0) (cid:0) 7 6 7 6 1 0 0 0 0 0 (cid:31)f’ (cid:31)f 0 (cid:31)f(cid:9)(cid:27) (cid:31)f’ 1 07 6 6(cid:0) (cid:0) 4 (cid:0) 2 (cid:0) 4 (cid:0) 3 7 7 6 6 6 0 (cid:0) 1 0 0 0 0 (cid:31)m 4 0 (cid:0) (cid:31)m 2 (cid:0) (cid:31)m 4 (cid:27) (cid:31)m 3 0 1 7 7 7 4 5 32

where (cid:9)=1 ’, (cid:0) 0 2 3 0 6 7 6 0 7 6 7 6 7 6 7 6 0 7 6 7 6 7 6 0 7 6 7 6 7 6 0 7 6 7 6 7 B =6 6 0 7 7 , 6 7 6 7 6 0 7 6 7 6 7 6 0 7 6 7 6 7 6 7 6 1 7 6 7 6 7 6 0 7 6 7 6 7 6 (cid:31)f(cid:9)(cid:27)7 6 6(cid:0) 3 7 7 6 6 6(cid:0) (cid:31)m 3 (cid:27) 7 7 7 4 5 and (cid:27) ((cid:19) ) 0 0 0 uf f 2 3 0 (cid:27) ((cid:19) ) 0 0 um m 6 7 6 0 0 (cid:27) 0 7 6 f 7 6 7 6 7 6 0 0 0 (cid:27) m7 6 7 6 7 6 0 0 0 0 7 C =6 7: 6 7 6 0 0 0 0 7 6 7 6 7 6 7 6 0 0 0 0 7 6 7 6 7 6 0 0 0 0 7 6 7 6 7 6 0 0 0 0 7 6 7 6 7 6 7 6 0 0 0 0 7 6 7 4 5 33

The targeting variables in the loss function are obtained from the structural equations via the following transformation: X t 2 3 Y t =D y t ; 6 7 6c~ 7 6 t7 6 7 4 5 where 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 3 0 0 0 0 0 0 0 0 0 0 1 0 0 (cid:27) (cid:0) 6 7 60 0 0 0 0 0 0 0 0 0 0 1 0 07 6 7 6 7 60 0 0 0 0 0 0 1 0 0 0 1 0 07 6 7 D =6 (cid:0) 7: 6 7 60 0 0 0 0 0 ’ 0 0 (1 ’)(cid:27) 0 0 0 07 6 (cid:0) 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 1 07 6 7 6 7 60 0 0 0 0 0 0 0 1 0 0 0 1 07 6 7 6 (cid:0) 7 6 7 60 0 0 0 0 0 1 0 0 (cid:27) 0 0 0 07 6 (cid:0) 7 4 5 In state-space form, I write the policy period loss function as 1 Ls = Y (cid:3)sY ; (24) t 2 t0 t where (cid:27) 0 0 0 0 0 0 0 2 3 0 ’(1 ’) 0 0 0 0 0 0 (cid:0) 6 7 60 0 X 0 0 0 0 0 7 6 1 7 6 7 60 0 0 X Y 0 0 0 7 (cid:3)s = 6 6 2 1 7 7; 6 7 60 0 0 Y 1 X 3 0 0 0 7 6 7 6 7 60 0 0 0 0 X 0 0 7 6 4 7 6 7 60 0 0 0 0 0 X Y 7 6 5 27 6 7 6 7 60 0 0 0 0 0 Y 2 X 67 6 7 4 5 34

(cid:18) (cid:11) X = f f ; 1 (1 (cid:11) (cid:12))(1 (cid:11) ) f f (cid:0) (cid:0) (cid:18) (cid:19) X = f f ; 2 (1 (cid:11) (cid:12))(1 (cid:11) )(1 (cid:19) ) f f f (cid:0) (cid:0) (cid:0) (cid:18) (1 (cid:11) )(cid:19) (cid:14)2 X = f (cid:0) f f f ; 3 (1 (cid:11) (cid:12))(1 (cid:19) ) f f (cid:0) (cid:0) ’(cid:18) (cid:11) X = m m ; 4 (1 (cid:11) (cid:12))(1 (cid:11) ) m m (cid:0) (cid:0) ’(cid:18) (cid:19) X = m m ; 5 (1 (cid:11) (cid:12))(1 (cid:11) )(1 (cid:19) ) m m m (cid:0) (cid:0) (cid:0) ’(cid:18) (1 (cid:11) )(cid:19) (cid:14)2 X = m (cid:0) m m m; 6 (1 (cid:11) (cid:12))(1 (cid:19) ) m m (cid:0) (cid:0) and (cid:18) (cid:19) (cid:14) Y = (cid:0) f f f ; 1 (1 (cid:11) (cid:12))(1 (cid:19) ) f f (cid:0) (cid:0) ’(cid:18) (cid:19) (cid:14) Y = (cid:0) m m m : 2 (1 (cid:11) (cid:12))(1 (cid:19) ) m m (cid:0) (cid:0) 35

