On Targeting Frameworks and Optimal Monetary Policy

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. On Targeting Frameworks and Optimal Monetary Policy Martin Bodenstein and Junzhu Zhao 2017-098 Please cite this paper as: Bodenstein, Martin, and Junzhu Zhao (2017). “On Targeting Frameworks and Optimal Monetary Policy,” Finance and Economics Discussion Series 2017-098. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.098. 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.

On Targeting Frameworks and Optimal Monetary Policy Martin Bodenstein Junzhu Zhao Federal Reserve Board Nanjing Audit University September 13, 2017 Abstract Speedlimitpolicy,amonetarypolicystrategythatfocusesonstabilizingin(cid:13)ationandthechangeintheoutput gap, consistently delivers better welfare outcomes than (cid:13)exible in(cid:13)ation targeting or (cid:13)exible price level targeting in empirical New Keynesian models when policymakers lack the ability to commit to future policies. Even if the policymakercancommitunderanin(cid:13)ationtargetingstrategy,thediscretionaryspeedlimitpolicyperformsbetter for most empirically plausible model parameterizations from a normative perspective. JEL classi(cid:12)cations: E52, E58 Keywords: in(cid:13)ation targeting, price level targeting, speed limit policy, optimal monetary policy, delegation. (cid:3) The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as re(cid:13)ecting theviewsoftheBoardofGovernorsoftheFederalReserveSystemoranyotherpersonassociatedwiththeFederalReserve System. WearegratefultoLarryChristiano,ChrisGust,PaulLevine,JosephPearlman,BenJohannsen,andRobertTetlow forhelpfulcommentsandsuggestions. (cid:3)(cid:3) Contact information: Martin Bodenstein (corresponding author), E-mail Martin.R.Bodenstein@frb.gov; Junzhu Zhao, E-mailjunzhuzhao@gmail.com 1

1 Introduction The optimal delegation problem in monetary policy studies how a central bank can best serve the interests of society when the optimal state-contingent plan derived under the true social objective function is time-inconsistent. Starting with Rogoff (1985), several authors have shown that assigning the central bank an objective that differs from the true social objective can lead to better normative outcomes under discretionary policymaking than otherwise.1 One such central bank objective is the speed limit policy under which, according to Walsh (2003), the policymaker focuses on stabilizing in(cid:13)ation and the change in the output gap. We show that in the discretionary Markov equilibrium, the speed limit policy framework consistently outperforms (cid:13)exible in(cid:13)ation targeting and often performs better than (cid:13)exible price level targeting in a set of New Keynesian models (NKM) ranging from the purely forward-looking textbook version of the NKM and its extensions to the medium-scale DSGE model in Christiano, Eichenbaum, and Evans (2005) as implemented and estimated in Smets and Wouters (2007) (CEE/SW model).2 ThespeedlimitpolicyperformsstronglyinthediscretionaryMarkovequilibriumasitcapturesarobust feature of the optimal monetary policy under commitment (henceforth, optimal commitment policy) in NKMs: The policymaker promises to keep future monetary policy tight in response to shocks that drive up in(cid:13)ation, such as a positive price markup shock, as evidenced by a slow closing of the negative output gap under the optimal commitment policy. The persistent rise in the policy interest rate deters excessive price and wage adjustments by the private sector in the impact period and reduces overall movements in in(cid:13)ation under the optimal commitment policy. Importantly, the price level is not necessarily stationary under the optimal commitment policy. The speed with and the extent to which nominally rigid prices and wages return to their long-run trend paths depend on the degree of price and wage indexation to past in(cid:13)ation. As the speed limit policy interprets the idea of stabilizing the real economy as preventing large changes intheoutputgapasopposedtodeviations oftheoutputgapfromzero,thepolicymakerprefersdelayingthe closing of the negative output gap after the in(cid:13)ationary shock by construction and keeps future monetary policy tight regardless of the policymaker’s ability to commit. If the private sector understands this behavior of the central bank, the rise in in(cid:13)ation is kept small while the price level rises permanently by a small amount. The price level targeting framework also incorporates the idea of keeping monetary policy tight after an in(cid:13)ationary shock albeit through a different mechanism. By assumption, the policymaker is determined to drive the price level back to its trend path under this framework and keeps the interest rate elevated to undo earlier changes induced by the shock. Anticipating such a policy move, households and 1Important contributions include King (1997), Svensson (1997), Svensson (1999), Clarida, Gali, and Gertler (1999), Walsh (2003), Woodford(2003b),NessenandVestin(2005),Vestin(2006),andBilbiie(2014). 2Consistentwiththeliterature,wede(cid:12)nethatundera(cid:13)exibletargetingframeworkthecentralbankminimizesthediscountedin(cid:12)nite sumofaperiodlossfunctionthatre(cid:13)ectsthecentralbank’spreferencesoverstabilizingpricesandtherealeconomysubjecttoitsmodelof theeconomy. Underin(cid:13)ationtargeting,thelossfunctionplacesweightonthesquareddeviationsofin(cid:13)ationfromitslong-runtargetand oftheoutputgapfromzeroasinSvensson(2010). Thepricelevel(indeviationfromadeterministictrend)takestheplaceofin(cid:13)ationin thelossfunctionunderpriceleveltargeting;inadditionthelossfunctionplacesweightonthesquarreddeviationsoftheoutputgapfrom zero. Finally as in Walsh (2003), the central bank’s loss function features an aversion to squarred deviations of in(cid:13)ation from its target andofthegrowth rate oftheoutputgapunderthespeedlimitpolicy. 2

(cid:12)rms feel deterred from implementing large changes in prices and wages in the (cid:12)rst place. By contrast, the in(cid:13)ation targeting framework lacks a built-in mechanism that facilitates implementing tight monetary policy after an in(cid:13)ationary shock in the discretionary Markov equilibrium. As the policymakerintendstostabilizein(cid:13)ationandtheleveloftheoutputgap,thepolicymakerwillnotbeexpectedto drive prices back to their trend level or to delay the closing of the output gap under the in(cid:13)ation targeting objective. In line with the \weight-conservative" central banker of Rogoff (1985), placing a high weight on stabilizing in(cid:13)ation helps improving the performance of the in(cid:13)ation targeting framework, but is generally too crude to make in(cid:13)ation targeting attractive relative to the speed limit policy under discretionary policymaking. Only in the simplest NKMs with a high degree of indexation to past in(cid:13)ation can in(cid:13)ation targeting perform best, since in this case the desirability of returning the price level to its previous trend vanishes under the optimal commitment policy. In more complex models featuring habit persistence in consumptionorstickynominalwages(unlesshighlyindexedtoin(cid:13)ationaswell)ortheempiricalCEE/SW model in(cid:13)ation targeting is undesirable irrespective of the degree of price indexation when policymakers cannot commit.3 Although, we view the case of discretionary policymaking as more realistic, we also report (cid:12)ndings for the case that the central bank can commit to future actions.4 Under commitment, the in(cid:13)ation targeting centralbankdoesdrivepricesandwagesbacktowardstheirlong-runtrendsifsodesiredundertheoptimal commitment policy and performs reliably best across models from the textbook NKM to the CEE/SW model with the speed limit policy a close second. Since under price level targeting the central bank will never allow for permanent changes in prices and wages, this framework performs worst when prices and wages are highly indexed to past in(cid:13)ation.5 Several experiments in the CEE/SW model lend further support to the speed limit policy framework when policymakers can only act under discretion. Beyond parameterizing the model at the mode of the posterior distribution reported in Smets and Wouters (2007), we consider alternative parameter choices drawn from the Laplace approximation to the posterior distribution. When the objective functions are parameterized optimally for each parameter draw, the speed limit policy dominates for almost all 30,000 empirically plausible draws when policymakers act under discretion. Surprisingly, the speed limit policy under discretion outperforms the in(cid:13)ation targeting framework under commitment for the majority of draws (including our benchmark parameterization). When we compare the targeting frameworks for selected speci(cid:12)cations of the objective functions that do not vary across the 30,000 parameterizations of the CEE/SW model, the speed limit policy almost always dominates regardless of the central bank’s ability to commit. Our (cid:12)ndings prevail in a version of the CEE/SW model that is estimated with euro area data instead ofUSdataoraversionthatreducestheimportanceofwagemarkupshocksrelativetolaborsupplyshocks 3Underhabitpersistencesmoothingaquasi-differenceoftheoutputgapentersinthetruesociallossfunctionsasamotivewhichiswell captured by the speed limit policy objective; under sticky nominal wages with a moderate degree or no in(cid:13)ation indexation, the optimal commitmentpolicypushesthelevelsofpricesandwagesbacktowardstheirdeterministictrendsevenifpricesarehighlyindexed. 4SeeBernankeandMishkin(1997)andKing(2004)forfurtherelaborationsonthisissue. 5In the case of commitment, adopting a simple objective function for the central bank can be justi(cid:12)ed on the grounds of improving transparency,accountabilityandthepursuitofthecentralbank’slegalmandate. 3

to address concerns about identi(cid:12)cation raised in Chari, Kehoe, and McGrattan (2009) and Justiniano, Primiceri, and Tambalotti (2013). Finally, we also account for the limitations of conventional monetary policy imposed by the zero lower bound constraint on the nominal interest rate. Unless long-lasting and frequent zero-bound episodes cannot be eliminated by raising the long-run in(cid:13)ation target, our results go unchallenged. In terms of scope and focus, our paper is closest to Walsh (2003). In a simple NKM with sticky prices and backward-looking elements in the form of lagged in(cid:13)ation and lagged output gap Walsh (2003) illustrates the potential advantages of the speed limit policy. However, the model in Walsh (2003) is not fully micro-founded and social welfare is measured by an ad hoc loss function that is not derived from the preferences of the representative household. Furthermore, the underlying model is calibrated rather than estimated and lacks many of the features found to be of empirical relevance in works such as Christiano, Eichenbaum, and Evans (2005) and Smets and Wouters (2005). In contrast to Walsh (2003), we (cid:12)nd that the speed limit policy outperforms in(cid:13)ation and price level targeting under discretion regardless of the degree of backward-looking in(cid:13)ation dynamics in the CEE/SW model. In Walsh (2003) and in simple NKMs, this conclusion applies only for the case of an intermediate degree of backward-looking behavior. Restricting attention to the case of a fully committed policymaker Debortoli, Kim, Lind(cid:19)e, and Nunes (2015) report strong support in favor of in(cid:13)ation targeting using the CEE/SW model, a result we con(cid:12)rm and extend to a range of other empirically relevant parameterizations of the CEE/SW model. However, as the optimal in(cid:13)ation targeting under commitment is dominated by the optimal speed limit policy under discretion for many empirically plausible parameterizations, our results appear more general. The remainder of the paper proceeds as follows. In Section 2, we analyze in(cid:13)ation targeting, price level targeting, and speed limit policy in a sequence of simple NKMs. We consider a wide range of parameterizations and variations of the CEE/SW model in Section 3. Concluding remarks are offered in Section 4. A technical appendix provides information on our methodology, details on the models, and additional results. 2 Baseline New Keynesian Model Throughout this paper, we refer to the NKM presented in Woodford (2003a), Gali (2008) or Walsh (2010) as the textbook NKM. This model features sticky nominal prices as in Calvo (1983) and a production technologythatrequiresonlylaborasinput. Salessubsidiesoffsetthedistortionsarisingfrommonopolistic competition in the steady state. Finally, the economy experiences technology and markup shocks. One at atime, weconsidertheroleoffeaturescommonlypresentinempiricalDSGEmodels: (i)intrinsicin(cid:13)ation inertia, (ii) steady state distortions, (iii) consumption habits, and (iv) sticky wages. Appendix A offers details on our computational approach. The models are described in Appendix B. 4

2.1 Simple objective functions and targeting frameworks Broadly speaking, analysis of monetary policy distinguishes between targeting frameworks and instrument rules. Underatargetingframework,thecentralbankoptimizesanobjectivefunction. Anin(cid:13)ationtargeting centralbank,forexample,isinstructedtokeepaselectedin(cid:13)ationmeasureintheneighborhoodofaspeci(cid:12)c target value. The central bank is granted some (cid:13)exibility in pursuing this goal and can deviate from its target in the short run to buffer the impact of shocks ((cid:13)exible in(cid:13)ation targeting).6 Given a speci(cid:12)c model of the economy, the policymaker derives a set of optimality conditions for the targeting variables to ful(cid:12)ll under the targeting framework. By contrast, an instrument rule as in Taylor (1993) is a is a formula that speci(cid:12)es directly the functional relationship between the central bank’s instrument and a set of variables. For model-based policy analysis, the central bank’s objective function under a targeting framework speci(cid:12)esthevariablesthatcharacterizethelong-rungoal(s)ofthecentralbankandtheweightsassignedto eachofthesevariablesasarguedinSvensson(2010). Inlinewiththeliterature,werepresentlossfunctions associated with the targeting frameworks of interest as: 1. in(cid:13)ation targeting (IT) LIT =(cid:25)2 +(cid:21)IT (xgap)2 (1) t p;t x t 2. price level targeting (PLT) LPLT =p^2+(cid:21)PLT (xgap)2 (2) t t x t 3. speed limit policy (SLP) ( ( )) LSLP =(cid:25)2 +(cid:21)SLP (xgap)(cid:0) xgap 2 (3) t p;t x t t(cid:0)1 where (cid:25) denotes deviations of the in(cid:13)ation measure from its value along the balanced growth path p;t (henceforth the long-run target), p^ is the log-deviation of the price level from its value along the balanced t growthpath (henceforth the long-run trend), and xgap measures the(model-speci(cid:12)c) output gap. Werefer t to (cid:21)TF as the weight on the activity measure under framework TF. x Each objective function implies a long-run commitment to price stability expressed in terms of a longrun in(cid:13)ation target, or equivalently, a deterministic trend in the price level to provide a nominal anchor. Thecentralbankminimizesthediscountedsumoflossessubjecttotheequationsthatdescribethebehavior of the economy. We consider both the case that in doing so the policymaker can commit to future policy actionsandthecasethatsuchacommitmentisnotfeasible(discretion). Atargetingframeworkisreferred to as optimal, when the objective function associated with this framework is parameterized to minimize the expected welfare loss under this objective relative to the social optimum. The social optimum is de(cid:12)nedbytheeconomicoutcomesundertheoptimalcommitmentpolicywhenthepolicymaker’spreferences 6In practice, a targeting framework full(cid:12)ls a list of formal criteria. State of the art in(cid:13)ation targeting, for example, is commonly characterized as featuring the following elements, see Hammond (2012): (1) price stability as the main goal of monetary policy, (2) public announcement of a quantitative target for in(cid:13)ation, (3) policy based on in(cid:13)ation forecast, (4) mechanisms for transparency and accountability. Suitably adapted, these elements would also be present in other targeting frameworks. By contrast, our discussion of targeting frameworks treats monetary policy as the solution to an optimal control problem under a speci(cid:12)c objective function for each framework. Given our broader perspective, the analysis in this paper is also of relevance for central banks that do not adopt a formal targeting framework, but rather search for monetary policy strategies that achieve the central bank’s mandate as in the case of the U.S. FederalReserve. 5

are consistent with the true social loss function. Following Woodford (1999), we adopt the concept of \optimality from a timeless perspective" to derive commitment policies throughout this paper. 2.2 Targeting frameworks in the textbook NKM WestartourdiscussionoftargetingframeworksusingthetextbookNKM.Atthecoreofthelinearversion of this model lies the New Keynesian Phillips Curve (NKPC) which links in(cid:13)ation, (cid:25) , to the (welfarep;t relevant) output gap, x , t ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )=(cid:20) p ((cid:27) L +(cid:27) C )x t +(cid:12)E t ((cid:25) p;t+1 (cid:0)(cid:19) p (cid:25) p;t )+u p;t : (4) Here and subsequently, all variables are expressed in deviation from their steady state values (relative if carrying a \hat", absolute otherwise). The markup shock, u , follows a known stochastic process. The p;t composite parameter (cid:20) ((cid:27) +(cid:27) ) measures the slope of the NKPC and the parameter (cid:19) represents the p L C p degree of indexation to past in(cid:13)ation as in Christiano, Eichenbaum, and Evans (2005). The aggregate demand curve ( ) 1 x =E x (cid:0) i (cid:0)E (cid:25) (cid:0)g (cid:3) (5) t t t+1 (cid:27) t t p;t+1 mu;t C provides the connection between the output gap, in(cid:13)ation, the nominal interest rate, i , and the natural t [ ] rate of interest, g(cid:3) =(cid:27) E y^(cid:3) (cid:0)y^(cid:3) . The natural level of output in this model mu;t C t t+1 t 1+(cid:27) y^ (cid:3) = L ^(cid:24) (6) t (cid:27) +(cid:27) A;t L C is obtained from a counterfactual economy without nominal rigidities and without markup shocks. The natural level of output responds to changes in technology, ^(cid:24) ; other shocks that could move the natural A;t level of output and thus the natural rate of interest, but from which we abstract for now, are shocks to household preferences or government spending. The output gap is de(cid:12)ned as the difference between actual output and the natural level of output, x = y^ (cid:0)y^(cid:3). As in Woodford (2003a), the preferences of the t t t representative household (or equivalently the social welfare function in this context) are approximated to the second-order as ( ) ∑1 1 E (cid:12)t(cid:0)t0L (7) t0 2 t t=t0 with the true (approximate) social loss function L satisfying t 1+(cid:18) L t = ((cid:27) L +(cid:27) C )(x t )2+ (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 (8) p p with (cid:27) , (cid:27) , (cid:18) being known parameters. L C p To (cid:12)x ideas, we consider (cid:12)rst the performance of each targeting framework in the fully forward-looking 6

NKM, i.e., (cid:19) = 0. The policies associated with each framework are obtained by replacing the true social p lossfunctionL inequation(7)withthelossfunctionsin(1)-(3). Eachframeworkisevaluatedforarangeof t weightsontheactivitymeasure,(cid:21)IT,(cid:21)PLT,and(cid:21)SLP,respectively,bothundercommitmentanddiscretion x x x with xgap = x . Table 1 provides the parameterization of the model (and of all its extensions). For each t t targetingframeworkweconsiderandfortheoptimalcommitmentpolicy,shocksthattransmitthroughthe natural real interest rate, such as the technology shock, have no welfare consequences as adjustments in the nominal interest rate prevent movements of in(cid:13)ation and the output gap so as to prevent any welfare consequences. BlanchardandGali(2007)refertothisfeatureofthetextbookNKMasdivinecoincidence.7 In the following, we restrict attention to markup shocks which by contrast cannot be neutralized. Figure 1 plots the unconditional welfare loss for each framework relative to the optimal commitment policyexpressedasconsumptionequivalentvariation(CEV).Theweightontheactivitymeasureforwhich thewelfarelossisminimizedunderatargetingframeworkisindicatedby\◦"forpriceleveltargeting(PLT), \(cid:3)" for speed limit policy (SLP), and \⋄" for in(cid:13)ation targeting (IT). The optimal weights on the activity measurearelowrelativetotheweightsonthein(cid:13)ationmeasure(whichisnormalizedto1)andthewelfare lossesofnotimplementingtheoptimalcommitmentpolicyaresmallbothundercommitmentanddiscretion for each framework. Figure 1 reproduces some well-known results. Under in(cid:13)ation targeting, a central bank acting under commitmentcanreplicatetheoptimalcommitmentpolicy;thesolidlineinthetoppanelassumesthevalue of zero for the optimal choice of the weight (cid:21)IT in the objective function. In the textbook NKM without x indexation, the true social loss function (8) is written solely in terms of contemporaneous in(cid:13)ation and the welfare-relevant output gap. The central bank’s preferences over in(cid:13)ation and the output gap under in(cid:13)ation targeting coincide with the true social loss function, if (cid:21)IT = (cid:21) (cid:17) ((cid:27) +(cid:27) )(cid:20) (cid:18)p . Thus, x x C L p1+(cid:18)p the welfare loss under optimal in(cid:13)ation targeting relative to the optimal commitment policy must be zero. Given the modi(cid:12)cations in the objective functions for price level targeting (p^ instead of (cid:25) ) and speed t p;t limit policy (xgap(cid:0)xgap instead of xgap) relative to the true social loss function the outcomes under these t t(cid:0)1 t two targeting frameworks are suboptimal by construction. The equivalence between in(cid:13)ation targeting and the optimal commitment policy breaks down for any change in the model environment, most notably if the central bank lacks commitment. For example, in response to a transitory markup shock, the optimal commitment policy manages to reduce deviations of in(cid:13)ation and the output gap from their target values in the impact period by allowing these variables to deviate from their target values also in future periods after the shock has ceased. A central bank acting under discretion with the objective in (8) for (cid:19) = 0, however, will (cid:12)nd it optimal to eliminate p these deviations from target in future periods to fully stabilize the economy earlier (stabilization bias). As households and (cid:12)rms correctly anticipate this behavior, the discretionary central bank will not be able to reapthebene(cid:12)tsoftheoptimalcommitmentpolicyintheimpactperiodtherebycausinglargermovements 7Thisfeatureofthemodelrequiresthatshocksaresufficientlysmallinorderforpolicynottobeconstrainedbythezerolowerbound onthenominalinterestrate. 7

in in(cid:13)ation and the output gap.8 Borrowing the idea of a \(weight-) conservative central banker" from Rogoff (1985), Clarida, Gali, and Gertler (1999) show that the optimal in(cid:13)ation targeting central bank puts lower weight on the activity measure than society does, i.e., (cid:21)IT < (cid:21) , which mitigates, but does not eliminate, the negative welfare x x consequencesofthestabilizationbias. Thus,theCEVinFigure1ispositiveforoptimalin(cid:13)ationtargeting under discretion.9 Changes to the functional form of the policymaker’s objective function can induce further welfare improvements: The welfare loss under optimal price level targeting is close to zero and is marginally higher under the optimal speed limit policy in Figure 1. To understand the strong performance of price level targeting and speed limit policy in the textbook NKM when the policymaker acts under discretion, we revisit the effects of a markup shock under the optimal commitment policy. Let the shock lead initially to an unexpected rise in in(cid:13)ation and a drop in the output gap. Over time the optimal commitment policy drives the price level back to its long-run trend by pushing in(cid:13)ation temporarily below its long-run target. The explanation for the optimality of price levelstability(relativetoitslong-runtrend)recognizesthelinkbetweenpricedispersionandin(cid:13)ation: the cross-sectional variation of prices is proportional to the squared value of in(cid:13)ation as shown in Woodford (2003a)andAppendixB.2. Byassumption, (cid:12)rmsthatdonotadjustpricesoptimallyinthecurrentperiod adjust prices by the value of the long-run in(cid:13)ation target instead. Suppose, that the central bank does not plan to return the price level to its long-run trend. Firms that have not adjusted optimally for some time will be far off the new price level and thus contribute to increased dispersion of prices. When such (cid:12)rms are (cid:12)nally called upon to adjust optimally, a sizable price adjustment will contribute to higher in(cid:13)ation. If the central bank does return the price level to its long-run trend, (cid:12)rms that have not adjusted optimally for some time will (cid:12)nd their prices to be close to the expected long-run price level; hence prices adjust little when these (cid:12)rms are called upon to do so. In addition, (cid:12)rms that happen to adjust optimally closer in time to the impact of the shock will be deterred from raising prices: if the price level will return to its long-run trend over time, larger price adjustments early on bear the risk of the (cid:12)rms’ prices to be far off the price level over time absent future optimal adjustments. As price level targeting under discretion will drive the price level back to its long-run trend by construction, whereas in(cid:13)ation targeting considers past deviations of in(cid:13)ation from its target bygones, the former outperforms the latter.10 Anequivalentdescriptionoftheoptimalcommitmentpolicyfocusesonthedynamicsoftheoutputgap after an in(cid:13)ationary markup shock: an increase of in(cid:13)ation above its target is subsequently countered by tighter monetary policy resulting in a negative output gap. Anticipating such a policy, forward-looking 8In the case of the textbook NKM with an efficient steady state and the central bank’s preferences coinciding with those of the representativehouseholdthetruesociallossfunctionisgivenbyequation(8)regardlessofthecentralbank’sabilitytocommit. 9Rogoff(1985)formulatestheideaofaconservativecentralbanktoovercomethein(cid:13)ationbiasthatarisesunderpolicydiscretionina model with product or labor market distortions akin to Barro and Gordon (1983). A subsidy to offset market distortions also eliminates the in(cid:13)ation bias under discretionary policy in this setting. Yet, in the textbook NKM, even with an efficient steady state due to such subsidies,theoptimalcommitmentpolicycontinuestobetime-inconsistentasdiscussedinthetext. 10Following Vestin (2006), we prove in Appendix B.3 that for purely transitory markup shocks, as opposed to the ARMA(1,1) shock underlyingFigure1,optimalpriceleveltargetingunderdiscretionreplicatestheoptimalcommitmentpolicy. Evenwhenthemarkupshock ispersistent,theresponseoftheeconomyundertheoptimalpriceleveltargetingandspeedlimitpolicyareclosetooptimal. Bilbiie(2014) shows how to construct a loss function for the central bank that replicates under discretion the optimal commitment policy regardless of thepersistenceofthemarkuppushshock. 8

(cid:12)rms restrain their price response in the (cid:12)rst place. Rewriting equation (5), we express the output gap as the sum of current and future real interest rates using 2 3 ∑1 x =(cid:0) 1 (i (cid:0)(cid:25) )(cid:0) 1 E 4 (i (cid:0)(cid:25) ) 5 ; (9) t (cid:27) t p;t+1 (cid:27) t t+j p;t+1+j C C j=1 where we have set g(cid:3) =0 for all j, and we express in(cid:13)ation as the discounted sum of output gaps mu;t+j 2 3 2 3 ∑1 ∑1 (cid:25) =(cid:20) ((cid:27) +(cid:27) )x +(cid:20) ((cid:27) +(cid:27) )E 4 (cid:12)jx 5 +E 4 (cid:12)ju 5 : (10) p;t p L C t p L C t t+j t p;t+j j=1 j=0 Following equation (9), tight future monetary policy in terms of higher future real interest rates affects negatively the contemporaneous and expected future values of the output gap. In turn, expectations of a slowly closing output gap reduce the trade off between contemporaneous in(cid:13)ation and the output gap in equation (10) for a given markup shock. As the speed limit policy assigns dislike to changes in the output gap, xgap(cid:0)xgap, it replicates the slow closing of the output gap under the optimal commitment policy. t t(cid:0)1 Yet, the speed limit policy cannot replicate the optimal commitment policy as it fails to drive the price level back to its long-run trend. As under in(cid:13)ation targeting the price level changes permanently underthe speed limit policy. However, the built-in mechanismof closing the output gap slowly byrunning tighter monetary policy after an in(cid:13)ationary shock reduces the initial increase in the price level under the discretionary speed limit policy compared to in(cid:13)ation targeting. The problem with in(cid:13)ation targeting is notthatdeviationsofin(cid:13)ationfromtargetareconsideredbygones, butthelackofamechanismtocommit to tight future monetary policy after an in(cid:13)ationary shock. Thesuperiorperformanceofpriceleveltargetingshouldnotbemistakenasageneralresult. Thespeed with and the extent to which the price level returns to its long-run trend under the optimal commitment policyissensitivetoarangeofmodelfeatures,buttheneedtopromisekeepingmonetarypolicytightafter in(cid:13)ationary shocks for longer is a general feature of the optimal commitment policy. Whether price level targeting or speed limit policy strikes a better balance between the path of the price level and other policy considerationswhenthepolicymakerlackscommitmentisthequantitativequestionexploredinthispaper. 2.3 Extensions to the textbook NKM The welfare ordering of the targeting frameworks in the textbook NKM is robust to the addition of other features. In(cid:13)ation targeting is the preferred framework under commitment; price level targeting and speed limit policy outperform in(cid:13)ation targeting under discretion. Figure 2 explores the performance of the speed limit policy and price level targeting relative to in(cid:13)ation targeting as a function of the degree of price indexation, (cid:19) , for (i) the textbook NKM, (ii) the textbook NKM with a distorted steady state, (iii) p a model with external consumption habit, (iv) and a model with sticky nominal wages. With the in(cid:13)ation targeting framework set to be the point of reference, a negative CEV indicates that the framework under investigation is inferior to in(cid:13)ation targeting and superior otherwise. We turn to a detailed discussion of 9

each model variation. 2.3.1 The role of price indexation in the textbook NKM The textbook NKM with price indexation is given by equations (4)-(8) with 0 < (cid:19) (cid:20) 1. The lagged p in(cid:13)ation rate enters equation (4) through the behavior of those (cid:12)rms that are not selected to reset prices optimally in the current period. Following the literature, we assume that these non-selected (cid:12)rms adjust prices by the geometric average of the steady state in(cid:13)ation rate and the in(cid:13)ation rate that prevailed in the previous period. Theweight(cid:19) governsthesocialdesirabilityofundoingearlierchangesinthepricelevel. Ifnon-selected p (cid:12)rms adjust prices by the steady state in(cid:13)ation rate ((cid:19) =0), prices of these (cid:12)rms grow along the long-run p trendofthepricelevel. Theoptimalcommitmentpolicylimitswelfare-costlypricedispersionbypromising to drive the price level back to its long-run trend over the medium run. Bycontrast,whenin(cid:13)ationisfullyindexed((cid:19) =1),thepricesofnon-selected(cid:12)rmsre(cid:13)ectthedeviations p of the price level from its previous trend. The optimal commitment policy contains price dispersion, which is proportional to ((cid:25) p;t (cid:0)(cid:25) p;t(cid:0)1 )2 for (cid:19) p = 1, by considering past deviations of in(cid:13)ation from its long-run target bygones and by allowing the price level to change permanently. If monetary policies attempted to revert the price level to its previous trend, it would cause unnecessary price dispersion in future periods. In analogy to the case without indexation, the optimal commitment policy under full indexation promises to return in(cid:13)ation (rather than prices) back to its long-run trend while it is the change in in(cid:13)ation (rather thanthechangeinprices)thatentersthetruesociallossfunction. Thispromiseofthecentralbankdeters (cid:12)rmsthatadjustpricesoptimallyinagivenperiodfromchoosingapricethatisfaroffthepriceunderthe automatic indexation scheme for non-selected (cid:12)rms. If the degree of price indexation falls strictly between 0 and 1, the price level is stationary under the optimal commitment policy, but the horizon over which the price level returns to its long-run trend lengthens with the degree of indexation. As in the case of the textbook NKM without indexation, a shock that calls for monetary tightening in the current period under the optimal commitment policy also calls for tighter policy in future periods as evidenced by a slow closing of the output gap.11 Turning to the evaluation of targeting frameworks, note that in the presence of indexation to past price in(cid:13)ation, the in(cid:13)ation targeting objective cannot be parameterized to match the true social loss function in equation (8). Nevertheless, as shown in the (cid:12)rst row of panels in Figure 2, optimal in(cid:13)ation targeting outperforms price level targeting and speed limit policy under commitment for any degree of price indexation, (cid:19) , owing to the fact that the objective functions for price level targeting and speed limit p policy depart even more from the true social loss function. The dominance of in(cid:13)ation targeting is most strikingwhenindexationishighandthepricelevelreturnstoitslong-runtrendveryslowly,ifatall,under the optimal commitment policy. In particular, price level targeting performs poorly in this case given its 11Stationarity of the price level (or the lack thereof) under the optimal commitment policy can be shown by writing the (cid:12)rst order conditions as (cid:0) 1+ (cid:18)p (cid:18)p xt =p^t (cid:0)(cid:19)pp^t(cid:0)1. For (cid:19)p <1, the price level must return to its long-run trend for the output gap to be closed and in(cid:13)ationtobeatitslong-runtarget. For(cid:19)p=1,theoutputgapisclosedifandonlyifp^t (cid:0)p^t(cid:0)1=(cid:25)p;t=0. 10

tendency to force the price level back to trend too quickly. Undertheoptimalcommitmentpolicy,themonetaryauthorityrelatesacceptabledeviationsofin(cid:13)ation from target to the change in the output gap and past in(cid:13)ation: (cid:18) (cid:25) p;t =(cid:0) 1+ p (cid:18) (x t (cid:0)x t(cid:0)1 )+(cid:19) p (cid:25) p;t(cid:0)1 : (11) p Anin(cid:13)ationtargetingpolicymakeralsoaspirestosetin(cid:13)ationinaccordancewiththechangeintheoutput gap. But such a policymaker responds to expected future changes in the output gap and discards the role of past in(cid:13)ation: (cid:21)IT (cid:18) (cid:25) p;t =(cid:0) (cid:21) x 1+ p (cid:18) ((x t (cid:0)x t(cid:0)1 )(cid:0)(cid:12)(cid:19) p E t (x t+1 (cid:0)x t )): (12) x p For a markup shock with a strong transitory component as under our parameterization, the optimal commitment policy allows in(cid:13)ation to rise and the output gap to turn negative initially followed by a period of below-target in(cid:13)ation and a gradual closing of the output gap. Under commitment, in(cid:13)ation targeting induces dynamics similar to those under the optimal commitment policy, when the central bank places a higherweightonstabilizingtheoutputgap, (cid:21)IT >(cid:21) =((cid:27) +(cid:27) )(cid:18)p(cid:20)p. Thehigherweightontheactivity x x L C 1+(cid:18)p measure compensates for the fact that the expected (positive) output gap growth term in equation (12) operates in the opposite direction of the lagged in(cid:13)ation term in equation (11). Finally, in(cid:13)ation targeting under commitment performs strongly although it fails to drive the price level back fully to its original trend. Under discretion, price level targeting and speed limit policy deliver better outcomes than in(cid:13)ation targetingforlowandmoderatedegreesofpriceindexation((cid:19) <0:8), butnotforahighdegreeasin(cid:13)ation p becomes increasingly persistent irrespective of policy. High in(cid:13)ation persistence feeds into higher expected in(cid:13)ation after an in(cid:13)ationary shock; an in(cid:13)ation targeting central bank will thus be expected to keep interest rates high to curb in(cid:13)ation. This feature of the textbook NKM with (high) indexation allows the discretionary central bank to indirectly commit to running tight future monetary policy and to preventing the output gap from closing too quickly thereby containing the initial response of in(cid:13)ation. The higher the degree of indexation, the more powerful is the fact that the in(cid:13)ation targeting objective replaces the quasi-difference in in(cid:13)ation in the true social loss function with in(cid:13)ation. In the limiting case of (cid:19) = 1, p optimal in(cid:13)ation targeting under discretion can even implement the optimal commitment policy under suitable assumptions for the nature of the underlying stochastic shocks|just as price level targeting can implement the optimal commitment policy for the case of (cid:19) =0. p More formally, provided that shocks are sufficiently small to prevent the zero lower bound constraint from binding, note that in the model without indexation, (cid:19) = 0, the price level targeting central bank p adopts the objective function LPLT =p^2+(cid:21)PLT (x )2 and faces the NKPC of the form t t x t (p^ t (cid:0)p^ t(cid:0)1 )=(cid:20) p ((cid:27) L +(cid:27) C )x t +(cid:12)E t (p^ t+1 (cid:0)p^ t )+u p;t : (13) 11

In the case of full indexation, (cid:19) = 1, the in(cid:13)ation targeting central bank adopts the objective function p LIT =(cid:25)2 +(cid:21)IT (x )2 and faces the NKPC of the form t p;t x t ((cid:25) p;t (cid:0)(cid:25) p;t(cid:0)1 )=(cid:20) p ((cid:27) L +(cid:27) C )x t +(cid:12)E t ((cid:25) p;t+1 (cid:0)(cid:25) p;t )+u p;t : (14) Substituting (cid:25) with p^ reveals that in(cid:13)ation targeting under discretion in the model with (cid:19) = 1 is p;t t p isomorphicwithpriceleveltargetingunderdiscretioninthemodelwith(cid:19) =0. Astheoptimalcommitment p policy stabilizes the price level absent indexation, but stabilizes the in(cid:13)ation rate under full indexation, in(cid:13)ation targeting performs close to optimal when (cid:19) =1 by analogy. Price level targeting and speed limit p policy impose too tight monetary policy in future periods when prices are fully indexed.12 Finally, this discussion shows that for a high degree of indexation optimal in(cid:13)ation targeting under discretion can outperform in(cid:13)ation targeting under commitment. This observation raises the question under what conditions it is desirable to assign the central bank a (simple) loss function that departs from the true social loss function when policymakers can fully commit to future actions. 2.3.2 Inefficient steady state TheoreticalworksbuildingontheNewKeynesianparadigmoftenassumethatthesteadystateofthemodel is efficient as subsidies/taxes offset the distortions from monopolistic competition. By contrast, works on empiricalDSGEmodels|includingtheseminalcontributionsofChristiano,Eichenbaum,andEvans(2005) and Smets and Wouters (2007)|tend to abstract from such subsidies and taxes. The (in-)efficiency of the steady state affects the welfare ranking of policies through the de(cid:12)nition of the output gap. Following Benigno and Woodford (2005) the true social loss function in the model with an inefficient steady state satis(cid:12)es 1+(cid:18) L t = ((cid:27) L +(cid:27) C )(x~ t )2+ (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 (15) p p where x~ denotes the welfare-relevant output gap. The structural equations are given by t (cid:27) +(cid:27) ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 ) = (cid:20) p ((cid:27) L +(cid:27) C )x~ t +(cid:12)E t ((cid:25) p;t+1 (cid:0)(cid:19) p (cid:25) p;t )+ (cid:27) +(cid:27) + L ((cid:8)(cid:0) C 1)(1+(cid:27) ) u p;t (16) L C L 1 x~ = E x~ (cid:0) (i (cid:0)E (cid:25) (cid:0)g~ ) (17) t t t+1 (cid:27) t t p;t+1 mu;t C 1+(cid:27) ((cid:8)(cid:0)1) 1+(cid:27)L y~ = L ^(cid:24) (cid:0) (cid:27)L+(cid:27)C u (18) t (cid:27) +(cid:27) A;t (cid:27) +(cid:27) +((cid:8)(cid:0)1)(1+(cid:27) ) p;t L C L C L with g~ = (cid:27) [E y~ (cid:0)y~]. At (cid:12)rst glance, it appears that we have merely replaced the output gap mu;t C t t+1 t term \x " with \x~ " and rescaled the impact of the markup shock. However, the two de(cid:12)nitions of the t t output gap respond differently to the markup shock. Under the de(cid:12)nition x (cid:17)y^ (cid:0)y^(cid:3), the target output t t t 12For ( anint ) ermediatedegreeofindexation,0<(cid:19)p<1,hybridpriceleveltargetingwiththeobjectivefunctionLh t PLT =(p^t (cid:0)(cid:19)pp^t(cid:0)1)2+ (cid:21) xgap 2 canbeshowntoperformatleastaswellasin(cid:13)ationorpriceleveltargeting. SeeRoisland(2005)andGaspar,Smets,and hPLT t Vestin(2007)foradditionaldiscussion. 12

level y^(cid:3) de(cid:12)ned in equation (6) does not respond to the markup shock; all else equal under the de(cid:12)nition t x~ (cid:17) y^ (cid:0)y~, the output gap will respond by less to a markup shock since the relevant output level y~ t t t t de(cid:12)ned in equation (18) moves in the same direction as actual output. Absent steady state distortions, i.e., (cid:8)=1, thetwode(cid:12)nitionsoftheoutputgapcoincide. Furthermore, inresponsetoatechnologyshock, the divine coincidence continues to apply under the optimal commitment policy regardless of steady state distortions. Applying this change in the de(cid:12)nition of the relevant output gap to the three targeting frameworks, i.e. xgap =x~ ,thesecondrowofpanelsinFigure2plotstheresultsforthecaseofadistortedsteadystatewith t t thesalessubsidysetequaltozero. Bothundercommitmentanddiscretion, priceleveltargetingandspeed limit policy appear closer to in(cid:13)ation targeting than in the case of an efficient steady state. The reason for this(cid:12)ndingisthereducedimpactofthemarkupshockinthemodelwithaninefficientsteadystate((cid:8)>1): in the NKPC the markup shock is scaled by a term smaller than unity and movements in the output gap are curtailed by the adjustments in y~. With the effective magnitude of the markup shock reduced the t welfare losses under each targeting framework relative to the optimal commitment policy shrink. The behavior of the output gap, and thus the ranking of targeting frameworks, is sensitive to the de(cid:12)nition of potential output. If xgap = x despite the distorted steady state the measured output gap t t is larger after a markup shock all else equal, and calls for a larger adjustment in policy than under the output gap de(cid:12)nition of x~ . When using x as the output gap measure despite the presence of steady state t t distortions,in(cid:13)ationtargetingimprovesitsperfomanceanddominatespriceleveltargetingandspeedlimit policy already for the moderate degree of price indexation of (cid:19) =0:4. p 2.3.3 Habit persistence Whenthehousehold’sutilityfunctiondependsonaquasi-differenceinconsumption(habitpersistence),the implied output gap enters with its quasi-difference into the (approximate) true social loss function. Under external consumption habits as in Smets and Wouters (2007), the linear-quadratic form of the model is given by the loss function (cid:27) 1+(cid:18) L t = (cid:27) L (x t )2+ (1(cid:0)h)( C 1(cid:0)h(cid:12)) (x t (cid:0)hx t(cid:0)1 )2+ (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 (19) p p and the structural equations ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 ) = (cid:20) p mcc t +(cid:12)E t ((cid:25) p;t+1 (cid:0)(cid:19) p (cid:25) p;t )+u p;t (20) (cid:27) h(cid:12) mcc t = (cid:27) L x t + 1(cid:0) C h (x t (cid:0)hx t(cid:0)1 )+ 1(cid:0)h(cid:12) g m (cid:3) u;t (21) 1(cid:0)h ( ) (x t (cid:0)hx t(cid:0)1 ) = E t (x t+1 (cid:0)hx t )(cid:0) (cid:27) i t (cid:0)E t (cid:25) p;t+1 (cid:0)g m (cid:3) u;t (22) C 13

[ ( ) ( )] where g m (cid:3) u;t is de(cid:12)ned as g m (cid:3) u;t = 1 (cid:27) (cid:0) C h E t y^ t (cid:3) +1 (cid:0)hy^ t (cid:3) (cid:0) y^ t (cid:3)(cid:0)hy^ t (cid:3) (cid:0)1 : The efficient output level satis(cid:12)es the difference equation ( ) ( ) (cid:27) (cid:27) (cid:27) y^ (cid:3) + C y^ (cid:3)(cid:0)hy^ (cid:3) (cid:0)h(cid:12) C E y^ (cid:3) (cid:0)hy^ (cid:3) =(1+(cid:27) )^(cid:24) : (23) L t (1(cid:0)h)(1(cid:0)h(cid:12)) t t(cid:0)1 (1(cid:0)h)(1(cid:0)h(cid:12)) t t+1 t L A;t The degree of habit persistence is measured by the parameter h2[0;1). The model with habit persistence features endogenous persistence, since the lagged value of the output gap enters into the NKPC and the aggregate demand curve, which in turn affects the dynamics of in(cid:13)ation.13 The presence of the lagged output gap term in the true social loss function (19) strengthens the motive for smoothing the evolution of the output gap under the optimal commitment policy. AsshowninthethirdrowofpanelsinFigure2,thespeedlimitpolicycanoutperformin(cid:13)ationtargeting under commitment for a moderate degree of habit persistence (h = 0:7) and low in(cid:13)ation inertia due to little or no price indexation. Abstracting from price indexation, the true social loss function resembles the objective function of the speed limit policy framework: A reasonably high degree of habit persistence implies that most of the weight is placed on the term (x t (cid:0)hx t(cid:0)1 )2 in the true social loss function and the optimal speed limit policy under commitment mimics the optimal commitment policy. Overall, under commitment, the differences between speed limit policy and in(cid:13)ation targeting are much reduced for any degree of price indexation. Price level targeting performs relatively poorly under commitment for a high degree of price indexation as in the previous two model variations. When policy is conducted under discretion, in(cid:13)ation targeting never outperforms the other two frameworks regardless of the degree of in(cid:13)ation indexation. Compared to the textbook NKM the differences between frameworks are of much larger magnitude. The advantage of speed limit policy and price level targeting over in(cid:13)ation targeting narrows considerably as the degree of price indexation (cid:19) approaches 1. p However, the isomorphism of in(cid:13)ation targeting for (cid:19) = 1 with price level targeting for (cid:19) = 0 under p p discretion no longer applies in the presence of consumption habits. Higher in(cid:13)ation persistence as a result of indexation allows the discretionary in(cid:13)ation targeting central bank to commit indirectly to tighter monetary policy in the future after an in(cid:13)ationary shock. Yet, the expected future policy under in(cid:13)ation targeting is not tight enough. When consumption experiences habit persistence, the optimal commitment policy engages in more smoothing of the output gap which strengthens the motive of keeping monetary policy tight after an in(cid:13)ationary shock. The in(cid:13)ation targeting objective does not capture this additional motive and provides less stabilization of the economy. 13Whenhabitsareexternal,thedecisionstakenbythehouseholdmembersarenotefficientunder(cid:13)exiblepricesevenifasalessubsidy removesthedistortionsfrommonopolisticcompetitioninthegoodsmarket. Torenderthesteadystateofthemodelefficient,weintroduce a consumption tax; yet, the dynamics remain inefficient even for technology shocks. With the term h(cid:12) g(cid:3) entering equation (20) 1(cid:0)h(cid:12) mu;t through the de(cid:12)nition of the marginal cost term, mcct, the central bank is unable to perfectly stabilize in(cid:13)ation and the welfare-relevant output gap in response to technology shocks. As discussed in Leith, Moldovan, and Rossi (2012) and Woodford (2003a), consumption habitshavetobespeci(cid:12)edasinternalinorderforthedivinecoincidencetore-emerge. 14