References Amato, Je⁄ery D. and Thomas Laubach (2003), "Rule-of-thumb Behavior and Monetary Policy," European Economic Review, 47, 791-831. Bernanke, Ben S., Thomas Laubach, Frederic S. Mishkin, and Adam S. Posen (1999), In(cid:135)ation Targeting, Princeton University Press, Princeton, New Jersey. Blinder, Alan S., Elie R. D. Canetti, David E. Lebow and Jeremy B. Rudd (1998), Asking About Prices: A New Approach to Understanding Price Stickiness, Russell Sage Foundation, New York. Calvo, Guillermo (1983), "Staggered Prices in a Utility-Maximizing Framework," Journal of Monetary Economics, 12, 383-398. Carlton, Dennis W. (1986), "The Rigidity of Prices," American Economic Review, 76, 637-658. Clarida, Richard, Jordi Gal(cid:237), and Mark Gertler (1999), "The Science of Monetary Policy: A New Keynesian Perspective," Journal of Economic Literature, 37, 1661-1707. Clark, Todd E. (1999), "The Responses of Prices at Di⁄erent Stages of Production to Monetary Policy Shocks," The Review of Economics and Statistics, 81, 420-433. Cooley, Thomas F. and Edward C. Prescott (1995), "Economic Growth and Business Cycles," in ThomasF.Cooley,ed.,FrontiersofBusinessCycleResearch,PrincetonUniversityPress,Princeton, New Jersey. Dixit,AvinashK.andJosephE.Stiglitz(1977),"MonopolisticCompetitionandOptimumProduct Diversity," American Economic Review, 67, 297-308. Fuhrer, Je⁄rey C. (1997), "The (Un)Importance of Forward-Looking Behavior in Price Speci(cid:133)cations," Journal of Money, Credit and Banking, 29, 338-350. Gal(cid:237), Jordi and Mark Gertler (1999), "In(cid:135)ation Dynamics: A Structural Econometric Analysis," Journal of Monetary Economics, 44, 195-222. 36

Gomme, Paul and Peter Rupert (2007), "Theory, Measurement and Calibration of Macroeconomic Models," Journal of Monetary Economics, 54, 460-497. Hansen, Gary D. (1985), "Indivisible Labor and the Business Cycle," Journal of Monetary Economics, 16, 309-327. Huang, Kevin X. D. and Zheng Liu (2005), "In(cid:135)ation Targeting: What In(cid:135)ation Rate to Target?" Journal of Monetary Economics, 52, 1435-1462. NessØn,MarianneandDavidVestin(2005),"AverageIn(cid:135)ationTargeting,"JournalofMoney,Credit and Banking, 37, 837-863. Roberts, John M. (2005), "How Well Does the New Keynesian Sticky-Price Model Fit the Data?" The B.E. Journal of Macroeconomics: Contributions to Macroeconomics, 5, Issue 1, Article 10. Rudebusch, Glenn D. (2002), "Assessing Nominal Income Rules for Monetary Policy with Model and Data Uncertainty," The Economic Journal, 112, 402-432. S(cid:246)derlind, Paul. (1999) "Solution and Estimation of RE Macromodels with Optimal Policy." European Economic Review 43, 813-823. Steinsson,J(cid:243)n(2003),"OptimalMonetaryPolicyinanEconomywithIn(cid:135)ationPersistence,"Journal of Monetary Economics, 50, 1425-1456. Strum, Brad E. (2009), "Monetary Policy in a Forward-Looking Input-Output Economy," Journal of Money, Credit and Banking, 41, 619-650. Walsh, Carl E. (2003), "Speed Limit Policies: The Output Gap and Optimal Monetary Policies," American Economic Review, 93, 265-278. Woodford, Michael (2003), Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press, Princeton, New Jersey. Woodford, Michael (2007), "Interpreting In(cid:135)ation Persistence: Comments on the Conference on (cid:146)Quantitative Evidence on Price Determination,(cid:146)" Journal of Money, Credit, and Banking, 39, 203-210. 37

Yang,WonY.,WenwuCao,Tae-SangChung,andJohnMorris,(2005),AppliedNumericalMethods Using MATLAB, John Wiley and Sons, Hoboken, New Jersey. 38