2.3.4 Sticky wages Sticky nominal wages as in Erceg, Henderson, and Levin (2000) are the (cid:12)nal feature that we consider in isolation. In detail, the loss function can be shown to satisfy 1+(cid:18) 1+(cid:18) L t = ((cid:27) L +(cid:27) C )(x t )2+ (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2+ (cid:18) (cid:20) w ((cid:25) w;t (cid:0)(cid:19) w (cid:25) p;t(cid:0)1 )2 (24) p p w w while the structural equations are summarized by ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 ) = (cid:20) p mcc t +(cid:12)E t ((cid:25) p;t+1 (cid:0)(cid:19) p (cid:25) p;t )+u p;t (25) mcc = !^ (cid:0)^(cid:24) (26) t t A;t ((cid:25) w;t (cid:0)(cid:19) w (cid:25) p;t(cid:0)1 ) = (cid:20) w (mdrs t (cid:0)!^ t )+(cid:12)E t ((cid:25) w;t+1 (cid:0)(cid:19) w (cid:25) p;t )+u w;t (27) mdrs (cid:0)!^ = ((cid:27) +(cid:27) )x (cid:0)(!^ (cid:0)!^ (cid:3) ) (28) t t L C t t t ( ) ( ) (!^ t (cid:0)!^ (cid:3) t ) = !^ t(cid:0)1 (cid:0)!^ (cid:3) t(cid:0)1 +(cid:25) w;t (cid:0)(cid:25) p;t (cid:0) !^ (cid:3) t (cid:0)!^ (cid:3) t(cid:0)1 (29) ( ) 1 x = E x (cid:0) i (cid:0)E (cid:25) (cid:0)g (cid:3) : (30) t t t+1 (cid:27) t t p;t+1 mu;t C TheNKPCforwages,equation(27),linkswagein(cid:13)ation,(cid:25) ,tothegapbetweenthemarginalrateofsubw;t stitution (between consumption and leisure), mdrs , and the real wage, !^ . The policymaker places weight t t on stabilizing price and wage in(cid:13)ation with the weights inversely related to the slopes of the respective NKPCs. To maintain comparability with the previous models we focus on price markup shocks. As in the model with (cid:13)exible wages, a policy that promises to be tight in the future|summarized by the discounted sum of future (negative) output gaps in equation (31)|acts towards stabilizing the output gap, and (a weighted average of price and wage) in(cid:13)ation in the impact period: 2 3 2 3 ∑1 ∑1 (cid:25) + (cid:20) p(cid:25) =(cid:20) ((cid:27) +(cid:27) )x +(cid:20) ((cid:27) +(cid:27) )E 4 (cid:12)jx 5 +E 4 (cid:12)ju 5 : (31) p;t (cid:20) w;t p L C t p L C t t+j t p;t+j w j=1 j=0 Theoptimalsplitbetweenmovementsinwageandpricein(cid:13)ationdependsontherelativestickinessbetween prices and wages as captured by the slope coefficients (cid:20) and (cid:20) and the evolution of the real wage. p w According to equation (25), 2 3 2 3 ∑1 ∑1 (cid:25) =(cid:20) E 4 (cid:12)j! 5 +E 4 (cid:12)ju 5 : (32) p;t p t t+j t p;t+j j=0 j=0 If the central bank allows the real wage to fall persistently, it can lean against the initial rise in in(cid:13)ation. However,adeclineinthefuturerealwagealsorequiresthatpricesrisefasterthanwages. Ifthepolicymaker places a high weight on stabilizing price in(cid:13)ation, the adjustment process has to operate more through wage in(cid:13)ation. Under the optimal commitment policy, tight monetary policy in the periods following an in(cid:13)ationaryshockundoesalmostalloftheearlierchangesinthepriceandwagelevel,butpricesandwages arenotstationaryunlessthereisnoin(cid:13)ationindexation, i.e., (cid:19) =(cid:19) =0. Thespeedwithwhichpriceand p w 15

wage changes are undone depends on the degree of indexation. Unless both prices and wages are highly indexed, this process is rather fast. When prices and wages are fully indexed ((cid:19) = (cid:19) = 1), there is no p w partial undoing of earlier changes in prices and wages at all. With these features of the optimal commitment policy in mind, we return to Figure 2. The fourth row of the (cid:12)gure shows that in(cid:13)ation targeting outperforms the other frameworks, when the policymaker can commit. To induce outcomes that are close to the optimal commitment policy, in(cid:13)ation targeting under commitment must place a sufficiently low weight on price in(cid:13)ation to prevent wages from carrying too much of the burden of the real wage adjustment. Overall, when the central bank implements its objective under commitment, the welfare differences across targeting frameworks are small and comparable to those in the previous models. When the targeting frameworks are implemented under discretion, speed limit policy and price level targeting dominate in(cid:13)ation targeting|and for the case of no wage indexation depicted in Figure 2|this (cid:12)nding does not depend on the degree of price indexation. Given the features of the optimal commitment policy, price level targeting is best suited to stabilize the economy although it pushes prices and wages back to their long-run trends. Discretionary in(cid:13)ation targeting views all changes to prices and wages as permanent; promising to revert price in(cid:13)ation to its long-run target is not a sufficient deterrent against changes in prices and wages. Finally, the speed limit policy keeps the initial response of prices and wages incheckastheprivatesectorexpectschangesintheoutputgaptobesmoothre(cid:13)ectingonceagaintheidea to keeping future monetary policy tight after an in(cid:13)ationary shock. Overall, the welfare outcomes under the speed limit policy are close to those under price level targeting. Incontrasttothepreviousmodels, therelativeperformanceofdiscretionaryin(cid:13)ationtargetingworsens when prices are increasingly indexed while keeping the degree of wage indexation unchanged. More price indexation implies more persistent price in(cid:13)ation after a markup shock, which leads the in(cid:13)ation targeting centralbankwantingtostabilizepricein(cid:13)ationmoreaggressivelyandtherebytoputmoreburdenonwage in(cid:13)ation in the adjustment process. The performance of in(cid:13)ation targeting improves for a higher degree of price indexation, when wage indexation is also high|in this case the optimal commitment policy ends up stabilizing in(cid:13)ation rates and does little to push prices and wages back towards their previous trends.14 Finally, if we keep the degree of price indexation constant and low, a higher degree of wage indexation implies a better relative performance of the optimal in(cid:13)ation targeting under discretion. An increase in wage indexation has little impact on the persistence of price in(cid:13)ation and on the optimal parameterization of the in(cid:13)ation targeting objective. Furthermore, changes in prices and wages are quickly pushed back under the optimal commitment policy. However, the welfare losses under each framework relative to the optimalcommitmentpolicyshrinksincewagedispersion,measuredby(cid:25) (cid:0)(cid:19) (cid:25) ,dropsforhighervalues w;t w p;t of (cid:19) . While the welfare differences become smaller, the ranking of targeting frameworks is preserved.s w 14If wage indexation is kept (cid:12)xed at a high value, the advantage of speed limit policy and price level targeting over in(cid:13)ation targeting (cid:12)rstincreasesasthedegreeofpriceindexationrisesfrom0beforeeventuallyfalling(andpossiblyturningnegative)asthedegreeofprice indexationapproaches1. 16

2.3.5 Comparison with Walsh (2003) Walsh (2003) concludes that a high degree of price indexation is necessary in order for in(cid:13)ation targeting to outperform speed limit policy and price level targeting when policymakers cannot commit to future policy paths.15 Our analysis generalizes this insight to the case of sticky wages: all prices and wages that experience nominal rigidities must be highly indexed for in(cid:13)ation targeting to perform strongly under discretion. Furthermore, our (cid:12)ndings point to the role of consumption habits as increasing the central bank’smotiveforkeepingfuturemonetarypolicytightafteranin(cid:13)ationaryshocktocurbthedispersionof prices and wages. This feature is not captured in Walsh (2003) who assumes a model-invariant social loss functionoftheform(cid:25)2 +(cid:21)(xgap)2 indeparturefromthelinear-quadraticapproximationofthepreferences p;t t of the representative household. 3 Empirical models of the business cycle Moving beyond the textbook NKM, we extend our analysis to the medium scale CEE/SW model which features sticky nominal prices and wages both with partial indexation to past in(cid:13)ation, physical capital andinvestmentwithcapitalutilizationandinvestmentadjustmentcosts,habitpersistenceinconsumption, a variable elasticity of substitution between intermediate goods as in Kimball (1995) and the same for labor types, a distorted steady state, and shocks to technology, the risk premium, government spending, investment, price and wage markups, and monetary policy as detailed in Appendix D. An important step in extending our analysis is to obtain a second-order accurate approximation to the preferences of the representative household. We follow a numerical approach. Let the N (cid:2)1 vector of endogenous variables in the CEE/SW model be denoted by x , with the partition x = (x~′;i )′. The t t t t variable i is the policy instrument of the central bank. The vector (cid:16) refers to the set of exogenous t t variables. Given the central bank’s choice of the policy instrument for all periods t (cid:21) t , fi g1 , the 0 t t=t0 remaining N (cid:0)1 endogenous variables satisfy the N (cid:0)1 structural model equations E t g(x t(cid:0)1 ;x t ;x t+1 ;(cid:16) t )=0 (33) in equilibrium. ∑ With the intertemporal preferences of society given by U = E 0 1 t=t0 (cid:12)t(cid:0)t0U(x t(cid:0)1 ;x t ;(cid:16) t ), the optimal commitment policy is derived from the maximization program ∑1 fx m t g a 1 t= x t0 E 0 t=t0 (cid:12)t(cid:0)t0U(x t(cid:0)1 ;x t ;(cid:16) t ) s:t: E t g(x t(cid:0)1 ;x t ;x t+1 ;(cid:16) t )=0 g(x t0 (cid:0)2 ;x t0 (cid:0)1 ;x t0 ;(cid:16) t0 (cid:0)1 )=g(cid:22) t0 : (34) 15SeeAppendixCformodeldetails. Figure14replicatesouranalysisforthemodelinWalsh(2003)forboththecaseofdiscretionand commitmentwiththelatteronenotbeingincludedinWalsh(2003). 17

The constraint g(x t0 (cid:0)2 ;x t0 (cid:0)1 ;x t0 ;(cid:16) t0 (cid:0)1 )=g(cid:22) t0 captures the policymaker’s ability to pre-commit before the beginning of time in t = t to embed the idea of optimality from a timeless perspective as in Woodford 0 (2003a).16 Using the toolbox developed in Bodenstein, Guerrieri, and LaBriola (2014), the (cid:12)rst-order conditions associated with the program in (34) can be used to obtain the purely quadratic approximation to the intertemporal preferences of society. The true social loss function [ ] ∑1 E (cid:12)t(cid:0)t0 1 x^ ′ A(L)x^ +x^ ′ B(L)(cid:16) +(cid:12) (cid:0)1φ^ (cid:3)′ C(0)x^ (35) t0 2 t t t t+1 t0 (cid:0)1 t0 t=t0 correctly ranks (the (cid:12)rst-order accurate) outcomes fx^ g1 obtained under any monetary policy from the t t=t0 perspective of the optimal commitment policy (from a timeless perspective). The matrices A(L) and B(L) representtheapproximationofthepreferenceswith\L"denotingthelag-operator. AsdiscussedinBenigno and Woodford (2012), the term (cid:12) (cid:0)1φ^ (cid:3)′ C(0)x^ punishes violations of the pre-commitment constraint t0 (cid:0)1 t0 under the assessed policy in the case of discretion.17 Appendix A provides the details of obtaining and evaluatingthewelfarecriterion(35)andofsolvingforthedecisionrulesunderdiscretionandcommitment. As in Section 2, we compare the welfare implications under in(cid:13)ation targeting, speed limit policy, and price level targeting both under commitment and discretion. At times, we also report results from two nominal income targeting frameworks included in Walsh (2003): 1. nominal income targeting 1 (NIT) LN t IT =(cid:25)2 p;t +(cid:21)N x IT ((cid:25) p;t +y^ t (cid:0)y^ t(cid:0)1 )2 (36) 2. nominal income targeting 2 (NIT-II) LN t IT-II =(xg t ap)2+(cid:21)N x IT-II((cid:25) p;t +y^ t (cid:0)y^ t(cid:0)1 )2: (37) The optimal parameterization of a targeting framework, i.e., the optimal choice of (cid:21)TF, minimizes the x welfare distance between the targeting framework and the optimal commitment policy as measured by the welfare criterion in equation (35). In this section, we follow Smets and Wouters (2007) in measuring the output gap as the difference between actual output and the potential output de(cid:12)ned as the output level that would have prevailed absent nominal rigidities and inefficient markup shocks to prices and wages. Our analysis of targeting frameworks in the CEE/SW model proceeds as follows. First, we (cid:12)x the parameters of the model at their posterior mode estimated in Smets and Wouters (2007). We then explore alternativeparameterizationoftheCEE/SWmodelobtainedbydrawningfromtheLaplaceapproximation 16Benigno and Woodford (2012) and Debortoli and Nunes (2006) show that assuming policy to be conducted under suitable precommitments is generally needed to obtain a purely quadratic approximation to the preferences of the representative household. For the modelsinSection2,theassumptionofthetimelessperspectiveiskeyforderivingthetruesociallossfunctionwhenthesteadystateisnot efficient;seealsoAppendixB. 17In practice, the correction term tends to be small. Although we did not emphasize this term in Section 2, we did include it in our computationswhenneeded. 18

to the posterior distribution in Smets and Wouters (2007). We close by assessing robustness of our (cid:12)ndings along three dimensions. First, we compute optimal targeting frameworks for the CEE/SW model when the model is estimated with data for the euro area instead of the United States. Second, we investigate how our (cid:12)ndings are affected by the difficulties of distinguishing between wage markup shocks and preference shocks that shift the marginal utility of labor. And (cid:12)nally, we explore the implications resulting from the zero lower bound on nominal interest rates. 3.1 Targeting frameworks in the CEE/SW model Figure3 summarizesour (cid:12)ndingsfortheCEE/SW model. Asbefore, weconsider variationsin thedegrees of price and wage indexation. The top row of panels shows how the degree of price indexation (cid:19) impacts p the relative ordering of the (cid:12)ve targeting frameworks in the CEE/SW model. A vertical line marks the posterior mode of (cid:19) = 0:22. The results nicely relate to our earlier (cid:12)ndings. With consumption habits p at 0:71 and sticky nominal wages, the optimal speed limit policy is a close second to in(cid:13)ation targeting when the policymaker acts under commitment. As price level targeting places too much importance on pricestabilityanddisregardstheneedtosmooththeevolutionoftheoutputgap,thewelfareoutcomesare somewhat inferior. The two nominal income targeting frameworks are strictly outperformed by the speed limit policy and the price level targeting framework. The overall magnitude of the welfare differences is signi(cid:12)cantly larger in the CEE/SW model than in the simple NKMs, re(cid:13)ecting the presence of additional model features and shocks that introduce welfare-relevant policy trade-offs. Underdiscretion,thespeedlimitpolicyframeworkstrictlyoutperformsallotherframeworksirrespective of the degree of price indexation. At the posterior mode parameterization of the model, the optimal speed limit policy exceeds welfare under in(cid:13)ation targeting by more than 0:30% of steady state consumption, whereas the advantage of the price level targeting framework over in(cid:13)ation targeting is a bit smaller with 0:25%. As in the textbook NKM with sticky wages, the advantage of the optimal speed limit policy over in(cid:13)ation targeting is larger when the degree of price indexation is higher while keeping the degree of wage indexation constant. Even the two nominal income targeting frameworks strongly outperform in(cid:13)ation targeting in the discretionary Markov equilibrium. AsshowninFigure4,discretionaryspeedlimitpolicyandpriceleveltargetingcapturekeyfeaturesofthe optimal commitment policy in the CEE/SW model in response to price markup and wage markup shocks. Giventheestimatedmoderatedegreeofindexation((cid:19) =0:22and(cid:19) =0:59),priceandwagedispersionare p w closelyrelatedtopriceandwagein(cid:13)ation,whicharekeptlowbythepromiseoftightfuturemonetarypolicy after an in(cid:13)ationary shock under the optimal commitment policy. As a result, the price and wage levels return slowly towards their pre-shock trends, although not completely. Noticeably, the speed limit policy considers deviations of price and wage in(cid:13)ation from their long-run target values bygones. However, given the built-in promise of keeping future policy tight after an in(cid:13)ationary shock this policy reduces overall in(cid:13)ationandtheriseinthepriceandwagelevels. Priceleveltargetingasamonetarypolicystrategysignals tight monetary policy in response to in(cid:13)ationary shocks through explicitly promising to return prices and 19

wages to their earlier trends. For a moderate degree of indexation, the resulting stabilization of price and wage in(cid:13)ation is close to optimal. By contrast, the in(cid:13)ation targeting objective does not include built-in featuresthatwouldallowthecentralbanktopromisetightfuturemonetarypolicyinanenvironmentwith low to moderate in(cid:13)ation indexation under discretion. Thus, the in(cid:13)ation targeting central bank is less effective at stabilizing the economy: In(cid:13)ation is persistently higher and the output gap drops by more on impact compared to the optimal commitment policy and the other targeting frameworks in Figure 4. The CEE/SW model abstracts from taxes/subsidies that could correct the distortions associated with monopolistic competition in the production of intermediate goods and the labor market. The second row of panels in Figure 3 reveals that if these distortions are removed, in(cid:13)ation targeting improves its relative performance slightly. AsforthetextbookNKMwithstickywages,wevarythedegreeofwageindexationinthebottomrowof panels. Varyingthedegreeofwageindexationawayfromitsposteriormodeof(cid:19) =0:59whilekeepingthe w degree of price indexation at its posterior mode of (cid:19) =0:22 reveals that a lower degree of wage indexation p goes along with a relatively poorer performance of in(cid:13)ation targeting under discretion as in the previous section. Under commitment, changing the degree of wage indexation impacts the relative performance of the frameworks in a manner similar to changes in price indexation. 3.2 Deconstructing the results While the outcomes in the CEE/SW model resemble those in Section 2, we also consider one of the many sequences of expanding the textbook NKM step-by-step to the CEE/SW model. We present results for the case of discretion. Figure 5 plots the CEV values for each framework relative to the in(cid:13)ation targeting frameworkunderdiscretion. StartingfromthetextbookNKMwithpreferencesbeingspeci(cid:12)edasinSmets and Wouters (2007)|titled SW{Woodford|and using the parameters estimated by Smets and Wouters (2007) where applicable we introduce the following changes step-by-step: (cid:15) remove taxes/subsidies for intermediate goods, (cid:15) government spending, physical capital and investment, including capital utilization and investment adjustment costs, and related shocks, (cid:15) sticky wages (with a wage subsidy to offset distortions in the labor market and no wage markup shock), (cid:15) a wage markup shock, (cid:15) remove the wage subsidy, (cid:15) habit persistence, (cid:15) a higher degree of nominal rigidities measured by the probabilities of not adjusting prices or wages optimallyfrom(cid:24) =0:65and(cid:24) =0:73to(cid:24) =0:85and(cid:24) =0:88,respectively,inordertomatchthe p w p w slopes of the NKPC between a model with and without a variable elasticity of substitution (Kimball aggregator), 20

(cid:15) a variable elasticity of substitution as in Kimball (1995). The(cid:12)gurecon(cid:12)rmstheimportanceofindexation,stickywages,andhabitpersistenceindeterminingthe rankingoftargetingframeworksunderdiscretion. Absentstickywages, ahigherdegreeofpriceindexation plays out in favor of in(cid:13)ation targeting under discretion. In the presence of sticky nominal wages this (cid:12)nding is overturned. Furthermore, the magnitude of welfare differences increases with sticky wages and the associated wage markup shocks. Habit persistence in consumption raises the overall welfare costs of not implementing the optimal commitment policy and thus the advantage of speed limit policy and price level targeting over in(cid:13)ation targeting. With the true social loss function featuring an explicit motive for smoothing the quasi-difference in the output gap, the speed limit policy gets even closer to the price level targetingframework. Theroleofcapitalaccumulationandinvestmentadjustmentcostsonthequantitative differences between targeting frameworks is relatively minor. In addition to the features discussed in Section 2, the variable elasticity of substitution is the other feature of quantitative importance as it increases the strategic complementarity in price setting. The Kimball aggregator impacts our outcomes mostly through changing the slope of the NKPCs. Moving from the bottom left panel in the (cid:12)gure (constant elasticity of substitution and (cid:24) = 0:65 and (cid:24) = 0:73) to p w the bottom right panel (variable elasticity of substitution and (cid:24) = 0:65 and (cid:24) = 0:73) directly, the p w welfare differences between price level targeting (or speed limit policy) and in(cid:13)ation targeting triple. Yet, considering the intermediate step of the middle panel (constant elasticity of substitution and (cid:24) = 0:85 p and (cid:24) =0:88) reveals that this increase could also be obtained by raising the degree of nominal rigidities w whilekeepingtheslopesoftheNKPCsthesamebetweenthelasttwopanels. Similarconclusionsregarding the importance of the various model features emerge when we change the sequence of introducing them or when policymakers act under commitment. 3.3 Robustness to alternative parameterizations To explore the sensitivity of our (cid:12)ndings to alternative, yet empirically plausible, parameter choices. We draw 30000 parameter speci(cid:12)cations from the Laplace approximation to the posterior distribution Smets and Wouters (2007) and we 1. compute the optimal weights on the activity measure in the objectivefunctions, (cid:21)TF, associated with x in(cid:13)ation targeting, speed limit policy, and price level targeting for each parameter draw and compare welfare for each parameter draw under these optimal weights, 2. comparewelfareacrosstargetingframeworksforeachparameterdrawwhentheweightsontheactivity measure in the objective function are (cid:12)xed at speci(cid:12)c values. We exclude the NIT and NIT-II framework from this exercise as they were strictly dominated by price level targeting and speed limit policy. The (cid:12)rst experiment, referred to as the \optimal weights case," con(cid:12)rms that the ordering of targeting frameworks is robust to alternative empirically plausible parameterizations of the CEE/SW model. Figure 6 plots the distribution of welfare losses relative to the optimal commitment policy (expressed in CEV) 21

for each draw of parameters and targeting framework. Under commitment (the top row of panels), the distribution of welfare losses is similar across targeting frameworks, although the losses tend to be slightly smaller under in(cid:13)ation targeting. The median loss under in(cid:13)ation targeting is -0.0288, whereas it reaches -0.0538 under price level targeting and -0.0454 under the speed limit policy. Large losses are rare for all frameworks. Table 2 Panel (a) reports the frequency with which each of the frameworks performs better than the remaining two. The optimal in(cid:13)ation targeting framework emerges as the winner for 97% of the parameter draws. Table 2 Panel (d) is designed to shed light on the magnitude of the welfare differences. For each draw of parameters we compute the welfare difference between a given targeting framework and thebestperformingframeworkoftheremainingtwoandreportthepercentilesoftheresultingdistribution of welfare differences in increasing order. Since in(cid:13)ation targeting almost always performs best, when policymakerscancommit,thedifferencesreportedincolumns3and4basicallycoincidewiththedifferences between price level targeting and in(cid:13)ation targeting and between the speed limit policy and in(cid:13)ation targeting, respectively. Only for 5% of the parameter draws does the difference between the price level targeting and the in(cid:13)ation targeting framework exceed -0.0493; for the speed limit policy framework, the value is even smaller with -0.0280. For the in(cid:13)ation targeting framework, the advantage over the next best targeting framework is smaller than 0.0280 for about 95% of the draws. The values at the nth percentile for column 2 (IT) and the (100(cid:0)n)th percentile for column 4 (SLP) indicate that the speed limit policy framework is the second-best performing framework for most parameter draws. Under discretion, the distributions of welfare losses induced by the three targeting frameworks look much less alike. In Figure 6 (the middle row of panel), the distribution of welfare losses relative to the optimal commitment policy is noticeably more dispersed for price level targeting and, in particular, for in(cid:13)ation targeting than under commitment. By contrast, the distribution under the speed limit policy is more concentrated, an observation leading us to speculate whether the optimal speed limit policy under discretion may deliver better welfare outcomes (1) than the optimal speed limit policy under commitment, and(2)thanoptimalin(cid:13)ationtargetingundercommitment. The(cid:12)rstclaimistrueforanyparameterization we consider; the second claim is true for more than 50% of the parameter draws and in particular it is true when the parameters in the CEE/SW model are (cid:12)xed at their posterior mode. Table 2, Panel (a) further reveals the superiority of speed limit policy under discretion. It is found to perform better than in(cid:13)ationtargetingandpriceleveltargetingformostparameterdraws(around98%). AsshowninPanel(d), the advantage of the speed limit policy framework over the in(cid:13)ation targeting framework can be sizeable (column 5). Although price level targeting performs consistently better than in(cid:13)ation targeting under discretion, it rarely performs best (column 6). The (cid:12)nal row of Figure 6 plots the cumulative distribution functions of the optimal weights on the activity measure. For each framework, the optimal weights tend to be larger and the distributions of weights are more dispersed under commitment than under discretion. For example, the median weight under the speed limit policy framework is (cid:21)SLP = 11:86 for commitment, but only (cid:21)SLP = 3:39 for x x discretion. Therobustperformanceofthespeedlimitpolicyframeworkacrosscommitmentanddiscretionnotonly 22

appliestoawiderangeofempiricallyplausibleparameterizationsoftheCEE/SWmodelwhentheweights on the activity measure are set optimally for each draw and framework. Our second set of experiments (cid:12)nds that the performance of the speed limit policy framework is also less sensitive to the exact choice of the weight on the activity measure: (1) We (cid:12)x the weight on the activity measure for each targeting framework at the value found to be optimal when the parameters in the CEE/SW model are (cid:12)xed at their posterior mode (under commitment and discretion, respectively) and compute the welfare losses relative to the optimal commitment policy for each of the 30000 parameter draws. (2) We (cid:12)x the weight on the activity measure for each targeting framework at the value found to be optimal under commitment (discretion)whentheparametersintheCEE/SWmodelare(cid:12)xedattheirposteriormodeandcomputethe welfare losses relative to the optimal commitment policy for each of the 30000 parameter draws, but solve the model under the assumption that the policymaker acts under discretion (commitment). Subsequently, we refer to (1) as the \(cid:12)xed weights case" and to (2) as the \exchanged weights case." As reported in Table 2, Panel (b), in the (cid:12)xed weights case, the speed limit policy performs best for 16.5% of the parameter draws under commitment|up from 2.7% in the original experiment|and it maintainsitssuperiorperformanceunderdiscretionbyoutperformingtheotherframeworksfor98%ofthe draws. Figure 7 plots the distribution of welfare losses under the (cid:12)xed weights case relative to the optimal weights case. The welfare losses that are caused by the policymaker using the optimal weights for a given parameter draw are small under commitment across regimes, but are often sizeable under discretion for both price level targeting and, in particular, in(cid:13)ation targeting. The exchanged weights case explores the sensitivity of the targeting frameworks to both parameter uncertainty and uncertainty about the ability of the policymaker to commit. As shown in Table 2, Panel (c)whenpolicyisconductedundercommitment,butthepolicymakerusestheweightsfoundtobeoptimal under discretion for the posterior mode parameterization of the CEE/SW model, the speed limit policy framework performs best for 99% of the parameter draws. Under discretion, the speed limit policy framework performs best for 98% of the draws. Figure 8 also plots the distribution of welfare losses under the exchanged weights case relative to the optimal weights case for each framework. The in(cid:13)ation targeting framework is very sensitive to getting the weight on the activity measure right as evidenced by the high share of large welfare losses exceeding 1% (measured as CEV) for more than 50% of the parameter draws. Under the speed limit policy framework such large losses are never observed. ThespeedlimitpolicyframeworkemergesasthemostdesirablesettinginouranalysisoftheCEE/SW model. Across parameterizations, the optimal speed limit policy consistently outperforms the in(cid:13)ation targeting and the price level targeting framework under discretion; under commitment the speed limit policyframeworkisaveryclosesecondtothein(cid:13)ationtargetingframework;theoptimalspeedlimitpolicy framework implemented under discretion delivers higher social welfare than optimal in(cid:13)ation targeting under commitment. Finally, the performance of the economy under a speed limit policy is much less sensitive to the exact parameterization of the objective function which is of relevance if the policymaker faces uncertainty about the correct speci(cid:12)cation of the economy. 23

3.4 Additional robustness checks Weconcludeouranalysiswithrobustnesschecksregardingthedatausedtoestimatethemodel,theroleof therelativeimportanceoflaborsupplyshocksversuswagemarkupshocks,andthelimitationsofmonetary policy imposed by the zero lower bound constraint on the nominal interest rate. 3.4.1 Robustness to alternative data Smets and Wouters (2007) estimated the CEE/SW model using U.S. data. Figure 9 compares the performance of all (cid:12)ve targeting frameworks when the CEE/SW model is estimated using data for the euro area instead.18 Qualitatively,theresultsfortheeuroareaaresimilartothosederivedfromU.S.data. Fromaquantitative perspective, the case for price level targeting and speed limit policy is even stronger. Their advantage over in(cid:13)ation targeting measured in terms of steady state consumption doubles under discretion. Under commitment the in(cid:13)ation targeting framework maintains a small advantage over speed limit policy and price level targeting. 3.4.2 Shocks to labor supply and wage markups Chari, Kehoe, and McGrattan (2009) point to an identi(cid:12)cation problem in the CEE/SW model that preference shocks shifting the marginal disutility of labor cannot be easily distinguished from wage markup shocks. Gali, Smets, and Wouters (2011) and Justiniano, Primiceri, and Tambalotti (2013) impose assumptions to overcome this identi(cid:12)cation problem.19 While in comparison to the CEE/SW model wage markupshocksplayalessimportantroleinboththesepapers,wagemarkupshockscontinuetocontribute signi(cid:12)cantlytothe(cid:13)uctuationsinin(cid:13)ationinGali,Smets,andWouters(2011). Giventhedifferentwelfare implications of the inefficient wage markup shocks, which creates a monetary policy trade off, and the efficient labor supply shocks, the relative importance of these two shocks may in(cid:13)uence the ranking of targeting frameworks. Figure 10 provides a preliminary inquiry into the importance of the issues raised by Chari, Kehoe, and McGrattan (2009) for the ranking of frameworks. We compute the welfare differences between targeting frameworks by changing the relative importance of wage markup and labor supply shocks. Following Gali, Smets, and Wouters (2011) and Justiniano, Primiceri, and Tambalotti (2013), we model the labor supply shock as a shock to the marginal disutility of labor. The labor supply shock is speci(cid:12)ed to match the unconditional variance of the wage markup shock and to induce responses similar in magnitude to those 18Smets and Wouters (2005) estimate a medium-scale DSGE model for the euro area, but the details of the model differ from those in Smets and Wouters (2007). To maintain comparability of results, we estimate the model speci(cid:12)ed in Smets and Wouters (2007) using data for the euro area from the Area Wide Model database (see Fagan, Henry, and Mestre (2005)). Data on consumption, investment, GDP,hoursandwagesareexpressedin100timesthelog. In(cid:13)ationisthe(cid:12)rstdifferenceofthelogGDPde(cid:13)ator. Theinterestrateisthe short-terminterestintheAWMdatabase. AsstatedinSmetsandWouters(2005),totalemploymentdataisusedinplaceofhoursworked duetotheabsenceofhoursworkeddatafortheeuroarea. 19Gali,Smets,andWouters(2011)obtainidenti(cid:12)cationbyembedingatheoryofunemploymentandbyincludingdataonunemployment. Justiniano,Primiceri,andTambalotti(2013)donotexploittheconnectionbetweenunemploymentandwagemarkupsandassumeinstead aparticularstochasticstructureforthelatter(whitenoise)toobtainidenti(cid:12)cation. 24

induced by the wage markup shock. The relative weight on the labor supply shock depicted along the horizontal axis governs the relative importance of the two shocks. Both for the commitment and the discretion case, the ranking of targeting frameworks is independent of the relative importance of wage markup and labor supply shocks with the exception of the NIT and the NIT-II framework for the case of discretion and a high importance of the labor supply shock. As the importance of the inefficient wage markup shock is reduced, the welfare differences between targeting frameworks shrink by construction. Monetary policy can mostly offset the welfare consequences of the labor supply shock; when wage markup shocks are absent from the model, price markup shocks are the only remaining source of inefficient (cid:13)uctuations. Aslongasonebelieveswagemarkupshockstoplaysomeroleindrivingbusinesscycle(cid:13)uctuationasin Gali, Smets, and Wouters (2011), the speed limit policy framework under discretion strongly outperforms all other frameworks under discretion (as well as the in(cid:13)ation targeting framework under commitment). ButevenfortheassessmentinJustiniano,Primiceri,andTambalotti(2013),whichassignslittleimportance to wage markup shocks, the speed limit policy framework performs well. Absent certainty about the true data-generating process, adopting the speed limit policy framework may turn out to be a prudent choice. 3.4.3 Zero lower bound on nominal interest rates Following earlier work on optimal policy design, we have abstracted from the implications for monetary policy imposed by the zero lower bound on the nominal interest rate. This way of preceding allows us to include larger models and to consider aspects of parameter uncertainty. Furthermore, the probability of the policy rate reaching zero (and staying at zero for several periods) is low in the CEE/SW model. As long as the time that the economy spends at the zero bound is short, economic outcomes when the zero boundisenforcedbarelydifferfromtheoutcomeswhenthepolicyrateisallowedtoviolatethezerobound. Thus, the optimal parameterization of each targeting framework is expected to change by little if we were to impose the zero bound in our analysis. Nevertheless, we want to touch on the challenges for monetary policy design presented by the zero bound at least in closing. Figure 11 plots the impulse responses of selected variables to a combination of contractionary demand shocksundertheoptimalcommitmentpolicy. The(cid:12)gurealsoplotstheresponsesunderin(cid:13)ationtargeting, priceleveltargeting, and the speed limitpolicy: the policymakeractsunder discretion, the model parameters are (cid:12)xed at the posterior mode, and the objective functions are parameterized as found to be optimal absent the zero bound constraint.20 In response to the shock, the optimal commitment policy lowers the short-term interest rate to zero, although not for long, and allows for mild de(cid:13)ation of prices and wages. The output gap turns negative and closes slowly. Further out, the optimal commitment policy allows for onlyveryminorovershootingofpriceandwagein(cid:13)ationabovetheirlong-runtargetvaluesandtheoutput 20Initially, the economy is assumed to be growing along the balanced growth path. In period 1 the economy experiences a negative one-standard deviation risk-premium shock together with a negative 10-standard deviation shock to government spending. In addition, we lowered the value of the nominal interest rate along the balanced growth path to 4%. The problem is solved using the piece-wise linearapproachinEggertssonandWoodford(2003),Coibion,Gorodnichenko,andWieland(2012),andGuerrieriandIacoviello(2015);we abstractfrommodi(cid:12)cationsofthesociallossfunctionthatcouldresultfromthezeroboundconstraint. 25

gap hardly rises above zero. Although operated under discretion, all three targeting frameworks perform closely to the optimal commitment policy. The in(cid:13)ation and price level targeting central banks are more aggressive at stabilizing priceandwagein(cid:13)ationandtheoutputgap. Astheshockpushespriceandwagein(cid:13)ation, andtheoutput gapinthesamedirection,thehighrelativeweightonpricein(cid:13)ationintheobjectivefunctionofthein(cid:13)ation targeting central bank allows the in(cid:13)ation targeting central bank to mimic the behavior of the price level targeting central bank.21 The optimal speed limit policy computed in Section 3.1 allows for larger deviations of in(cid:13)ation and the output gap than the optimal commitment policy. Under the speed limit policy, the policymaker seeks to adjust the output gap gradually. While such gradualism is of advantage in response to price and wage markupshocks|keepingtheoutputgapnegativeafteranin(cid:13)ationaryshockssignalstightfuturemonetary policy and reduces the initial rise in in(cid:13)ation|it is of potential disadvantage after large demand shocks that push the policy interest rate to zero. The slow closing of the output gap under the speed limit policy prevents price and wage in(cid:13)ation from a fast return to their long-run targets. Shocks that are more contractionary than the ones underlying Figure 11 can exacerbate this feature of the speed limit policy. This potential drawback of the speed limit policy can be ameliorated by reducing the weight on the activity measure in the objective function. To convey this idea, Figure 11 also plots the impulse responses under a speed limit policy with a reduced weight on the output gap under the label Alt. SLP (that is one tenth of the weight found to be optimal in Section 3.1). Under the reduced weight, the speed limit policy closely resembles the optimal commitment policy. While the dramatic reduction in weight on the activity measure worsens the performance of the speed limit policy to price and wage markup shocks in particular, this speci(cid:12)c parameterization of the speed limit policy still outperforms the optimal in(cid:13)ation and the optimal price level targeting framework under discretion computed in Section 3.1 for the posterior mode parameterization of the model.22 The optimal parameterization let alone the ranking of targeting regimes in the CEE/SW model may hardly be affected if we enforced the zero bound constraint on nominal interest rates. If shocks that call for lowering the policy interest rates to zero are more frequent than in the CEE/SW model, price level targeting might be preferred to the speed limit policy under discretionary policymaking given a low value of the long-run in(cid:13)ation target. However, raising the long-run in(cid:13)ation target may constitute a viable alternative: the monetary authority can adopt a speed limit policy which is effective in ameliorating the time inconsistency problem associated with price and wage markup shocks while signi(cid:12)cantly reducing the likelihoodofzeroboundevents. Whetherthesebene(cid:12)tsoutweighthecostsofachievingalong-runin(cid:13)ation 21This result is not at odds with Adam and Billi (2007) or Bodenstein, Hebden, and Nunes (2012) who point out the importance of commitment at the zero lower bound when the central bank maximizes the discounted utility of the representative household. In our application,thediscretionarycentralbankplacesahigherweightonstabilizingpricein(cid:13)ationthanunderthetruesociallossfunctionand isthereforemuchbetterpositionedtostabilizetheeconomythroughaccommodativemonetarypolicythaninthosepapersforthecaseof discretion. 22Abstractingfromthezerolowerbound,theoptimalparameterizationofeachframeworkisprimarilydeterminedbythepriceandwage markup shocks. Ironically, the optimal weight on the activity measure under the speed limit policy is higher when these markup shocks aremoreimportantwhichinturnimpedesthecentralbank’sabilitytostabilizetheeconomyinthefaceoflargenegativedemandshocks andzerointerestrates. 26

target is an empirical question beyond the scope of this section.23 4 Conclusion The debate on targeting frameworks has often focused on the differences between in(cid:13)ation and price level targeting. In models that follow the New Keynesian paradigm, the optimal commitment policy tends to undo most, if not all, changes of price and wage in(cid:13)ation from their long-run targets over time to realign pricesandwageswiththeirlong-runtrends. Giventhisinsight,priceleveltargetingappearstobeanatural contender to in(cid:13)ation targeting when policymakers act under discretion. However, we argue that speed limit policy is a clear alternative to both the in(cid:13)ation targeting and the price level targeting framework. The objective function underlying the speed limit policy framework with its long-run commitment to stable in(cid:13)ation and its short-run focus on in(cid:13)ation and smooth changes in the output gap leads to better outcomes than all other frameworks when policymakers act under discretion in many circumstances. When policymakers act under commitment, the differences between the three targetingframeworksarenegligible. Mostimportantly,thespeedlimitpolicyunderdiscretionoutperforms in(cid:13)ationtargetingundercommitmentinnumerouscases. Weshowtherelativesuperiorityofthespeedlimit policyframeworkinasequenceofsimpleNKmodels,thatintroducein(cid:13)ationindexation,habitpersistence in consumption, and sticky wages, and in the CEE/SW model. The optimal speed limit policy is more robust to empirically-relevant alternative parameterizations of the CEE/SW model and to unclarity about theabilityofthecentralbanktocommit. Unlesstheeconomycanexperiencelargeandpersistentnegative (demand) shocks and the costs of raising the long-run in(cid:13)ation target are high, the speed limit policy will also outperform in(cid:13)ation and price level targeting under discretion when the zero lower bound constraint on nominal interest rates is enforced in the model. Sincespeedlimitpolicieshavenotyetbeenasthoroughlyexaminedasin(cid:13)ationandpriceleveltargeting, a range of open questions remain to be addressed. How would a speed limit policy perform under model settings that included informational rigidities, or (cid:12)nancial frictions? How does a central bank’s ability to measure the output gap accurately in real-time|an issue explored in Orphanides (2003)| in(cid:13)uence the relative performance of targeting frameworks? What about central bank communication of current and futurepolicygoals? Giventhepromisingperformanceofspeedlimitpoliciesshowninthispaper,itappears worth to continue exploring the implications of this policy and (cid:12)nd answers to the preceding questions. 23Pursuinghigherin(cid:13)ationtargetshascapturedtheimaginationofpolicymakersintheaftermathoftheGreatRecession,seeWilliams (2016). Coibion, Gorodnichenko, and Wieland (2012) compute the optimal in(cid:13)ation target for a discretionary central bank to fall just below3%;Billi(2011)reportssigni(cid:12)cantlyhighernumbers. 27

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Table 1: Parameter values for the textbook NKM and its extensions Parametersgoverning stickyprices stickywages other Model description (cid:24) p (cid:19)p (cid:18)p (cid:28)(cid:22)p (cid:24) w (cid:19)w (cid:18)w (cid:28)(cid:22)w h (cid:27)C (cid:27)L (cid:12) 1 textbookmodel 0:8 0 0:61 0:61 (cid:0) (cid:0) (cid:0) (cid:0) 0 1.39 1.92 0.9984 2 priceindexation 0:8 [0;1] 0:61 0:61 (cid:0) (cid:0) (cid:0) (cid:0) 0 1.39 1.92 0.9984 3 inefficientsteadystate 0:8 [0;1] 0:61 0 (cid:0) (cid:0) (cid:0) (cid:0) 0 1.39 1.92 0.9984 4 consumptionhabits 0:8 [0;1] 0:61 0:61 (cid:0) (cid:0) (cid:0) (cid:0) 0:7 1.39 1.92 0.9984 5 stickywages 0:8 [0;1] 0:61 0:61 0:8 0 0:5 0:5 0 1.39 1.92 0.9984 Note: The table documents the parameter values of the textbook NKM and its extensions underlying Figures 1 and 2. Model 1 is the textbookNKMwithoutindexation. InModel2weaugmentthetextbookNKMtoallowforpriceindexation. Model3featuresdistortions inthesteadystate. HabitpersistenceinconsumptionisintroducedinModel4. Finally,Model5allowsforstickynominalpricesandwage. Inallmodels,anARMA(1,1)pricemarkupshockisthesolesourceof(cid:13)uctuationswiththeautocorrelationcoefficient(cid:26) =0:9,themoving u averagecoefficient(cid:26) uϵ =0:74,andthestandarddeviationforinnovations(cid:27)u=0:0014. 31

Table 2: Performance of targeting frameworks under parameter uncertainty a: Frequencyofbeingthebestframework: optimalweightscase IT PLT SLP Commitment 0.9723 0.0004 0.0273 Discretion 0.0000 0.0167 0.9833 b: Frequencyofbeingthebestframework: (cid:12)xedweightscase IT PLT SLP Commitment 0.8281 0.0073 0.1646 Discretion 0.0000 0.0168 0.9832 c: Frequencyofbeingthebestframework: exchangedweightscase IT PLT SLP Commitment 0.0036 0.0056 0.9908 Discretion 0.0000 0.0162 0.9838 d: PercentilesofCEVdifferences Commitment Discretion Quantile IT PLT SLP IT PLT SLP 5% 0.0022 -0.0493 -0.0280 -0.7717 -0.2169 0.0114 10% 0.0055 -0.0413 -0.0244 -0.6086 -0.1621 0.0193 15% 0.0081 -0.0368 -0.0222 -0.5291 -0.1350 0.0258 20% 0.0098 -0.0338 -0.0207 -0.4772 -0.1172 0.0310 25% 0.0110 -0.0316 -0.0194 -0.4349 -0.1035 0.0362 30% 0.0119 -0.0297 -0.0184 -0.4013 -0.0927 0.0409 35% 0.0128 -0.0280 -0.0174 -0.3728 -0.0837 0.0458 40% 0.0135 -0.0264 -0.0166 -0.3475 -0.0757 0.0510 45% 0.0142 -0.0250 -0.0158 -0.3253 -0.0685 0.0565 50% 0.0150 -0.0237 -0.0150 -0.3053 -0.0624 0.0624 55% 0.0158 -0.0226 -0.0143 -0.2856 -0.0565 0.0685 60% 0.0165 -0.0213 -0.0136 -0.2674 -0.0510 0.0757 65% 0.0174 -0.0202 -0.0128 -0.2483 -0.0458 0.0837 70% 0.0183 -0.0191 -0.0120 -0.2294 -0.0409 0.0927 75% 0.0193 -0.0178 -0.0110 -0.2105 -0.0362 0.1035 80% 0.0206 -0.0165 -0.0099 -0.1910 -0.0310 0.1172 85% 0.0222 -0.0149 -0.0082 -0.1675 -0.0258 0.1350 90% 0.0243 -0.0130 -0.0056 -0.1388 -0.0193 0.1621 95% 0.0279 -0.0094 -0.0023 -0.0970 -0.0114 0.2169 Note: Thetablesummarizestheperformanceofin(cid:13)ationtargeting(IT),priceleveltargeting(PLT),andspeedlimitpolicy(SLP)whenthe parametersoftheCEE/SWmodelaredrawnfromtheLaplaceapproximationtotheposteriordistributioninSmetsandWouters(2007). Panel (a) states the frequency of each targeting regime being the best performing one for both the case of commitment and discretion. Theweightontheactivitymeasure(cid:21)TF ischosenoptimallyforeachframeworkandeachparameterdraw. InPanel(b)theweightonthe x activitymeasure(cid:21)TF is(cid:12)xedforeachframeworkatthevaluethatisoptimalwhenthemodelisparameterizedattheposteriormode. All x other parameters are drawn from the Laplace approximation to the posterior distribution. In Panel (c) when policy is conducted under commitment(discretion)theweightontheactivitymeasure(cid:21)TF is(cid:12)xedforeachframeworkatthevaluethatisoptimalunderdiscretion x (commitment)fortheposteriormodeparameterizationofthemodel. AllotherparametersaredrawnfromtheLaplaceapproximationto theposteriordistribution. InPanel(d),we(cid:12)rstcomputetheCEVdifferencebetweenthebestperformingandthesecondbestperforming frameworkforeachparameterization;wethenrankthedifferencesbysizeforeachframeworkandcomputepercentiles. 32

Figure 1: Targeting frameworks in the textbook NKM 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 weight on output gap λ x VEC*001 Commitment 0.8 PLT SLP IT 0.7 OPT−PLT OPT−SLP OPT−IT 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 weight on output gap λ x VEC*001 Discretion 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Note: The (cid:12)gure plots the welfare loss for each targeting framework against the optimal commitment policy under different values for (cid:21)TF. The only source of (cid:13)uctuations is an ARMA(1,1) markup shock. Welfare is reported in terms of consumption equivalent variation x multipliedby100. Theweight(cid:21)TF forwhichthewelfarelossisminimizedisindicatedby\◦"underpriceleveltargeting(PLT),\(cid:3)"under x speedlimitpolicy(SLP),and\⋄"underin(cid:13)ationtargeting(IT),respectively. 33

Figure 2: Welfare evaluation of targeting frameworks in extensions of the textbook NKM 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 Textbook NKM 0 -0.05 -0.1 PLT SLP -0.15 IT -0.2 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 Textbook NKM 0.2 0.1 0 -0.1 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 Distorted SS 0 -0.02 -0.04 -0.06 -0.08 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 Distorted SS 0.1 0.05 0 -0.05 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 External Habit 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 External Habit 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 Sticky Wage 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 COMMITMENT DISCRETION Changing Price Indexation Sticky Wage 4 3 2 1 0 Note: Welfare performance of price level targeting and speed limit policy relative to in(cid:13)ation targeting in the textbook NKM and its extensionswithvaryingdegreeofpriceindexation(cid:19)p undercommitmentanddiscretion. Theonlysourceof(cid:13)uctuationsisanARMA(1,1) markupshock. Welfareisreportedintermsofconsumptionequivalentvariationmultipliedby100. Thetoprowdepictstheresultsinthe textbookNKMwithanefficientsteadystateandpriceindexation. EachofthefollowingrowsdiffersfromthetextbookNKMbyasingle feature: distortedsteadystate(secondrow),externalconsumptionhabits(thirdrow),andstickynominalwages(lastrow). 34

Figure 3: Welfare evaluation of targeting frameworks in the CEE/SW model 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 Baseline CEE/SW 0 -5 -10 -15 PLT -20 SLP NIT -25 NIT−II IT -30 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 Baseline CEE/SW 60 50 40 30 20 10 0 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 CEE/SW with Subsidies and Taxes 0 -5 -10 -15 -20 -25 -30 -35 -40 0 0.2 0.4 0.6 0.8 1 price indexation ι p VEC∆*001 CEE/SW with Subsidies and Taxes 40 35 30 25 20 15 10 5 0 0 0.2 0.4 0.6 0.8 1 wage indexation ι w VEC∆*001 Baseline CEE/SW 0 -5 -10 -15 0 0.2 0.4 0.6 0.8 1 wage indexationι w VEC∆*001 COMMITMENT DISCRETION Changing Price Indexation est. posterior mode: ι = 0.22 p est. posterior mode: ι = 0.22 p est. posterior mode: ι = 0.22 p est. posterior mode: ι = 0.22 p Changing Wage Indexation Baseline CEE/SW 50 40 est. posterior mode: ι = 0.59 w est. posterior mode: ι = 0.59 30 w 20 10 0 Note: Welfare performance of price level targeting (PLT), speed limit policy (SLP), and the two nominal income targeting frameworks (NIT,NIT-II)relativetoin(cid:13)ationtargeting(IT)intheCEE/SWmodelundercommitmentanddiscretion. Parametersaresetatthemode of the posterior distribution reported in Smets and Wouters (2007). Welfare is measured in terms of consumption equivalent variation multipliedby100. Inthe(cid:12)rsttworowsofpanels,wevarythedegreeofpriceindexation. ThesecondrowdeviatesfromSmetsandWouters (2007)bycorrectingsteadystateinefficienciesduetoexternalhabitsandmonopolisticcompetition. Thethirdrowconsidersvariationsin thedegreeofwageindexation. 35

Figure 4: Impulse responses in the CEE/SW model to price and wage markup shocks 0 20 40 60 ss morf .ved tnecrep Price Level 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 20 40 60 ss morf .ved tnecrep Wage Level 1.5 1 0.5 0 0 20 40 60 tnecrep Output Gap -0.1 -0.2 Optimal Commitment Policy -0.3 Optimal IT Optimal PLT Optimal SLP -0.4 -0.5 0 20 40 60 quarters ss morf .ved tnecrep 0 20 40 60 Real Wage -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 ss morf .ved tnecrep Price Level 6 5 4 3 2 1 0 20 40 60 ss morf .ved tnecrep Wage Level 6 5 4 3 2 1 0 20 40 60 tnecrep Output Gap 0 -0.2 -0.4 -0.6 -0.8 0 20 40 60 quarters ss morf .ved tnecrep Price Markup Shock Wage Markup Shock Real Wage 0.4 0.3 0.2 0.1 0 Note: The (cid:12)gure compares the impulse responses to a price and wage markup shock under the optimal commitment policy, in(cid:13)ation targeting (IT), price level targeting (PLT), and speed limit policy (SLP). The two markup shocks follow ARMA(1,1) processes. See also AppendixD. 36

Figure5: UnderstandingthewelfarerankingsintheCEE/SWmodelunderdiscretion: introducingfeaturessequentially 0 0.2 0.4 0.6 0.8 price index ι p VEC∆*001 SW--Woodford 0.4 0.2 0 -0.2 -0.4 -0.6 0 0.2 0.4 0.6 0.8 price index ι p VEC∆*001 Inefficient SS 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 0 0.2 0.4 0.6 0.8 price index ι p VEC∆*001 K and G 0.2 0 -0.2 -0.4 -0.6 0 0.2 0.4 0.6 0.8 price index ι p VEC∆*001 Sticky wage with subsidy 6 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 price index ι p VEC∆*001 PLT SLP NIT NIT−II IT Wage markup shock 6 4 2 0 -2 -4 0 0.2 0.4 0.6 0.8 price index ι p VEC∆*001 No wage subsidy 6 4 2 0 -2 -4 0 0.2 0.4 0.6 0.8 price index ι p VEC∆*001 h = 0.71 12 10 8 6 4 2 0 0 0.2 0.4 0.6 0.8 price index ι p VEC∆*001 DISCRETION ξ = 0.85 and ξ = 0.88 p w 80 70 60 50 40 30 20 10 0 0 0.2 0.4 0.6 0.8 price index ι p VEC∆*001 Kimball (Full SW) 60 50 40 30 20 10 0 Note: Welfare performance of price level targeting (PLT), speed limit policy (SLP), and the two nominal income targeting frameworks (NIT, NIT-II) relative to in(cid:13)ation targeting (IT) in the CEE/SW model under discretion. From top left to bottom right we augment the textbook NKM step-by-step by the features in Smets and Wouters (2007): Goods subsidies are removed to render the steady state inefficient,capitalandgovernmentspendingareaddedintoprightpanel. Inthesecondrow,stickywageswithawagesubsidytoremove distortionsinthelabormarketareintroduced,awagemarkupshockisadded,and(cid:12)nally,thewagesubsidyisremoved. Inthe(cid:12)nalrow, we introduce external consumption habits, increase the nominal rigidities to obtain the same slopes in the NKPCs in the model without variableelasticityofsubstitutionasinthefullCEE/SWmodelwithaKimball(1995)aggregatorinthebottomrightpanel. 37

Figure 6: Targeting frameworks in the CEE/SW model for alternative parameterizations: optimal weights case -0.8 -0.6 -0.4 -0.2 0 CEV ytisned IT under Commitment 40 35 30 25 20 15 10 5 0 -0.8 -0.6 -0.4 -0.2 0 CEV ytisned PLT under Commitment 25 20 15 10 5 0 -0.8 -0.6 -0.4 -0.2 0 CEV ytisned SLP under Commitment 30 25 20 15 10 5 0 -2.5 -2 -1.5 -1 -0.5 0 CEV ytisned IT under Discretion 4 3.5 3 2.5 2 1.5 1 0.5 0 -0.8 -0.6 -0.4 -0.2 0 CEV ytisned PLT under Discretion 15 10 5 0 -0.8 -0.6 -0.4 -0.2 0 CEV ytisned SLP under Discretion 50 40 30 20 10 0 0 0.2 0.4 0.6 optimized weight ytilibaborp CDF for IT 11 0.8 0.6 0.4 0.2 0 0 20 40 60 80 optimized weight ytilibaborp CDF for PLT 11 0.8 0.6 0.4 0.2 0 0 50 100 optimized weight ytilibaborp 95%: -0.0137 95%: -0.0281 95%: -0.0250 50%: -0.0288 50%: -0.0538 50%: -0.0454 5%: -0.1272 5%: -0.1390 5%: -0.1649 95%: -0.1284 95%: -0.0341 95%: -0.0130 50%: -0.3342 50%: -0.0903 50%: -0.0260 5%: -0.8637 5%: -0.3261 5%: -0.1329 CDF for SLP 1 0.8 0.6 0.4 0.2 0 Note: The (cid:12)gure plots the distribution of welfare and the optimized weights (cid:21)TF for in(cid:13)ation targeting (IT), price level targeting (PLT) x andspeedlimitpolicy(SLP)undercommitmentanddiscretionwhentheparametersoftheCEE/SWmodelaredrawnfromtheLaplace approximation to the posterior distribution in Smets and Wouters (2007). We simulate 30000 draws. The top row shows the density distribution of the consumption equivalent variation (CEV) under commitment, the middle row shows the results under discretion. The bottomrowofpanelsdepictsthecumulativedistributionfunction(CDF)oftheoptimalweightsunderdiscretionandcommitmentforeach frameworkinasinglepanel. 38

Figure 7: Targeting frameworks in the CEE/SW model for alternative parameterizations: (cid:12)xed weights case 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 CEV difference ytilibaborP CDF under Commitment 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 PLT SLP IT 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 CEV difference ytilibaborP CDF under Discretion 0.9 0.8 0.7 0.6 SLP 0.5 PLT IT 0.4 0.3 0.2 0.1 0 Note: The(cid:12)gureplotsthecumulativewelfaredistributionunderin(cid:13)ationtargeting(IT),priceleveltargeting(PLT),andspeedlimitpolicy (SLP)whentheweightsontheactivitymeasureare(cid:12)xedatthevaluesthatareoptimalundertheposteriormodeparameterizationofthe CEE/SW model relative to the case when the weights on the activity measure are set optimally for each parameter draw and targeting framework. All other parameters are drawn from the Laplace approximation to the posterior distribution in Smets and Wouters (2007). Wesimulate30000draws. Theupperpanelplotsthecumulativedistributionfunction(CDF)undercommitment;thebottompanelplots thecumulativedistributionfunction(CDF)underdiscretion. 39

Figure 8: Targeting frameworks in the CEE/SW model for alternative parameterizations: exchanged weights case 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 CEV difference ytilibaborP CDF under Commitment 1 0.9 0.8 0.7 0.6 SLP 0.5 PLT IT 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 CEV difference ytilibaborP CDF under Discretion 1 0.9 0.8 0.7 0.6 SLP 0.5 PLT IT 0.4 0.3 0.2 0.1 0 Note: The(cid:12)gureplotsthecumulativewelfaredistributionunderin(cid:13)ationtargeting(IT),priceleveltargeting(PLT),andspeedlimitpolicy (SLP)whentheweightsontheactivitymeasureundercommitment(discretion)are(cid:12)xedatthevaluesthatareoptimalundertheposterior modeparameterizationoftheCEE/SWmodelwithdiscretion(commitment)relativetothecasewhentheweightsontheactivitymeasure are set optimally for each parameter draw and targeting framework. All other parameters are drawn from the Laplace approximation to the posterior distribution in Smets and Wouters (2007). We simulate 30000 draws. The upper panel plots the cumulative distribution function(CDF)undercommitment;thebottompanelplotsthecumulativedistributionfunction(CDF)underdiscretion. 40

Figure 9: Welfare evaluation of targeting frameworks in the CEE/SW model estimated with euro area data 0 0.2 0.4 0.6 0.8 price index ι p VEC∆*001 CEE/SW Euro Data 0 -2 -4 -6 -8 -10 -12 -14 PLT SLP -16 NIT NIT−II -18 IT -20 0 0.2 0.4 0.6 0.8 1 price index ι p VEC∆*001 CEE/SW Euro Data 120 100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1 wage index ι w VEC∆*001 CEE/SW Euro Data 0 -5 -10 -15 0 0.2 0.4 0.6 0.8 1 wage index ι w VEC∆*001 COMMITMENT DISCRETION Changing Price Indexation posterior mode: ι =0.128 p posterior mode: ι =0.128 p Changing Wage Indexation CEE/SW Euro Data 80 70 60 50 40 30 posterior mode: ι w =0.374 posterior mode: ι =0.374 w 20 10 0 Note: Welfare performance of price level targeting (PLT), speed limit policy (SLP), and the two nominal income targeting frameworks (NIT,NIT-II)relativetoin(cid:13)ationtargetingintheCEE/SWmodelestimatedwitheuroareadata(1975Q4to2008Q3)undercommitment anddiscretion. Inthe(cid:12)rstrowofpanelsthedegreeofpriceindexationisvaried. Thesecondrowconsidersvariationsinthedegreeofwage indexation. Thedegreeofindexationattheposteriormodeisindicatedwith(cid:19)p=0:128forpricesand(cid:19)w =0:374forwages,respectively. 41

Figure 10: Welfare evaluation of targeting frameworks: relative importance of wage markup shocks and labor supply shocks 0 0.2 0.4 0.6 0.8 1 weight on labor supply shock VEC∆*001 COMMITMENT 0 -2 -4 -6 PLT -8 SLP NIT NIT−II IT -10 -12 -14 0 0.2 0.4 0.6 0.8 1 weight on labor supply shock VEC∆*001 DISCRETION 35 30 25 20 15 10 5 0 -5 Note: Welfare performance of price level targeting (PLT), speed limit policy (SLP), and the two nominal income targeting frameworks (NIT, NIT-II) relative to in(cid:13)ation targeting in the modi(cid:12)ed CEE/SW model when allowing for labor supply and wage markup shocks. This version of the model features preferences that are separable in consumption and leisure. The relative importance of the two shocks is controlled by the weight parameter indicated on the x-axis. \0" indicates the absence of the labor supply shock and \1" indicates the absenceofthewagemarkupshock. ThewagemarkupshockfollowsanARMA(1,1)processasinSmetsandWouters(2007),whereasthe laborsupplyshockisassumedtobeanAR(1)process. Thelaborsupplyshockisscaledtoensuresimilarmagnitudesoftheshockasthe ARMA(1,1)wagemarkupshockwhencomparingtheunconditionalvariancesoftheshocks. 42

Figure 11: Welfare evaluation of targeting frameworks under the zero lower bound constraint 0 2 4 6 8 10 12 14 16 18 20 ss morf .ved tniop egatnecrep Nominal Interest Rate -0.5 -1 -1.5 -2 -2.5 Optimal Commitment Policy Optimal IT -3 Optimal PLT Optimal SLP Alt. SLP -3.5 -4 0 2 4 6 8 10 12 14 16 18 20 ss morf .ved tniop egatnecrep Price Inflation 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 -0.45 0 2 4 6 8 10 12 14 16 18 20 quarters ss morf .ved tniop egatnecrep Wage Inflation 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 2 4 6 8 10 12 14 16 18 20 quarters tnecrep Output Gap 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 Note: The (cid:12)gure compares the impulse responses to a negative one-standard deviation risk premium shock and a negative 10-standard deviation shock to government spending under in(cid:13)ation targeting (IT), price level targeting (PLT), and speed limit policy (SLP) each underdiscretionandtheundertheoptimalcommitmentpolicy. Theshocksarelargeenoughforthepolicyinterestratetobeconstrained bythezerolowerbound. 43

Online Appendix A Methodology This Appendix discusses the computational details of our analysis. We describe how to: 1. obtain a valid second-order accurate welfare criterion for any nonlinear model 2. evaluate this welfare criterion given the (linear) decision rules under each monetary policy framework 3. compute the linear decision rules under each targeting framework for the case of (a) commitment (b) discretion 4. translatethewelfaredifferencesthatarisebetweenthemonetarypolicyframeworksintoconsumption units. Given a fully-speci(cid:12)ed model, our analysis proceeds as follows. First, we obtain a purely quadratic approximation of the welfare function that describes the preferences of society (in standard applications, the utility function of the representative household)|the true social loss function. This approximation is summarized by the matrices (A(L);B(L)). Next, we assume that the central bank optimizes a given, yet arbitrary, quadratic objective subject to the linearized structural equations of the underlying model of the economy. Thelinearized economyissummarized bythematrices (C(L);D(L)). Solving thesystem of(cid:12)rst order conditions delivers linear decision rules that describe the behaviour of the economy under the given objective function for monetary policy. Finally, we use the matrices (A(L);B(L)) and the linear decision rulestomeasurethewelfareimplicationsofeachpolicyobjective. Withineachclassofpolicyobjectiveswe searchforitsloss-minimizingparameterization. Whilewerestrictattentiontolinear-quadraticframeworks, i.e., quadraticobjectivefunctionsandlinearconstraints, forcomparabilitywiththe existingliterature, our approach can be implemented at higher orders of approximation without restrictions. A.1 Welfare criterion For a given model, let the N (cid:2) 1 vector of endogenous variables be denoted by x with the partition t x = (x~′;i )′. The variable i is the policy instrument of the central bank, typically a short-term interest t t t t rate. The vector (cid:16) refers to the complete set of exogenous variables. Given the central bank’s choice of t the policy instrument for all periods t(cid:21)t , fi g1 , the remaining N(cid:0)1 endogenous variables satisfy the 0 t t=t0 N (cid:0)1 structural model equations E t g(x t(cid:0)1 ;x t ;x t+1 ;(cid:16) t )=0 (38) in equilibrium. The system of equations in (38) is assumed to be differentiable up to the desired order of approximation. Werefrainfromsplittingg((cid:1))intoequationsthatcontainnoforward-lookingvariablesand equations that do contain forward-looking variables for ease of notation and proceed as if each equation in g((cid:1))containsatleastoneforward-lookingvariable.24 Theintertemporalpreferencesofsocietyaredescribed 24Whenimplementingournumericalprocedure,however,wecarefullyseparatetheequationsintothosewithandwithoutforward-looking variablesasinBenignoandWoodford(2012). 44

Online Appendix ∑ by U = E 0 1 t=t0 (cid:12)t(cid:0)t0U(x t(cid:0)1 ;x t ;(cid:16) t ) with the utility function U(x t(cid:0)1 ;x t ;(cid:16) t ). Within this setting, the optimal monetary policy under full commitment is derived from the maximization program ∑1 fx m t g a 1 t= x t0 E 0 t=t0 (cid:12)t(cid:0)t0U(x t(cid:0)1 ;x t ;(cid:16) t ) s:t: E t g(x t(cid:0)1 ;x t ;x t+1 ;(cid:16) t )=0: (39) As is well understood, the problem stated in (39) does not deliver time-invariant decision rules. Following alargebodyoftheliterature,weoptfortheoptimalmonetarypolicyundercommitmentfromthetimeless perspective as the reference point to evaluate the performance of different policies, henceforth referred to as the optimal commitment policy; see Woodford (2003a). Optimality from a timeless perspective assumes that the policymaker can \pre-commit" to a policy before period t of the form 0 g(x t0 (cid:0)2 ;x t0 (cid:0)1 ;x t0 ;(cid:16) t0 (cid:0)1 )=g(cid:22) t0 (40) to yield the new optimization program ∑1 fx m t g a 1 t= x t0 E 0 t=t0 (cid:12)t(cid:0)t0U(x t(cid:0)1 ;x t ;(cid:16) t ) s:t: E t g(x t(cid:0)1 ;x t ;x t+1 ;(cid:16) t )=0 g(x t0 (cid:0)2 ;x t0 (cid:0)1 ;x t0 ;(cid:16) t0 (cid:0)1 )=g(cid:22) t0 : (41) If not all equations in g((cid:1)) contain forward-looking variables, pre-commitments only need to be speci(cid:12)ed for those equations with forward-looking variables. As stressed in Benigno and Woodford (2012) and Debortoli and Nunes (2006) assuming that policy is conducted under suitable pre-commitments is generally key in order to obtain a purely quadratic approximation of the welfare function.25 Two important remarks are in order: 1. Includingthepre-commitmentconstraintsin(40)intoproblem(39)changestheoriginaloptimization problem. 2. Policies that violate the initial pre-commitments (40) are penalized with regard to welfare in accordance with the severity of the violation. In particular, the path of the endogenous variables derived from the original problem (39) may no longer be deemed optimal under the new program (41). There are two equivalent approaches to obtain the correct linear-quadratic approximation of the optimization problem stated in (41). The (cid:12)rst approach (LQ problem), described in Benigno and Woodford (2012), is often followed to obtain a compact characterization of the policy problem in small-scale models. Starting from a second-order Taylor-series expansion of the utility function U(x t(cid:0)1 ;x t ;(cid:16) t ), second-order Taylor-seriesexpansionsofthestructuralequations,E t g(x t(cid:0)1 ;x t ;x t+1 ;(cid:16) t )=0,andofthepre-commitment 25Most prominently, assuming optimality from a timeless perspective is necessary if the deterministic steady state of the model is inefficient. CompareBenignoandWoodford(2005)fordetails. 45

Online Appendix constraint,g(x t0 (cid:0)2 ;x t0 (cid:0)1 ;x t0 ;(cid:16) t0 (cid:0)1 )=g(cid:22) t0 ,areusedtosubstituteoutthelineartermsintheapproximation to the utility function. As a result, the approximation to the welfare function involves quadratic terms only and it can be maximized subject to the linear approximation of the constraints in (38) and (40) to get a (cid:12)rst-order accurate approximation to the problem in (41). The alternative approach computes the (cid:12)rst order conditions of the problem in (41) and then seeks the approximation of the resulting system of equations to the desired order. Both approaches can be implemented numerically. We utilize the toolbox developed in Bodenstein, Guerrieri, and LaBriola (2014) which follows the second approach. The (cid:12)rst order conditions associated with the program (41) ∑1 fx m t g a 1 t= x t0 E 0 t=t0 (cid:12)t(cid:0)t0U(x t(cid:0)1 ;x t ;(cid:16) t ) s:t: E t g(x t(cid:0)1 ;x t ;x t+1 ;(cid:16) t )=0 g(x t0 (cid:0)2 ;x t0 (cid:0)1 ;x t0 ;(cid:16) t0 (cid:0)1 )=g(cid:22) t0 imply that under the assumptions of full commitment and optimality from a timeless perspective the equilibrium process fx ;φ g1 satis(cid:12)es t t t=t0 D x U(x t(cid:0)1 ;x t ;(cid:16) t )+(cid:12)E t D x(cid:0)U(x t ;x t+1 ;(cid:16) t+1 ) { } +(cid:12)E t φ ′ t+1 D x(cid:0)g(x t ;x t+1 ;x t+2 ;(cid:16) t+1 ) +E t fφ ′ t D x g(x t(cid:0)1 ;x t ;x t+1 ;(cid:16) t )g +(cid:12) (cid:0)1φ ′ t(cid:0)1 D x+ g(x t(cid:0)2 ;x t(cid:0)1 ;x t ;(cid:16) t(cid:0)1 )=0 (42) and the structural equations E t g(x t(cid:0)1 ;x t ;x t+1 ;(cid:16) t )=0 (43) ateachdatet(cid:21)t . ThenotationD denotesthevectorofpartialderivativesofanyfunctionswithrespect 0 x to the elements of x t ; likewise do D x(cid:0) and D x+ for derivatives with respect to x t(cid:0)1 and x t+1 , respectively. Taking a (cid:12)rst order approximation of the equations in (42) around the deterministic steady state of the model delivers [ ] D x 2 x(cid:0) { U(cid:22)x^ t(cid:0)1 + D x 2 x U(cid:22) +(cid:12)D x 2 (cid:0)x(cid:0) U(cid:22) x^ t +(cid:12)D x 2 (cid:0)x U(cid:22)E t x^ t+1 +D x 2 (cid:16) U(cid:22)(cid:16) t } +(cid:12)D x 2 (cid:0)(cid:16) U(cid:22)E t (cid:16) t+1 +(cid:12)φ(cid:22) D2 g(cid:22)x^ +D2 g(cid:22)E x^ +D2 g(cid:22)E x^ +D2 g(cid:22)E (cid:16) x(cid:0)x(cid:0) t x(cid:0)x t t+1 x(cid:0)x+ t t+2 x(cid:0)(cid:16) t t+1 { } +φ(cid:22) D x 2 {x(cid:0) g(cid:22)x^ t(cid:0)1 +D x 2 x g(cid:22)x^ t +D x 2 x+ g(cid:22)E t x^ t+1 +D x 2 (cid:16) g(cid:22)(cid:16) t } +(cid:12) (cid:0)1φ(cid:22) D x 2 +x(cid:0) g(cid:22)x^ t(cid:0)2 +D x 2 +x g(cid:22)x^ t(cid:0)1 +D x 2 +x+ g(cid:22)x^ t +D x 2 +(cid:16) g(cid:22)(cid:16) t(cid:0)1 +(cid:12)E t D x(cid:0)g(cid:22) ′ φ^ t+1 +D x g(cid:22) ′ φ^ t +(cid:12) (cid:0)1D x+ g(cid:22) ′ φ^ t(cid:0)1 =0: (44) ThenotationD2 marksthematrixofsecondderivativesofafunctionwithrespecttoxandx(cid:0). U(cid:22) andg(cid:22) xx(cid:0) areusedasshort-handtoindicatethatafunction(oritspartialderivatives)isevaluatedatthesteady-state values fx(cid:22);φ(cid:22)g. \Hatted" variables refer to the deviation of the original variable from its steady-state value. 46

Online Appendix Regrouping terms delivers [ ] { [ ]} φ(cid:22) (cid:12) (cid:0)1D x 2 +x(cid:0) g(cid:22) x^ t(cid:0)2 + D x 2 x(cid:0) U(cid:22) +φ(cid:22) D x 2 x(cid:0) g(cid:22)+(cid:12) (cid:0)1D x 2 +x g(cid:22) x^ t(cid:0)1 {[ ] [ ]} + D2 U(cid:22) +(cid:12)D2 U(cid:22) +φ(cid:22) D2 g(cid:22)+(cid:12)D2 g(cid:22)+(cid:12) (cid:0)1D2 g(cid:22) x^ xx x(cid:0)x(cid:0) xx x(cid:0)x(cid:0) x+x+ t { [ ]} + (cid:12)D2 U(cid:22) +(cid:12)φ(cid:22) D2 g(cid:22)+(cid:12) (cid:0)1D2 g(cid:22) ′ E x^ [ xx(cid:0) ] xx(cid:0) { x+x t t+1 } +(cid:12)2φ(cid:22) (cid:12) (cid:0)1D2 g(cid:22) ′ E x^ + (cid:12)D2 U(cid:22) +(cid:12)φ(cid:22)D2 g(cid:22) E (cid:16) x+x(cid:0) t t+2 x(cid:0)(cid:16) x(cid:0)(cid:16) t t+1 { } + D2 U(cid:22) +φ(cid:22)D2 g(cid:22) (cid:16) +(cid:12) (cid:0)1φ(cid:22)D2 g(cid:22)(cid:16) x(cid:16) x(cid:16) t x+(cid:16) t(cid:0)1 +(cid:12)E t D x(cid:0)g(cid:22) ′ φ^ t+1 +D x g(cid:22) ′ φ^ t +(cid:12) (cid:0)1D x+ g(cid:22) ′ φ^ t(cid:0)1 =0 (45) which coincides with the (cid:12)rst order conditions of the following LQ problem, where we have turned the maximization problem of the utility function into a minimization problem of the (approximated) true social loss function [ ] ∑1 1 min E (cid:12)t(cid:0)t0 x^ ′ A(L)x^ +x^ ′ B(L)(cid:16) fx^t g1 t=t0 t0 t=t0 2 t t t t+1 s:t: E C(L)x^ +D(L)(cid:16) =0 t t+1 t C(L)x^ =d t0 t0 (cid:16) =(cid:0)(cid:16) +(cid:7)(cid:24) (46) t t(cid:0)1 t where [ ] A = (cid:0)2φ(cid:22) (cid:12) (cid:0)1D2 g(cid:22) 2 x+x(cid:0) ( [ ]) A = (cid:0)2 D2 U(cid:22) +φ(cid:22) D2 g(cid:22)+(cid:12) (cid:0)1D2 g(cid:22) 1 xx(cid:0) xx(cid:0) x+x [ ] [ ] A = (cid:0) D2 U(cid:22) +(cid:12)D2 U(cid:22) (cid:0)φ(cid:22) D2 g(cid:22)+(cid:12)D2 g(cid:22)+(cid:12) (cid:0)1D2 g(cid:22) 0 xx x(cid:0)x(cid:0) xx x(cid:0)x(cid:0) x+x+ A(L) = A +A L+A L2 0 1 2 { } { } B(L) = (cid:0) (cid:12)D2 U(cid:22) +(cid:12)φ(cid:22)D2 g(cid:22) (cid:0) D2 U(cid:22) +φ(cid:22)D2 g(cid:22) L(cid:0)(cid:12) (cid:0)1φ(cid:22)D2 L2 x(cid:0)(cid:16) x(cid:0)(cid:16) x(cid:16) x(cid:16) x+(cid:16) C(L) = D x+ g(cid:22)+D x g(cid:22)L+D x(cid:0)g(cid:22)L2 D(L) = D g(cid:22): (cid:16) where x^ measures the (log-) deviation of variable \x" from its value assumed in the deterministic steady t state. The matrices (A(L);B(L)) capture the second-order approximation of the welfare function, where \L" denotes the lag-operator. The matrices C(L) and D(L) capture the linear approximation of the constraints. The linear constraints C(L)x^ =d implementthe timeless perspective through the appropriate t0 t0 choice of d . The model description is completed by the evolution of the exogenous variables, the last t0 equation in (46). The innovations (cid:24) follow iid standard normal distributions. To a (cid:12)rst-order approxit mation, the output of the toolbox in Bodenstein, Guerrieri, and LaBriola (2014) is equivalent to that of the LQ approach studied in Benigno and Woodford (2012) and using the above de(cid:12)nitions, it is easy to compute the matrices for the LQ problem from the numerical output produced by the toolbox described 47

Online Appendix in Bodenstein, Guerrieri, and LaBriola (2014). ∑1 [ ] The criterion E (cid:12)t(cid:0)t0 1x^′A(L)x^ +x^′B(L)(cid:16) ranks outcomes fx g1 obtained from policies t0 2 t t t t+1 t t=t0 t=t0 that satisfy the initial pre-commitment constraints C(L)x^ = d correctly by their welfare implications. t0 t0 However, if the policies considered do not respect the initial pre-commitment constraints, the criterion needs to be augmented to include a penalty for violations of the initial pre-commitment. As discussed in detailinBenignoandWoodford(2012),thecorrectcriterionthatallowsformeaningfulwelfarecomparisons of arbitrary policies against the optimal commitment policy is given by [ ] ∑1 E (cid:12)t(cid:0)t0 1 x^ ′ A(L)x^ +x^ ′ B(L)(cid:16) +(cid:12) (cid:0)1φ^ (cid:3)′ C(0)x^ : (47) t0 2 t t t t+1 t0 (cid:0)1 t0 t=t0 φ^ (cid:3)′ denotes the values of the Lagrange multipliers associated with the pre-commitment constraints t0 (cid:0)1 under the optimal commitment policy. C(0) is the coefficient matrix going along with the forward-looking variables in the (cid:12)rst order approximation of the equations in g((cid:1)). Finally, x^ contains the values of the t0 endogenousvariablesattimet underthepolicythatisactuallyimplemented. Intuitively,insuringthatthe 0 optimal commitment policy is the best policy among all feasible policies requires a change in preferences. ∑ Rather than viewing preferences as being described by E 0 1 t=t0 (cid:12)t(cid:0)t0U(x t(cid:0)1 ;x t ;(cid:16) t ), preferences need to be viewed as ∑1 ( ) E 0 (cid:12)t(cid:0)t0U(x t(cid:0)1 ;x t ;(cid:16) t )+(cid:12) (cid:0)1φ ′ t0 (cid:0)1 g(x t0 (cid:0)2 ;x t0 (cid:0)1 ;x t0 ;(cid:16) t0 (cid:0)1 )(cid:0)g(cid:22) t0 : (48) t=t0 The optimal policy problem is then given by ∑1 ( ) fx m t g a 1 t= x t0 E 0 t=t0 (cid:12)t(cid:0)t0U(x t(cid:0)1 ;x t ;(cid:16) t )+(cid:12) (cid:0)1φ ′ t0 (cid:0)1 g(x t0 (cid:0)2 ;x t0 (cid:0)1 ;x t0 ;(cid:16) t0 (cid:0)1 )(cid:0)g(cid:22) t0 s:t: E t g(x t(cid:0)1 ;x t ;x t+1 ;(cid:16) t )=0: (49) Approximating this problem following the same steps as above yields the criterion function in (47). By construction, the problem in (49) implies the same (cid:12)rst-order conditions as the optimization program in (41). In (cid:12)nding a second-order approximation of the augmented utility function one only needs to include the second-order expansion of the penalty term, which after eliminating (cid:12)rst-order terms, is simply given by (cid:12) (cid:0)1φ^ ′ D g(cid:22)′x^ =(cid:12) (cid:0)1φ^ ′ C(0)x^ . t0 (cid:0)1 x+ t0 t0 (cid:0)1 t0 A.2 Applying the welfare criterion We focus on unconditional welfare, but similar steps apply for computing conditional welfare. In doing so, we integrate out initial conditions with the help of the invariant unconditional distribution over possible initial conditions | including the pre-commitments. Consideranarbitrarypolicyregime, indexedbyTF, andsupposethatthe(linear)equilibriumdecision rules can be summarized by zTF =PTFzTF +QTF(cid:24) : (50) t t(cid:0)1 t 48

Online Appendix If policy is conducted under commitment (from a timeless perspective), the vector zTF contains the ent dogenous variables x^TF, the exogenous shocks (cid:16) and (cid:16) , and a set of Lagrange multipliers φ^TF. Under t t t(cid:0)1 t discretion, Lagrange multipliers are not part of the state space and will be omitted from the vector zTF. t The same applies if one were to include instrument rules in the analysis. We denote the decision rules under the optimal commitment policy by a star, \(cid:3)", instead of TF. The unconditional variance-covariance matrix Cov satis(cid:12)es zTF;zTF [ ] Cov =PTF Cov PTF′ +QTFQTF′ (51) zTF;zTF zTF;zTF whichcanbecomputedefficientlyusingthedoublingalgorithmsuggestedinAnderson,McGrattan,Hansen, and Sargent (1996). The ((cid:12)rst) auto-covariance term is obtained by recognizing that Cov = zTF;z(cid:0) TF 1 PTFCov . zTF;zTF To compute the unconditional welfare implied by the policy TF we simplify the two terms in equation (47) as follows. The (cid:12)rst term of the welfare criterion can be written in terms of the unconditional covariances and auto-covariances between the endogenous variables, x^TF, and exogenous variables, (cid:16) , t t since { [ ]} 1 tr E 1 x^TF′ A(L)x^TF +x^TF′ B(L)(cid:16) 1(cid:0)(cid:12) 2 t t t t+1 { } ∑2 ∑2 1 1 = tr A(i)Cov + B(i)Cov (52) 1(cid:0)(cid:12) 2 x^T (cid:0) F i ;x^TF (cid:16)(cid:0)i+1 ;x^TF i=0 i=0 where tr(M) denotes the trace of the matrix M. Because the second term in (47) involves the Lagrange multipliers associated with the optimal commitmentpolicy,evaluationofthetermrequiresknowledgeofthevariance-covariancematrixoftheendogenous (cid:3) variablesundertheoptimalcommitmentpolicy,φ^ . Thepre-commitmentsaredrawnfromtheinvariant t0 (cid:0)1 distribution of the endogenous variables under the optimal policy. If the policy TF is conducted under commitment, the second term can be written as [ ] { } E (cid:12) (cid:0)1φ^ (cid:3) t0 ′ (cid:0)1 C(0)x^T t0 F =(cid:12) (cid:0)1tr C(0)S x P (cid:3) Cov z(cid:3);φ^(cid:3) +C(0)S x Q (cid:3) Cov (cid:24);φ^(cid:3) (cid:0)1 (53) from the unconditional perspective. The matrix S selects the elements in z that coincide with those in x t the vector x^ . If the policymaker respects the pre-commitments consistent with the optimal commitment t policy, it must be that C(0)x^TF =C(0)x^ . Thus, the term (53) does not depend on the decision rules of t0 t0 the policy regime under consideration as long as the policymaker respects pre-commitments. When pre-commitments are not honoured, in particular under discretion or an instrument rule, the secondtermdoesdependonthedecisionrulesofthepolicyimplementedbythecentralbankandtherefore the correction term satis(cid:12)es [ ] { } E (cid:12) (cid:0)1φ^ (cid:3) t0 ′ (cid:0)1 C(0)x^T t0 F =(cid:12) (cid:0)1tr C(0)S x PTFCov z(cid:3);φ^(cid:3) +C(0)S x QTFCov (cid:24);φ^(cid:3) (cid:0)1 (54) with S de(cid:12)ned appropriately to select the elements in z that coincide with those in the vector x^ under x t t 49

Online Appendix discretion. A.3 Decision rules under commitment and discretion For each targeting framework, we consider the case of the central bank optimizing its assigned objective under full commitment from a timeless perspective and the case of optimization under discretion. We assume that the central bank is committed to an explicit long-run in(cid:13)ation target. Thus, our analysis abstracts from the in(cid:13)ationary bias under discretion; our work focuses purely on the stabilization bias. Each targeting framework is represented by a quadratic loss function: 1. in(cid:13)ation targeting (IT) LIT =(cid:25)2 +(cid:21)IT (xgap)2 (55) t p;t x t 2. price level targeting (PLT) LPLT =p^2+(cid:21)PLT (xgap)2 (56) t t x t 3. speed limit policy (SLP) ( ( )) LSLP =(cid:25)2 +(cid:21)SLP (xgap)(cid:0) xgap 2 (57) t p;t x t t(cid:0)1 4. nominal income targeting (NIT) LN t IT =(cid:25)2 p;t +(cid:21)N x IT ((cid:25) p;t +y^ t (cid:0)y^ t(cid:0)1 )2 (58) 5. nominal income targeting II (NIT-II) LN t IT-II =(xg t ap)2+(cid:21)N x IT-II((cid:25) p;t +y^ t (cid:0)y^ t(cid:0)1 )2 (59) where (cid:25) denotes deviations of the in(cid:13)ation measure from its value along the balanced growth path, p^ p;t t is the log-deviation of the price level from its value along the balanced growth path, and xgap measures t the output gap. We follow Smets and Wouters (2007) and de(cid:12)ne the output gap as the difference between actual output (in deviations from the balanced growth path), y^, and the output level that would prevail t absent nominal rigidities and markup shocks. A.3.1 Targeting frameworks under commitment For a given parameterization of a targeting framework, a central bank, that formulates policy under commitmentandrespectsthesamepre-commitmentsastheoptimalcommitmentpolicy,solvestheoptimization problem ∑1 min E (cid:12)t(cid:0)t0 1 x^TF′ ATF(L)x^TF fx^TFg1 t0 2 t t t t=t0 t=t0 s:t: E C(L)x^TF +D(L)(cid:16) =0 t t+1 t C(L)x^TF =d t0 t0 (cid:16) =(cid:0)(cid:16) +(cid:7)(cid:24) : (60) t t(cid:0)1 t 50

Online Appendix ThematrixATF(L)isparameterizedtore(cid:13)ectthelossfunctionthatcharacterizesthetargetingframework under consideration with TF = fIT;PLT;SLP;NIT;NIT-IIg. The entries into ATF(L) are zero except for those diagonal elements that correspond to the positions of the targeting variables in the vector x^ t for the targeting regime TF. Thus, the problem resembles the one of obtaining the optimal commitment policy in (46) with (A(L);B(L)) being replace by ATF(L). The (cid:12)rst-order conditions associated with this linear quadratic program can be solved using standard algorithms to obtain the decision rules of the endogenous variables and the Lagrange multipliers. These decision rules are then used to compute the relevant variance-covariance matrices to evaluate the welfare criterion (47). A.3.2 Targeting frameworks under discretion To (cid:12)nd the (Markov equilibrium) decision rule of a central bank acting under discretion we follow the methodology suggested in Dennis (2007). Today’s central bank is viewed as the Stackelberg leader; households and (cid:12)rms as well as future policymakers are the Stackelberg followers. De(cid:12)ne z~ t ( ) x^TF;ni z~ t = t (cid:16) (61) t to be the vector that contains the endogenous variables, x^TF, except for the vector of policy instrut ments, i = x^TF;i, and the exogenous shocks. We start by writing the linearized equilibrium conditions t t E C(L)x^TF +D(L)(cid:16) =0 as t t+1 t M 0 z~ t =M 1 z~ t(cid:0)1 +M 2 E t z~ t+1 +M 3 i t +M 4 E t i t+1 +M 5 (cid:24) t (62) with [ ] M = (cid:0) Cni(1) 0 (63) 0 [ ] M = Cni(2) D(0)(cid:0)+D(1) (64) 1 [ ] M = Cni(0) 0 (65) 2 M = c(1) (66) 3 M = c(0) (67) 4 M = D(0)(cid:7): (68) 5 ThematrixCni(1)isderivedfromC(1)byeliminatingfromC(1)thecolumnc(1)whichisassociatedwith the policy instrument and similarly for C(0) and C(2). We assume c(2) to be a vector of zeros. Similarly, we write the objective function of the central bank|originally characterized by ATF(L)|to conformwiththeinclusionoftheexogenousvariablesintothevectorz~ andtheseparatingoutofthepolicy t instrument ∑1 [ ] 1 E (cid:12)t(cid:0)t0 z~ ′ WTFz~ +i ′ KTFi (69) t0 2 t t t t t=t0 51

Online Appendix where [ ] WTF = ATF;ni 0 (70) 0 0 KTF = aTF: (71) We proceed under the conjecture that the solution will be of the form z~ t =H 1 z~ t(cid:0)1 +H 2 (cid:24) t (72) i t =F 1 z~ t(cid:0)1 +F 2 (cid:24) t : (73) Substituting this conjecture into equation (62) we obtain [M 0 (cid:0)M 2 H 1 (cid:0)M 4 F 1 ]z~ t =M 1 z~ t(cid:0)1 +M 3 i t +M 5 (cid:24) t : (74) Similarly, the objective function (69) can be written as ∑1 [ ] E (cid:12)t(cid:0)t0 1 z~ ′ WTFz~ +i ′ KTFi t0 2 t t t t t=t0 ( ) (cid:12) = z~ ′ NTFz~ +i ′ KTFi + tr H ′ NTFH +F ′ KTFF (75) t t t t 1(cid:0)(cid:12) 2 2 2 2 since ( ) ∑1 [ ] ∑1 ( ) ( ) E (cid:12)t(cid:0)t0 z~ ′ WTFz~ = z~ ′ (cid:12)t(cid:0)t0 H ′t(cid:0)t0 WTF Ht(cid:0)t0 z~ t0 t t t 1 1 t t=t0 t=t0 ∑1 ∑1 ( ( ) ( ) ) +(cid:12) (cid:12)(t(cid:0)t0)+(t~(cid:0)t0)tr H ′ H ′t(cid:0)t0 WTF Ht(cid:0)t0 H 2 1 1 2 t=t0t~=t0 (cid:12) ′ ′ = z~Sz~ + tr(H SH ) (76) t t 1(cid:0)(cid:12) 2 2 and ( ) ∑1 [ ] ∑1 ( ) ( ) E (cid:12)t(cid:0)t0 i ′ KTFi = i ′ KTFi +(cid:12)z~ ′ (cid:12)t(cid:0)t0 H ′t(cid:0)t0 F ′ KTFF Ht(cid:0)t0 z~ t0 t t t t t 1 1 1 1 t t=t0 t=t0 ( ) (cid:12) + tr F ′ KTFF 1(cid:0)(cid:12) 2 2 ∑1 ∑1 ( ( ) ( ) ) +(cid:12) (cid:12)(t(cid:0)t0)+(t~(cid:0)t0)tr H ′ H ′t(cid:0)t0 F ′ KTFF Ht(cid:0)t0 H 2 1 1 1 1 2 t=t0t~=t0 (cid:12) ( ) (cid:12)2 = i ′ KTFi +(cid:12)z~ ′ Rz~ + tr F ′ KTFF + tr(H ′ RH ): t t t t 1(cid:0)(cid:12) 2 2 1(cid:0)(cid:12) 2 2 (77) The matrices S, R and NTF are de(cid:12)ned implicitly as S = WTF +(cid:12)H ′ SH (78) 1 1 R = F ′ KTFF1+(cid:12)H ′ RH (79) 1 1 1 NTF = S+(cid:12)R: (80) 52

Online Appendix S and R are (cid:12)xed points provided that the spectral radius of H is less than one. In our application, 1 KTF =0 and the second term of the objective function drops out. Under discretion, the policymaker optimizes the objective function (75) subject to the conditions in (74). Taking (cid:12)rst-order conditions and applying the method of undetermined coefficients yields M(cid:22) (cid:17) M (cid:0)M H (cid:0)M F (81) 0 2 1 4 1 NTF (cid:17) WTF +(cid:12)F ′ KTFF +(cid:12)H ′ NTFH (82) 1 1 1 1 F = (cid:0)(KTF +M ′ M(cid:22)′(cid:0)1NTFM(cid:22)(cid:0)1M ) (cid:0)1M ′ M(cid:22)′(cid:0)1NTFM(cid:22)(cid:0)1M (83) 1 3 3 3 1 F = (cid:0)(KTF +M ′ M(cid:22)′(cid:0)1NTFM(cid:22)(cid:0)1M ) (cid:0)1M ′ M(cid:22)′(cid:0)1NTFM(cid:22)(cid:0)1M (84) 2 3 3 3 5 H = M(cid:22)(cid:0)1(M +M F ) (85) 1 1 3 1 H = M(cid:22)(cid:0)1(M +M F ): (86) 2 5 3 2 Equations (72) and (73) can be combined to deliver the law of motion to the full vector z under t discretionary policies as in equation (50). In order to evaluate the (cid:12)ve targeting frameworks under discretionary policymaking, we do not need to characterize the optimal policy under discretion when the central bank’s objective is derived from the utility function of the representative household. Each targeting framework can be evaluated by applying thecriterionstatedin(47)toassessthewelfareimplicationsofthepolicypathsunderdiscretion|thetrue social loss function. The reason for condition (47) to suffice for welfare evaluations lies in the fact that absent shocks, the central bank chooses the same policy path under each objective regardless of policy being conducted under commitment or discretion. In particular, an in(cid:13)ationary bias cannot arise even if the steady state is not efficient.26 A.4 Welfare comparison WecomputewelfareunderthetargetingregimeWTF andtheoptimalcommitmentpolicyW(cid:3) andconvert the difference into consumption units. More concretely, the difference is expressed in terms of the consumption equivalent variation (CEV). The CEV is de(cid:12)ned as the amount of (steady state) consumption that the representative household|with preferences over consumption and leisure U(C;N)|would need to give up to be indifferent between the optimal commitment policy and the targeting framework being implemented. Algebraically, the CEV is de(cid:12)ned as WTF (cid:0)W (cid:3) = U((1+CEV)C(cid:22);N(cid:22))(cid:0)U(C(cid:22);N(cid:22)) (cid:12) (cid:12) = @U(cid:12) (cid:12) [(1+CEV)C(cid:22)(cid:0)C(cid:22)] @C (cid:12)C=C(cid:22) (cid:12) = @U(cid:12) (cid:12) C(cid:22)CEV @C C=C(cid:22) 26AspointedoutinWoodford(2003a),Chapter7,page470,footnote4,characterizingtheoptimalpolicyunderdiscretionisacomplicated task, in particular when the steady state is distorted. Assigning to a central bank acting under discretion the objective in (47) does not yieldtheoptimalpolicyunderdiscretionasthederivationsunderlyingexpression(47)assume thatpolicyisconductedundercommitment fromatimelessperspective. 53

Online Appendix or solved for the CEV WTF (cid:0)W(cid:3) (cid:12) CEV = (cid:12) : (87) @U(cid:12) (cid:12) C(cid:22) @C C=C(cid:22) When households experience habit persistence in consumption|here of the form U(C t ;C t(cid:0)1 ;L t ) = U(C t (cid:0)hC t(cid:0)1 ;N t )|we follow the approach in Otrok (2001). In this case, we obtain WTF (cid:0)W (cid:3) = U((1+CEV)C(cid:22);(1+CEV)C(cid:22);N(cid:22))(cid:0)U(C(cid:22);C(cid:22);N(cid:22)) (cid:12) (cid:12) = @U(cid:12) (cid:12) [((1+CEV)C(cid:22)(cid:0)(1+CEV)hC(cid:22))(cid:0)(C(cid:22)(cid:0)hC(cid:22))] @C (cid:12)C=C(cid:22) (cid:12) = @U(cid:12) (cid:12) (1(cid:0)h)C(cid:22)CEV: @C C=C(cid:22) Under additive separable preferences, as conventionally assumed in the textbook NKM, it is WTF (cid:0)W(cid:3) CEV = : ((1(cid:0)h)C(cid:22))1(cid:0)(cid:27)C Under the preferences assumed in Smets and Wouters (2007) the CEV is given by WTF (cid:0)W(cid:3) CEV = ( ): (cid:27) (cid:0)1 ((1(cid:0)h)C(cid:22))1(cid:0)(cid:27)Cexp C N(cid:22)1+(cid:27)L 1+(cid:27) L 54

Online Appendix B Baseline New Keynesian model B.1 Model description Forcompleteness,welayouttheassumptionsofthetextbookNKmodelanditsvariationsdiscussedinthe text. We then derive the linear-quadratic framework for versions of the NK model with price indexation, external habits, inefficient steady state, and sticky wages. B.1.1 Household Household j chooses consumption C (j), labor supply N (j), bond holdings to maximize expected dist t counted lifetime utility taking prices, wages, taxes, and transfers as given. The household’s preferences over consumption and leisure are given by {( ) } E ∑1 (cid:12)t(cid:0)t0 C t (j)(cid:0)hC t A (cid:0)1 1(cid:0)(cid:27)C (cid:0) N t (j)(1+(cid:27)L) : (88) t0 1(cid:0)(cid:27) 1+(cid:27) C L t=t0 Consumption habits are external; CA refers to the aggregate level of consumption in the previous period t(cid:0)1 and the degree of habit persistence is governed by the parameter h. The inverse of the intertemporal elasticityofsubstitutionofconsumptionisdenotedby(cid:27) ,andtheparameter(cid:27) istheinverseoftheFrisch C L elasticity of labor supply. We assume that (cid:12)nancial markets are complete due to a set of Arrow securities. As a result consumption is equalized across households in equilibrium. In addition, each household can invest in a simple bond without state-contingent payoffs. The budget constraint of household j satis(cid:12)es B (j) P t C t (j)+ R t =W t N t (j)+B t(cid:0)1 (j)+Profits t +Transfers t : (89) t ThehouseholdearnsincomebysupplyinglaborservicesN (j)forthenominalwageW ,receivespayments t t fromholdingbondsB (j),receivesanaliquotshareofpro(cid:12)tsProfits andgovernmenttransferTransfers . t t t This income is used to purchase the consumption good and bonds. The notation abstracts from the household’s transactions in Arrow securities. B.1.2 Labor market We consider the case with and without (cid:13)exible wages. If wages are (cid:13)exible, workers receive the same nominal wage W in period t. The household chooses the labor supply optimally. t In modeling nominal sticky wages we follow in general Erceg, Henderson, and Levin (2000), but the details of the implementation are as in Gali (2008). Households supply their homogenous labor to labor unions. The labor union differentiates the labor services, and resells them to a labor bundler. These aggregated labor services are then hired out to (cid:12)rms. The labor bundlers aggregate the labor services provided by the labor unions according to [∫ ] 1 1+(cid:18)w 1 L t = L t (j)1+(cid:18)wdj : (90) 0 55

Online Appendix Labor bundlers buy labor services L (j) from labor union j, combine the differentiated services into L , t t andthenreselltheaggregatelaborservicetointermediategoodsproducers. TheLaborbundlersmaximize pro(cid:12)ts under perfect competition. The (cid:12)rst order conditions associated with this maximization problem canbecombinedtoobtainthelabordemandfunctionsforthelaborservicesL (j)offeredbylaborunionj t [ ] (cid:0)1+(cid:18)w L (j)= W t (j) (cid:18)w L (91) t W t t and the aggregate wage index [∫ ] 1 (cid:0)(cid:18)w (cid:0) 1 W t = W t (j) (cid:18)wdj : (92) 0 Labor unions take the household’s marginal rate of substitution between consumption and leisure as the costs of labor services. The labor unions act under monopolistic competition and wages are set using staggered contracts as in Calvo (1983). Each period, the union faces a constant probability 1(cid:0)(cid:24) to w re-optimize its wage W~ (j). This probability is independent across unions and time. Unions that cannot t adjust their wage optimally in the current period will increase their wage by the weighted average of (gross) in(cid:13)ation (cid:5) t = P P t(cid:0) t 1 in the previous period and steady state in(cid:13)ation (cid:5)(cid:22) with weights (cid:19) w and 1(cid:0)(cid:19) w , respectively. Let W~ (j) be the optimal wage set by union j in period t. The union charges t ( ) W (j)=W~ (j) (cid:5)(cid:19)w(cid:5)(cid:22)(1(cid:0)(cid:19)w) (93) t+1 t t inperiodt+1, ifitisnotallowedtoadjustthewageoptimallyinperiodt+1. Whentheunioncanchoose its wage optimally, the union solves the following optimization problem ∑1 (cid:12)s(cid:21) [ ] maxE ((cid:24) )s t+s (1+(cid:28)(cid:22) )W (j)(cid:0)Wh L (j) W~ t(j) t s=0 w (cid:21) t w t+s t+s t+s ( ) (cid:0)1+(cid:18)w s:t: L t+s (j) = W t+s (j) (cid:18)w L W t+s t+s W (j)=W~ (j)XW 8t+s t t;s >>< 1 for s=0 X t W ;s = >>: ∏s ( (cid:5)(cid:19) t w +l(cid:0)1 (cid:5)(cid:22)1(cid:0)(cid:19)w ) for s=1;:::;1: (94) l=1 B.1.3 Intermediate goods producer Each intermediate goods producer employs labor to produce a variety. The cost minimization problem of the producer is minW L (i) t t Lt(i) s:t: Y (i)=(cid:24) L (i): (95) t A;t t (cid:24) denotes a shock to total factor productivity which follows an exogenous stochastic process A;t ( ) ( ) log (cid:24) =(cid:26) log (cid:24) +" (96) A;t A A;t(cid:0)1 A;t 56

Online Appendix " is white noise following N(0;(cid:27)2). A;t A Prices are set using staggered contracts as in Calvo (1983). Each period, a (cid:12)rm faces a constant probability 1(cid:0)(cid:24) to re-optimize its price P~(i). This probability is independent across (cid:12)rms and time. p t A (cid:12)rm that does not re-optimize its price in period t, the price increases by the weighted average of (gross) in(cid:13)ation (cid:5) t = P P t(cid:0) t 1 in the previous period and steady state in(cid:13)ation (cid:5)(cid:22) with weights (cid:19) p and 1(cid:0)(cid:19) p , respectively. Finally, (cid:12)rms engage in monopolistic competition. Thus, the price setting problem of intermediate goods producer i can be stated as ∑1 ( ) (cid:21) maxE (cid:12)(cid:24) s t+s [(1+(cid:28)(cid:22) )P (i)(cid:0)MC ]Y (i) P~ t(i) t s=0 p (cid:21) t p t+s t+s t+s ( ) (cid:0)1+(cid:18)p s:t: Y (i)= P t+s (i) (cid:18)p Y t+s P t+s t+s P (i)=P~(i)XP 8t+s t t;s >>< 1 for s=0 X t P ;s = >>: ∏s ( (cid:5)(cid:19) t p +l(cid:0)1 (cid:5)(cid:22)1(cid:0)(cid:19)p ) for s=1;:::;1: (97) l=1 In the following, we will assume the presence of a price markup shock, often also referred to as markup shock. Intheliterature,severalwayshavebeensuggestedtomotivatethisshock: (i)ashocktothesubsidy (cid:28)(cid:22) , (ii) a shock to the elasticity of substitution (cid:18) , or (iii) a shock in the (cid:12)rst order condition associated p p with the maximization problem of the intermediate goods producers. While all three approaches lead to the same set of equations when the model is approximated to the (cid:12)rst order, this is not true, when the modelisapproximatedtothesecondorder. Weofferashortdiscussiononthistopiclaterinthisappendix. B.1.4 Final good bundlers Intermediate goods are combined into a composite (cid:12)nal good by a continuum of representative bundlers acting under perfect competition. The standard Dixit-Stiglitz aggregator implies [∫ ] 1 1+(cid:18)p 1 Y t = Y t (i)1+(cid:18)pdi (98) 0 where 1+(cid:18)p denotes the elasticity of substitution between the intermediate goods. (cid:18)p Eachbundlermaximizespro(cid:12)tsbychoosingtheamountofeachintermadiategoodtoobtainthe(cid:12)nalgood ∫ 1 max P Y (cid:0) P (i)Y (i)di t t t t Yt(i);Yt 0 [∫ ] 1 1+(cid:18)p 1 s:t: Y t = Y t (i)1+(cid:18)pdi : (99) 0 The(cid:12)rstorderconditionstothisproblemprovidethedemandfunctionforeachintermediategoodsand an expression for the aggregate price level ( ) (cid:0)1+(cid:18)p Y (i)= P t (i) (cid:18)p Y (100) t P t t 57

Online Appendix and [∫ ] 1 (cid:0)(cid:18)p (cid:0) 1 P t = P t (i) (cid:18)pdi ; (101) 0 respectively. B.1.5 Resource constraint Market clearing in the market for the (cid:12)nal good requires that Y =C : (102) t t When wages are (cid:13)exible, the supply of the (cid:12)nal good is given by ΩyY =(cid:24) N (103) t t A;t t where Ωy is the measure of price dispersion t [ ] ∫ 1 Y (i)di Ωy t = [ ∫ 1 0 t 1 ] 1+(cid:18)p : (104) 0 Y t (i)1+(cid:18)pdi We have made use of the fact that under (cid:13)exible wages the labor market clears when N =L : (105) t t Under sticky wages an additional term that captures wage dispersion arises in equation (103). Note that ∫ [ ∫ ( ) ] N = 1 L (j)dj = 1 W t (j) (cid:0)1+ (cid:18)w (cid:18)w dj L =ΩlL (106) t t W t t t 0 0 t where j is the index of a labor union. Since the labor supplied by the households is homogeneous, N (i)= t N . Similarly, aggregate output and manufactured varieties satisfy the relationship t 2 3 ∫ ∫ ( ) 1 Y (i)di= 4 1 P t (i) (cid:0)1+ (cid:18)p (cid:18)p di 5 Y =ΩyY : (107) t P t t t 0 0 t Market clearing implies ∫ 1 Y (i)di=(cid:24) L (108) t A;t t 0 or making use of the above relationships ΩlΩyY =(cid:24) N : (109) t t t A;t t B.2 Linear-quadratic frameworks We derive the linear-quadratic framework consistent with the NK model laid out in the preceding section. We begin with a version of the model that features (cid:13)exible wages and an efficient steady state. Then we 58

Online Appendix discussthederivationsofthelinear-quadraticframeworkforthecaseofadistortedsteadystateand(cid:13)exible wages. Finally, we move on to the case of sticky wages. B.2.1 NKM with external consumption habits Our model with external consumption habits and in(cid:13)ation persistence resembles Leith, Moldovan, and Rossi (2012).27 Following the steps outlined in Woodford (2003a) and Gali (2008), the second-order approximation of the household preferences around the efficient steady state can be shown to be of the form (cid:27) 1+(cid:18) L t =(cid:27) L (x t )2+ (cid:14) C (x t (cid:0)hx t(cid:0)1 )2+ (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2: (110) p p To arrive at this results, we (cid:12)rst approximate each of the utility contributions from consumption and labor in equation (88). In the private sector equilibrium, the utility from consumption can be written as (C t (cid:0)hC t(cid:0)1 )1(cid:0)(cid:27)C 1(cid:0)(cid:27) {( C ) ( ) } 1 1 (cid:27) = U c C(cid:22) c^ t + 2 c^2 t (cid:0)h c^ t(cid:0)1 + 2 c^2 t(cid:0)1 (cid:0) 2(1(cid:0) C h) (c^ t (cid:0)hc^ t(cid:0)1 )2 +t:i:p:+O(2) (111) ( ) with U = (1(cid:0)h)C(cid:22) (cid:0)(cid:27)C. Summing over all periods leads to the expression c E ∑1 (cid:12)t(cid:0)t0 (C t (cid:0)hC t(cid:0)1 )1(cid:0)(cid:27)C t0 1(cid:0)(cid:27) C t=t0 { ( ) } ∑1 1 (cid:27) =U c C(cid:22)E t0 (cid:12)t(cid:0)t0 (1(cid:0)h(cid:12)) c^ t + 2 c^2 t (cid:0) 2(1(cid:0) C h) (c^ t (cid:0)hc^ t(cid:0)1 )2 +t:i:p:+O(2): t=t0 (112) Given the linearity of production in labor, the disutility from labor can be written as { } N1+(cid:27)L 1+(cid:27) t =U N(cid:22) n^ + Ln^2 +t:i:p:+O(2) (113) 1+(cid:27) n t 2 t L where U n =N(cid:22)(cid:27)L. Applying the following result from Woodford (2003a) 1+(cid:18) n^ = y^ (cid:0)^(cid:24) + pvar (p (i)) (114) t t A;t 2(cid:18) i t p in equation (113), the disutility from labor can be expressed as E ∑1 (cid:12)t(cid:0)t0 N t 1+(cid:27)L t0 1+(cid:27) L t=t0 { } ∑1 1+(cid:27) 1+(cid:27) 1+(cid:18) = N(cid:22)1+(cid:27)LE (cid:12)t(cid:0)t0 y^ + Ly^2(cid:0)2 Ly^^(cid:24) + pvar (p (i)) t0 t 2 t 2 t A;t 2(cid:18) i t p t=t0 +t:i:p:+O(2): (115) Before re-combining the expressions for the utility from consumption and the disutility from labor, we 27Leith,Moldovan,andRossi(2012)abstractfromin(cid:13)ationpersistenceandfocusontheconceptuallymorechallengingderivationsunder variousformulationsofconsumptionhabits(internalversusexternal,deepversussuper(cid:12)cialhabits). 59

Online Appendix turn to three relationships that allow us to simplify the approximation. The market clearing condition implies 1 1 c^ = y^ + y^2(cid:0) c^2+O(2) (116) t t 2 t 2 t and the price dispersion term can be expressed as ∑1 ∑1 (cid:24) (cid:12)t(cid:0)t0var i (p t (i)) = (1(cid:0)(cid:12)(cid:24) ) p (1(cid:0)(cid:24) ) (cid:12)t(cid:0)t0((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2: (117) t=t0 p p t=t0 The third relationship is (1(cid:0)h(cid:12))U C(cid:22) =U N(cid:22) (118) c n whichisderivedasfollows. Thedeterministicsteadystateofthemarketeconomyisnotnecessarilyefficient underexternalhabitsasagentsdonotinternalizetheimpactoftoday’sconsumptionchoiceontomorrow’s marginal utility. To render the steady state efficient, we introduce a tax on consumption, which satis(cid:12)es 1+(cid:28)(cid:22) = 1 . With this tax in place, the (cid:12)rst order conditions for consumption and labor imply that in c 1(cid:0)h(cid:12) the steady state ( ) (1(cid:0)h)C(cid:22) (cid:0)(cid:27)C N(cid:22) =(1+(cid:28)(cid:22) ) (119) N(cid:22)(cid:27)L c C(cid:22) or (1(cid:0)h(cid:12))U C(cid:22) =U N(cid:22): (120) c n Combining the utility from consumption and the disutility from labor using these three relationships, we obtain the second-order approximation to household preferences as {( ) } ∑1 1 (cid:27) U n N(cid:22)E t0 (cid:12)t(cid:0)t0 y^ t + 2 y^ t 2 (cid:0) 2(1(cid:0)h)( C 1(cid:0)h(cid:12)) (c^ t (cid:0)hc^ t(cid:0)1 )2 t=t0 {( ) } ∑1 1 (cid:27) 1+(cid:27) 1+(cid:18) (cid:0)U N(cid:22)E (cid:12)t(cid:0)t0 y^ + y^2 + Ly^2(cid:0)2 Ly^^(cid:24) + pvar (p (i)) n t0 t 2 t 2 t 2 t A;t 2(cid:18) i t p t=t0 +t:i:p:+O(2) { } ∑1 1 (cid:27) 1+(cid:18) = (cid:0) 2 U n N(cid:22)E t0 (cid:12)t(cid:0)t0 (cid:27) L y^ t 2+ (cid:14) C (y^ t (cid:0)hy^ t(cid:0)1 )2(cid:0)2(1+(cid:27) L )y^ t ^(cid:24) A;t + (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 p p t=t0 +t:i:p:+O(2) (121) with (1(cid:0)(cid:12)(cid:24) )(1(cid:0)(cid:24) ) (cid:20) = p p p (cid:24) p (cid:14) = (1(cid:0)h)(1(cid:0)h(cid:12)): Our baseline model with sticky prices and external consumption habits can be summarized in linearquadratic form by the (hybrid) New Keynesian Phillips curve ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )=(cid:20) p mcc t +u p;t +(cid:12)E t ((cid:25) p;t+1 (cid:0)(cid:19) p (cid:25) p;t ) (122) 60

Online Appendix where u denotes a stationary markup shock.28 Marginal costs follow t (cid:27) mcc t = (cid:27) L y^ t + 1(cid:0) C h (y^ t (cid:0)hy^ t(cid:0)1 )(cid:0)(1+(cid:27) L )^(cid:24) A;t (123) and the aggregate demand curve satis(cid:12)es 1(cid:0)h (y^ t (cid:0)hy^ t(cid:0)1 )=E t (y^ t+1 (cid:0)hy^ t )(cid:0) (cid:27) (i t (cid:0)E t (cid:25) p;t+1 ) (124) C where i denotes the nominal interest rate. The social loss function satis(cid:12)es t ( ) ∑1 1 E (cid:12)t(cid:0)t0L (125) t0 2 t t=t0 with (cid:27) 1+(cid:18) L t = (cid:27) L y^ t 2+ (cid:14) C (y^ t (cid:0)hy^ t(cid:0)1 )2(cid:0)2(1+(cid:27) L )y^ t ^(cid:24) A;t + (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2: (126) p p To express the model consisting of equations (122) to (126) in terms of \gaps," we adopt the notion of the welfare-relevant output gap as in Woodford (2003a). The welfare-relevant output gap is computed by de(cid:12)ning potential output as the output level that would prevail absent nominal rigidities and markup shocks, but the internalization of consumption habits . Although the consumption tax 1+(cid:28)(cid:22) removes c all static distortions arising from external habits, the dynamics remain distorted. To obtain the (linear) equilibrium dynamics of this efficient output level, we solve the model under internal habits, as a social planner would do, to deliver [ d ](cid:3) mcc (cid:3) = W (cid:0)(y^ (cid:3)(cid:0)n^ (cid:3) ) t P t t t ( ) ( ) (cid:27) (cid:27) = (cid:27) n^ (cid:3) + C c^ (cid:3)(cid:0)hc^ (cid:3) (cid:0)h(cid:12) CE c^ (cid:3) (cid:0)hc^ (cid:3) (cid:0)(y^ (cid:3)(cid:0)n^ (cid:3) ) L ( t (cid:14) ) t t(cid:0) ( 1 (cid:14) ) t t+1 ( t t ) t (cid:27) (cid:27) = (cid:27) y^ (cid:3)(cid:0)^(cid:24) + C y^ (cid:3)(cid:0)hy^ (cid:3) (cid:0)h(cid:12) CE y^ (cid:3) (cid:0)hy^ (cid:3) (cid:0)^(cid:24) L t A;t (cid:14) t t(cid:0)1 (cid:14) t t+1 t A;t ( ) ( ) (cid:27) (cid:27) = (cid:27) y^ (cid:3) + C y^ (cid:3)(cid:0)hy^ (cid:3) (cid:0)h(cid:12) CE y^ (cid:3) (cid:0)hy^ (cid:3) (cid:0)(1+(cid:27) )^(cid:24) : (127) L t (cid:14) t t(cid:0)1 (cid:14) t t+1 t L A;t In the efficient economy ((cid:13)exible prices, no markup shocks) real marginal costs are constant and therefore efficient output evolves according to ( ) ( ) (cid:27) (cid:27) (cid:27) y^ (cid:3) + C y^ (cid:3)(cid:0)hy^ (cid:3) (cid:0)h(cid:12) CE y^ (cid:3) (cid:0)hy^ (cid:3) =(1+(cid:27) )^(cid:24) : (128) L t (cid:14) t t(cid:0)1 (cid:14) t t+1 t L A;t Equation (128) can be used to rewrite the model in terms of the welfare-relevant output gap. Applied to equation (121), we obtain { } ∑1 1 1+(cid:18) (cid:0) 2 U n N(cid:22)E t0 (cid:12)t(cid:0)t0 (cid:27) L y^ t 2(cid:0)2(cid:27) L y^ t y^ t (cid:3) + (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 p p t=t0 28Theliteratureoffersseveralwaysofmodellingmarkupshockssuchasvariationsindesiredmarkupsinthepricesettingrulesof(cid:12)rms, exogenousvariationsinwagemarkups,shockstothepricesubsidypaidtoproducers,orevenshockstotheelasticityofsubstitutionbetween varieties. To a (cid:12)rst order approximation all these models imply the same dynamic responses of the economy to the markup shock. As discussed below, the second-order approximation of the household preferences, however, is not identical across approaches if the steady stateisinefficient. 61

Online Appendix ∑1 { [ ( ) ( )]} 1 (cid:27) (cid:0) 2 U n N(cid:22)E t0 (cid:12)t(cid:0)t0 (cid:14) C (y^ t (cid:0)hy^ t(cid:0)1 )2(cid:0)2 y^ t y^ t (cid:3)(cid:0)hy^ t (cid:3) (cid:0)1 (cid:0)(cid:12)hy^ t y^ t (cid:3) +1 (cid:0)hy^ t (cid:3) t=t0 +t:i:p:+O(2) { } ∑1 1 1+(cid:18) = (cid:0) 2 U n N(cid:22)E t0 (cid:12)t(cid:0)t0 (cid:27) L y^ t 2(cid:0)2(cid:27) L y^ t y^ t (cid:3) + (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 p p t=t0 ∑1 { ( )} 1 (cid:27) (cid:0) 2 U n N(cid:22)E t0 (cid:12)t(cid:0)t0 (cid:14) C (y^ t (cid:0)hy^ t(cid:0)1 )2(cid:0)2(y^ t (cid:0)hy^ t(cid:0)1 ) y^ t (cid:3)(cid:0)hy^ t (cid:3) (cid:0)1 ] t=t0 +t:i:p:+O(2) (129) using ∑1 {( ) ( )} ∑1 ( ) E t0 (cid:12)t(cid:0)t0y^ t y^ t (cid:3)(cid:0)hy^ t (cid:3) (cid:0)1 (cid:0)h(cid:12) y^ t (cid:3) +1 +hy^ t (cid:3) =E t0 (cid:12)t(cid:0)t0(y^ t (cid:0)hy^ t(cid:0)1 ) y^ t (cid:3)(cid:0)hy^ t (cid:3) (cid:0)1 : (130) t=t0 t=t0 Let the welfare-relevant output gap be denoted by x =y^ (cid:0)y^(cid:3). Equations (122) to (126) can be stated t t t as ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )=(cid:20) p mcc t +u p;t +(cid:12)E t ((cid:25) p;t+1 (cid:0)(cid:19) p (cid:25) p;t ) (131) with marginal costs following (cid:27) h(cid:12) mcc t = (cid:27) L x t + 1(cid:0) C h (x t (cid:0)hx t(cid:0)1 )+ 1(cid:0)h(cid:12) g m (cid:3) u;t (132) and the aggregate demand curve 1(cid:0)h ( ) (x t (cid:0)hx t(cid:0)1 )=E t (x t+1 (cid:0)hx t )(cid:0) (cid:27) i t (cid:0)E t (cid:25) p;t+1 (cid:0)g m (cid:3) u;t : (133) C g(cid:3) is de(cid:12)ned as mu;t [ ( ) ( )] (cid:27) g (cid:3) = C E y^ (cid:3) (cid:0)hy^ (cid:3) (cid:0) y^ (cid:3)(cid:0)hy^ (cid:3) : (134) mu;t 1(cid:0)h t t+1 t t t(cid:0)1 The social loss function is now written as ( ) ∑1 1 E (cid:12)t(cid:0)t0L (135) t0 2 t t=t0 with (cid:27) 1+(cid:18) L t =(cid:27) L (x t )2+ (cid:14) C (x t (cid:0)hx t(cid:0)1 )2+ (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2: (136) p p The equilibrium efficient output follows y^ t (cid:3) =(cid:0) y^(cid:3)y^ t (cid:3) (cid:0)1 +(cid:0) ^(cid:24)A ^(cid:24) A;t (137) as derived from equation (128) with (cid:0) y^(cid:3) being the solution to ( ) (cid:27) (cid:27) (cid:27) (cid:0)h(cid:12) C(cid:0)2 + (cid:27) + C(1+h2(cid:12)) (cid:0)2 (cid:0)h C =0 (138) (cid:14) y^(cid:3) L (cid:14) y^(cid:3) (cid:14) 62

Online Appendix and (cid:0) being given by ^(cid:24)A 1+(cid:27) (cid:0) = L : (139) ^(cid:24)A (cid:27) L + (cid:27) (cid:14) C(1+h2(cid:12))(cid:0)h(cid:12)(cid:27) (cid:14) C ((cid:0) y^(cid:3) +(cid:26) A ) As the term h(cid:12) g(cid:3) appears in equation (132), the central bank is unable to perfectly stabilize 1(cid:0)h(cid:12) mu;t in(cid:13)ation and the welfare-relevant output gap under external consumption habits in response to technology shocks. AsdiscussedinLeith,Moldovan,andRossi(2012)andWoodford(2003a),consumptionhabitshave to be speci(cid:12)ed as internal in order for the \divine coincidence" to re-emeerge; also compare to Blanchard and Gali (2007). B.2.2 Linear quadratic framework with distorted steady state Inourdiscussionofthecaseofadistortedsteadystate,wereturntothesimpleNewKeynesianModelwith (cid:13)exible wages and no consumption habits (h=0) as in Benigno and Woodford (2005). In our derivations, weallowfor twosourcesthat could justifythe presence ofa shockinthe NKPC. The(cid:12)rst one isan ad hoc markupshock(cid:22) thatisintroducedintothe(cid:12)rstorderconditionofpricesetting(cid:12)rms. Thesecondoneis p;t a shock to the sales subsidy (cid:28) . If the subsidies to prices do not fully offset the monopolistic distortions p;t in the product market, the steady state relationship between consumption and labor is determined by C(cid:22)1(cid:0)(cid:27)C =N(cid:22)1+(cid:27)L(cid:8) (140) with the steady state markup satisfying 1 = 1+(cid:18)p = (cid:8). In combining the utility from consumption, mc 1+(cid:28)(cid:22)p equation (112), and the disutility from labor, equation (115), the linear term y^ does not drop out t {( ) } ∑1 1 (cid:27) (cid:8)U N(cid:22)E (cid:12)t(cid:0)t0 y^ + y^2 (cid:0) Cy^2 n t0 t 2 t 2 t t=t0 {( ) } ∑1 1 (cid:27) 1+(cid:27) 1+(cid:18) (cid:0)U N(cid:22)E (cid:12)t(cid:0)t0 y^ + y^2 + Ly^2(cid:0)2 Ly^^(cid:24) + pvar (p (i)) n t0 t 2 t 2 t 2 t A;t 2(cid:18) i t p t=t0 +t:i:p:+O(2) ∑1 { } 1 = (cid:0) U N(cid:22)E (cid:12)t(cid:0)t0 (cid:0)2((cid:8)(cid:0)1)y^ +[((cid:27) +(cid:27) )(cid:0)(1(cid:0)(cid:27) )((cid:8)(cid:0)1)]y^2 2 n t0 t L C C t t=t0 { } ∑1 1 1+(cid:18) (cid:0) 2 U n N(cid:22)E t0 (cid:12)t(cid:0)t0 (cid:0)2(1+(cid:27) L )y^ t ^(cid:24) A;t + (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 +t:i:p:+O(3): p p t=t0 (141) Absent distortions, (cid:8) = 1, and the linear term ((cid:8)(cid:0)1)y^ in equation (141) cancels out. With distortions, t we employ the second-order approximation to the nonlinear New Keynesian Phillips curve as in Benigno and Woodford (2005) to substitute out for this linear term. The (cid:12)rst order condition for price setting is given by H Popt = t (142) t G t 63

Online Appendix where ( ) ∑1 (cid:0)1+(cid:18)p H = 1 E ((cid:12)(cid:24) )t(cid:0)t0(1+(cid:22) )N(cid:27)L (cid:5)(t)P t0 (cid:18)p Y (143) t0 C t (cid:0) 0 (cid:27)C t0 t=t0 p p;t t P t t ( ) G = 1 E ∑1 ((cid:12)(cid:24) )t(cid:0)t0(1+(cid:28) )C (cid:0)(cid:27)C (cid:5)(t)P t0 1(cid:0)1+ (cid:18)p (cid:18)p Y (144) t0 C t (cid:0) 0 (cid:27)C t0 t=t0 p p;t t P t t ∏t ( ) and (cid:5)(t) = (cid:25)(cid:19) t p 0+l(cid:0)1 (cid:25)(cid:22)1(cid:0)(cid:19)p . Following the steps outlined in Benigno and Woodford (2005), this relal=1 tionship can be shown to be approximated by 2 ( )( )3 V t = (cid:20) E ∑1 (cid:12)t(cid:0)t0 4 h^ t (cid:0)g^ t + h^ t (cid:0)g^ t h^ t +g^ t 5 (cid:27) +(cid:27) p t (cid:27) +(cid:27) 2((cid:27) +(cid:27) ) L C L C L C t=t0 [ ] ∑1 1+(cid:18) +(cid:20) p E t (cid:12)t(cid:0)t0 2((cid:27) +(cid:27) p )(cid:20) (cid:18) ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 +t:i:p:+O(3) (145) L C p p t=t0 where the terms g^ and h^ are given by t t (cid:28)(cid:22) g^ = p (cid:28)^ (cid:0)((cid:27) (cid:0)1)y^ t 1+(cid:28)(cid:22) p;t C t p (cid:22)(cid:22) 1+(cid:18) h^ t = 1+ p (cid:22)(cid:22) (cid:22)^ p;t +(1+(cid:27) L )y^ t (cid:0)(1+(cid:27) L )^(cid:24) A;t +(cid:27) L2(cid:20) (cid:18) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2: p p p Substituting the de(cid:12)nitions g^ and h^ into equation (145), t t ( )( ) h^ (cid:0)g^ h^ t (cid:0)g^ t h^ t +g^ t 1+(cid:18) (cid:27) t +(cid:27) t + 2((cid:27) +(cid:27) ) + 2((cid:27) +(cid:27) p )(cid:20) (cid:18) ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 L C L C L C p p 2+(cid:27) (cid:0)(cid:27) 1+(cid:27) 1+(cid:18) = y^ t + L 2 Cy^ t 2+ (cid:27) +(cid:27) L 2(cid:20) (cid:18) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 L C p p (1+(cid:27) )2 (cid:22)(cid:22) 1+(cid:27) (cid:28)(cid:22) 1(cid:0)(cid:27) (cid:0) L ^(cid:24) y^ + p L (cid:22)^ y^ (cid:0) p C (cid:28)^ y^ +t:i:p: (146) (cid:27) +(cid:27) A;t t 1+(cid:22)(cid:22) (cid:27) +(cid:27) p;t t 1+(cid:28)(cid:22) (cid:27) +(cid:27) p;t t C L p C L p C L multiplying with (cid:0) 1 U N(cid:22)((cid:8)(cid:0)1) and adding into equation (141) we obtain the approximation (cid:20)p N { } ∑1 1 1+(cid:27) (cid:0) U N(cid:22) 1+((cid:8)(cid:0)1) L E (cid:12)t(cid:0)t0((cid:27) +(cid:27) )y^2 2 n (cid:27) +(cid:27) t0 L C t L C { } t=t0 ∑1 1 1+(cid:27) 1+(cid:27) (cid:0) U N(cid:22) 1+((cid:8)(cid:0)1) L E (cid:12)t(cid:0)t0 (cid:0)2((cid:27) +(cid:27) ) L y^^(cid:24) 2 n (cid:27) +(cid:27) t0 L C (cid:27) +(cid:27) t A;t L C L C { } t=t0 ∑1 1 1+(cid:27) 1+(cid:18) (cid:0) 2 U n N(cid:22) 1+((cid:8)(cid:0)1) (cid:27) +(cid:27) L E t0 (cid:12)t(cid:0)t0 (cid:20) (cid:18) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 L C p p t=t0 { } ∑1 2 (cid:22)(cid:22) p ((cid:8)(cid:0)1)(1+(cid:27) ) (cid:0) 1 U N(cid:22) 1+((cid:8)(cid:0)1) 1+(cid:27) L E (cid:12)t(cid:0)t0 1+(cid:22)(cid:22) p L (cid:22)^ y^ 2 n (cid:27) +(cid:27) t0 (cid:27) +(cid:27) +((cid:8)(cid:0)1)(1+(cid:27) ) p;t t L C L C L t=t0 { } (cid:0) 1 U N(cid:22) 1+((cid:8)(cid:0)1) 1+(cid:27) L E ∑1 (cid:12)t(cid:0)t0 (cid:0)2 1+ (cid:28)(cid:22)p (cid:28)(cid:22)p ((cid:8)(cid:0)1)(1(cid:0)(cid:27) C ) (cid:28)^ y^ 2 n (cid:27) +(cid:27) t0 (cid:27) +(cid:27) +((cid:8)(cid:0)1)(1+(cid:27) ) p;t t L C L C L t=t0 +t:i:p:+O(3): (147) 64

Online Appendix Therefore, the model with a distortionary steady state can be written as ( ) ∑1 1 E (cid:12)t(cid:0)t0L (148) t0 2 t t=t0 with 1+(cid:18) L t = ((cid:27) L +(cid:27) C )(y^ t (cid:0)y~ t )2+ (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 (149) p p and the target output level y~ t 1+(cid:27) y~ = L ^(cid:24) t (cid:27) +(cid:27) A;t L C ((cid:8)(cid:0)1) 1+(cid:27)L (cid:22)(cid:22) (cid:0) (cid:27)L+(cid:27)C p (cid:22)^ (cid:27) +(cid:27) +((cid:8)(cid:0)1)(1+(cid:27) )1+(cid:22)(cid:22) p;t L C L p ((cid:8)(cid:0)1) 1(cid:0)(cid:27)C (cid:28)(cid:22) + (cid:27)L+(cid:27)C p (cid:28)^ : (cid:27) +(cid:27) +((cid:8)(cid:0)1)(1+(cid:27) )1+(cid:28)(cid:22) p;t L C L p The linear New Keynesian Phillips curve is given by ( ) (1(cid:0)(cid:12)(cid:24) )(1(cid:0)(cid:24) ) (cid:22)(cid:22) (cid:28)(cid:22) ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 ) = p (cid:24) p ((cid:27) L +(cid:27) C )y^ t + 1+ p (cid:22)(cid:22) (cid:22)^ p;t (cid:0) 1+ p (cid:28)(cid:22) (cid:28)^ p;t (cid:0)(1+(cid:27) L )^(cid:24) A;t p p p +(cid:12)E ((cid:25) (cid:0)(cid:19) (cid:25) ) (150) t p;t+1 p p;t or written in terms of the welfare relevant output gap y^ (cid:0)y~ t t ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 ) = (cid:20) p ((cid:27) (L +(cid:27) C )(y^ t (cid:0)y~ t ) ) ((cid:8)(cid:0)1)(1+(cid:27) ) (cid:22)(cid:22) +(cid:20) 1(cid:0) L p (cid:22)^ p (cid:27) +(cid:27) +((cid:8)(cid:0)1)(1+(cid:27) ) 1+(cid:22)(cid:22) p;t ( L C L ) p ((cid:8)(cid:0)1)(1(cid:0)(cid:27) ) (cid:28)(cid:22) (cid:0)(cid:20) 1(cid:0) C p (cid:28)^ p (cid:27) +(cid:27) +((cid:8)(cid:0)1)(1+(cid:27) ) 1+(cid:28)(cid:22) p;t L C L p +(cid:12)E ((cid:25) (cid:0)(cid:19) (cid:25) ): (151) t p;t+1 p p;t In contrast to the model with an efficient steady state, the target output level y~ responds to the price t markup shock (cid:22)^ and the shock to the subsidy (cid:28)^ . Only when (cid:8) = 1 does the target level remain p;t p;t unchanged after such shocks. While under an undistorted steady state the two shocks have the same impact under the optimal policy, this is no longer true if the steady state is distorted. This can easily be seen if (cid:27) =1. In this case, the shock (cid:28)^ does not impact the target output level at all. C p;t Ignoringthemovementsintheoutputtargetlevelinducedbymarkup/subsidyshockswhenformulating policies leads to inefficiencies. Although the optimal commitment policy can be described as an in(cid:13)ation targeting framework, the de(cid:12)nition of the output gap is key. If the output gap measure applied by the policymaker rests on a de(cid:12)nition of potential output as y(cid:22) t = (cid:27) 1 L + + (cid:27) (cid:27) L C ^(cid:24) A;t |as would be the case under the de(cid:12)nition applied in Smets and Wouters (2007)|instead of y~ the central bank’s response will not be t optimal. 65

Online Appendix B.2.3 Linear quadratic framework with sticky wages When prices and wages are sticky, we follow the steps outlined in Gali (2008) and Erceg, Henderson, and Levin (2000) to approximate the utility function of the household to the second-order. Our derivations includeashocktothemarginaldisutilityoflabortoillustratethediscussioninChari,Kehoe,andMcGrattan (2009). In comparison to the previous section, the approximations of the utility from consumption shown in equation (112) and the disutility from labor given in equation (113) remain unchanged with the small quali(cid:12)er that the latter expression is augmented by a term to capture the labor supply shock { ( ) } ∑1 1 (cid:27) U c C(cid:22)E t0 (cid:12)t(cid:0)t0 (1(cid:0)h(cid:12)) c^ t + 2 c^2 t (cid:0) 2(1(cid:0) C h) (c^ t (cid:0)hc^ t(cid:0)1 )2 +t:i:p:+O(2): t=t0 (152) and { } ∑1 1+(cid:27) U N(cid:22)E (cid:12)t(cid:0)t0 n^ + Ln^2+^(cid:24) n^ +t:i:p:+O(2): (153) n t0 t 2 t L;t t t=t0 Absent sticky wages aggregate labor supply N is related to (cid:12)nal output Y and the level of technology t t via a term that measures price dispersion. Under sticky wages an additional term that captures wage dispersion arises in this relationship. Note that ∫ [ ∫ ( ) ] N = 1 L (j)dj = 1 W t (j) (cid:0)1+ (cid:18)w (cid:18)w dj L =ΩlL (154) t t W t t t 0 0 t where j is the index of a labor union. Similarly, aggregate output and manufactured varieties satisfy the relationship 2 3 ∫ ∫ ( ) 1 Y (i)di= 4 1 P t (i) (cid:0)1+ (cid:18)p (cid:18)p di 5 Y =ΩyY : (155) t P t t t 0 0 t Market clearing implies ∫ 1 Y (i)di=(cid:24) L (156) t A;t t 0 or making use of the above relationships ΩlΩyY =(cid:24) N : (157) t t t A;t t ApplyingresultsfromWoodford(2003a)andGali(2008)regardingthesecond-orderapproximationsofΩy t and Ωl we obtain t 1+(cid:18) 1+(cid:18) n^ = y^ (cid:0)^(cid:24) + pvar (p (i))+ wvar (w (f)): (158) t t A;t 2(cid:18) i t 2(cid:18) f t p w 66

Online Appendix Thus, the disutility from labor can be approximated by { } N1+(cid:27)L 1+(cid:18) 1+(cid:18) (cid:24) t = U N(cid:22) y^ (cid:0)^(cid:24) + pvar (p (i))+ wvar (w (f)) L;t1+(cid:27) n t A;t 2(cid:18) i t 2(cid:18) f t L { ( p ) } w 1+(cid:27) 2 +U N(cid:22) L y^ (cid:0)^(cid:24) +^(cid:24) y^ +t:i:p:+O(2): (159) n 2 t A;t L;t t Assuming that the steady state is efficient due to appropriately chosen subsidies, i.e., (1(cid:0)h(cid:12))U C(cid:22) = c U N(cid:22), the utility function of the representative household can be approximated as n { } ∑1 1 (cid:27) U n N(cid:22)E t0 (cid:12)t(cid:0)t0 y^ t + 2 y^ t 2(cid:0) 2(1(cid:0)h)( C 1(cid:0)h(cid:12)) (y^ t (cid:0)hy^ t(cid:0)1 )2 t=t0 { } ∑1 1+(cid:18) 1+(cid:18) (cid:0)U N(cid:22)E (cid:12)t(cid:0)t0 y^ (cid:0)^(cid:24) + pvar (p (i))+ wvar (w (f)) n t0 t A;t 2(cid:18) i t 2(cid:18) f t p w t ∑ = 1 t0 { ( ) } 1+(cid:27) 2 (cid:0)U N(cid:22)E (cid:12)t(cid:0)t0 L y^ (cid:0)^(cid:24) +^(cid:24) y^ +t:i:p:+O(2) (160) n t0 2 t A;t L;t t t=t0 or after simplifying { } ∑1 1 (cid:27) 1+(cid:18) (cid:0) 2 U n N(cid:22)E t0 (cid:12)t(cid:0)t0 (cid:27) L y^ t 2+ (cid:14) C (y^ t (cid:0)hy^ t(cid:0)1 )2(cid:0)2(1+(cid:27) L )y^ t ^(cid:24) A;t + (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2 p p t=t0 { } ∑1 1 1+(cid:18) (cid:0) 2 U n N(cid:22)E t0 (cid:12)t(cid:0)t0 2y^ t ^(cid:24) L;t + (cid:18) (cid:20) w ((cid:25) w;t (cid:0)(cid:19) w (cid:25) p;t(cid:0)1 )2 +t:i:p:+O(2) (161) w w t=t0 where (1(cid:0)(cid:12)(cid:24) )(1(cid:0)(cid:24) ) (cid:20) = p p p (cid:24) p (1(cid:0)(cid:12)(cid:24) )(1(cid:0)(cid:24) ) (cid:20) = w w w (cid:24) w (cid:14) = (1(cid:0)h)(1(cid:0)h(cid:12)): To obtain the (linear) equilibrium dynamics of the efficient output level in the model with labor supply shocks, note that ( ) ( ) mcc (cid:3) = ^(cid:24) +(cid:27) n^ (cid:3) + (cid:27) C c^ (cid:3)(cid:0)hc^ (cid:3) (cid:0)h(cid:12) (cid:27) CE c^ (cid:3) (cid:0)hc^ (cid:3) (cid:0)(y^ (cid:3)(cid:0)n^ (cid:3) ) t L;t L t (cid:14) t t(cid:0)1 (cid:14) t t+1 t t t ( ) ( ) (cid:27) (cid:27) = (cid:27) y^ (cid:3) + C y^ (cid:3)(cid:0)hy^ (cid:3) (cid:0)h(cid:12) CE y^ (cid:3) (cid:0)hy^ (cid:3) (cid:0)(1+(cid:27) )^(cid:24) +^(cid:24) : (162) L t (cid:14) t t(cid:0)1 (cid:14) t t+1 t L A;t L;t and ( ) ( ) (cid:27) (cid:27) (cid:27) y^ (cid:3) + C y^ (cid:3)(cid:0)hy^ (cid:3) (cid:0)h(cid:12) CE y^ (cid:3) (cid:0)hy^ (cid:3) =(1+(cid:27) )^(cid:24) (cid:0)^(cid:24) : (163) L t (cid:14) t t(cid:0)1 (cid:14) t t+1 t L A;t L;t Efficient output is therefore a function of lagged efficient output, technology, and the labor supply shock. By substituting this last expression back into equation (161), we can approximate the utility function in terms of the welfare-relevant output gap, price and wage in(cid:13)ation, and the labor supply shock ∑1 { } 1 (cid:0) U N(cid:22)E (cid:12)t(cid:0)t0 (cid:27) y^2(cid:0)2(cid:27) y^y^ (cid:3) 2 n t0 L t L t t t=t0 67

Online Appendix { } ∑1 1 1+(cid:18) 1+(cid:18) (cid:0) 2 U n N(cid:22)E t0 (cid:12)t(cid:0)t0 (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2+ (cid:18) (cid:20) w ((cid:25) w;t (cid:0)(cid:19) w (cid:25) p;t(cid:0)1 )2 p p w w t=t0 ∑1 { [ ( ) ( )]} 1 (cid:27) (cid:0) 2 U n N(cid:22)E t0 (cid:12)t(cid:0)t0 (cid:14) C (y^ t (cid:0)hy^ t(cid:0)1 )2(cid:0)2 y^ t y^ t (cid:3)(cid:0)hy^ t (cid:3) (cid:0)1 (cid:0)(cid:12)hy^ t y^ t (cid:3) +1 (cid:0)hy^ t (cid:3) t=t0 +t:i:p:+O(2): (164) Applyingequation(130)oncemoreallowsustode(cid:12)nethesociallossfunctionforthemodelwithsticky wages and prices, indexation, labor supply shocks, and external habits as ( ) ∑1 1 E (cid:12)t(cid:0)t0L (165) t0 2 t t=t0 with (cid:27) L t = (cid:27) L (x t )2+ (cid:14) C (x t (cid:0)hx t(cid:0)1 )2 1+(cid:18) 1+(cid:18) + (cid:18) (cid:20) p ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )2+ (cid:18) (cid:20) w ((cid:25) w;t (cid:0)(cid:19) w (cid:25) p;t(cid:0)1 )2 (166) p p w w x =y^ (cid:0)y^(cid:3) denotes the welfare-relevant output gap. The structural equations of the model are given by t t t the New Keynesian Phillips curve for prices ((cid:25) p;t (cid:0)(cid:19) p (cid:25) p;t(cid:0)1 )=(cid:20) p mcc t +u p;t +(cid:12)E t ((cid:25) p;t+1 (cid:0)(cid:19) p (cid:25) p;t ) (167) with mcc =!^ (cid:0)^(cid:24) =!^ (cid:0)!^ (cid:3) (168) t t A;t t t and the price markup shock u , the New Keynesian Phillips curve for wages p;t ((cid:25) w;t (cid:0)(cid:19) w (cid:25) p;t(cid:0)1 )=(cid:20) w (mdrs t (cid:0)!^ t )+u w;t +(cid:12)E t ((cid:25) w;t+1 (cid:0)(cid:19) w (cid:25) p;t ) (169) with (cid:27) mdrs t (cid:0)!^ t = (cid:27) L y^ t + 1(cid:0) C h (y^ t (cid:0)hy^ t(cid:0)1 )(cid:0)(cid:27) L ^(cid:24) A;t +^(cid:24) L;t (cid:0)!^ t (cid:27) = (cid:27) L y^ t + 1(cid:0) C h (y^ t (cid:0)hy^ t(cid:0)1 )(cid:0)(1+(cid:27) L )^(cid:24) A;t +^(cid:24) L;t (cid:0)(!^ t (cid:0)!^ (cid:3) t ) (cid:27) = (cid:27) L x t + 1(cid:0) C h (x t (cid:0)hx t(cid:0)1 )(cid:0)(!^ t (cid:0)!^ (cid:3) t ) ( ) (cid:27) +(cid:27) y^ (cid:3) + C y^ (cid:3)(cid:0)hy^ (cid:3) (cid:0)(1+(cid:27) )^(cid:24) +^(cid:24) L t 1(cid:0)h t t(cid:0)1 L A;t L;t (cid:27) h(cid:12) = (cid:27) L x t + 1(cid:0) C h (x t (cid:0)hx t(cid:0)1 )(cid:0)(!^ t (cid:0)!^ (cid:3) t )+ 1(cid:0)h(cid:12) g m (cid:3) u;t (170) and the wage markup shock u , the evolution of real wages w;t ( ) ( ) (!^ t (cid:0)!^ (cid:3) t )= !^ t(cid:0)1 (cid:0)!^ (cid:3) t(cid:0)1 +(cid:25) w;t (cid:0)(cid:25) p;t (cid:0) !^ (cid:3) t (cid:0)!^ (cid:3) t(cid:0)1 (171) 68

Online Appendix and the aggregate demand curve 1(cid:0)h ( ) (x t (cid:0)hx t(cid:0)1 )=E t (x t+1 (cid:0)hx t )(cid:0) (cid:27) i t (cid:0)E t (cid:25) p;t+1 (cid:0)g m (cid:3) u;t : (172) C g(cid:3) is de(cid:12)ned as mu;t [ ( ) ( )] (cid:27) g (cid:3) = C E y^ (cid:3) (cid:0)hy^ (cid:3) (cid:0) y^ (cid:3)(cid:0)hy^ (cid:3) : (173) mu;t 1(cid:0)h t t+1 t t t(cid:0)1 The efficient equilibrium output follows ( ) ( ) (cid:27) (cid:27) (cid:27) y^ (cid:3) + C y^ (cid:3)(cid:0)hy^ (cid:3) (cid:0)h(cid:12) CE y^ (cid:3) (cid:0)hy^ (cid:3) =(1+(cid:27) )^(cid:24) (cid:0)^(cid:24) (174) L t (cid:14) t t(cid:0)1 (cid:14) t t+1 t L A;t L;t and the efficient real wage is determined by !^ (cid:3) =^(cid:24) : (175) t A;t Figures 12 and 13 depict the impulse responses after a price markup shock in the model with sticky wages and no price indexation and with full price indexation, respectively. 69

Online Appendix Figure 12: Impulse responses to a price markup shock in a model with sticky wages and no price indexation 0 5 10 15 tnecrep 1. output gap 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 0 5 10 15 ss morf .ved tniop egatnecrep 2. price inflation 0.6 0.5 Optimal Commitment Policy Optimal IT 0.4 Optimal PLT Optimal SLP 0.3 0.2 0.1 0 0 5 10 15 quarters ss morf .ved tniop egatnecrep 3. wage inflation 0.1 0.05 0 -0.05 -0.1 -0.15 0 5 10 15 quarters ss morf .ved tnecrep 4. real wage 0 -0.05 -0.1 -0.15 -0.2 -0.25 Note: The(cid:12)gureshowstheimpulseresponsesofselectedvariablesinthestickywagemodelwithnopriceorwageindexation((cid:19)p=(cid:19)w =0) afterapricemarkupshock. Theresultsforfourpoliciesareshown: theoptimalcommitmentpolicy(Ramsey),optimalin(cid:13)ationtargeting framework(IT),theoptimalpriceleveltargeting(PLT),andtheoptimalspeedlimitpolicy(SLP). 70

Online Appendix Figure 13: Impulse responses to a price markup shock in a model with sticky wages and full price indexation 0 5 10 15 tnecrep 1. output gap 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 0 5 10 15 ss morf .ved tniop egatnecrep 2. price inflation 1 Optimal Commitment Policy Optimal IT 0.8 Optimal PLT Optimal SLP 0.6 0.4 0.2 0 -0.2 0 5 10 15 quarters ss morf .ved tniop egatnecrep 3. wage inflation 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0 5 10 15 quarters ss morf .ved tnecrep 4. real wage 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 Note: The (cid:12)gure shows the impulse responses of selected variables in the sticky wage model with full price indexation ((cid:19)p = 1) but no wageindexation((cid:19)w =0)afterapricemarkupshock. Theresultsforfourpoliciesareshown: theoptimalcommitmentpolicy(Ramsey), optimalin(cid:13)ationtargetingframework(IT),theoptimalpriceleveltargeting(PLT),andtheoptimalspeedlimitpolicy(SLP). 71

Online Appendix B.3 Inertia under price level targeting and speed limit policy in the textbook NKM When policymakers act under discretion, price level targeting and speed limit policy perform strongly in comparison to the optimal commitment policy as discussed in Walsh (2003) and Vestin (2006). This appendixreproducesthekeystepstoshowhowinertiaintheoutputgapandin(cid:13)ationariseinthetextbook NKM without price indexation or habit persistence under these two targeting frameworks. In the following, we assume that the markup shock is transitory, i.e., E (u ) = 0, and we abstract t p;t+1 from the zero lower bound on the nominal interest rate, which allows us to ignore the aggregate demand curve of the model. Under these assumptions, the optimal commitment policy implies the following dynamics for the output gap and in(cid:13)ation x t = (cid:7) x x t(cid:0)1 +(cid:7) u u p;t (176) (cid:21) (cid:21) (cid:25) p;t = (cid:20)~ (1(cid:0)(cid:7) x )x t(cid:0)1 (cid:0) (cid:20)~ (cid:7) u u p;t (177) p p with (cid:7) x being the solution to ( ) ((cid:20)~ )2 1+(cid:12)+ p (cid:7) (cid:0)(cid:12)(cid:7)2 (cid:0)1=0 (178) (cid:21) x x that satis(cid:12)es (cid:7) <1. The value for (cid:7) is x u (cid:0)(cid:20)~ (cid:7) = p : (179) u (cid:21)f1+(cid:12)(1(cid:0)(cid:7) )g+((cid:20)~ )2 x p (cid:21)istheweightontheoutputgapinthetruesociallossfunctionwhentheweightonin(cid:13)ationisnormalized tounityinthetextbookNKMwithoutpriceindexation,i.e.,(cid:21)= (cid:20)~p(cid:18)p. Recall,thatundertheassumptions 1+(cid:18)p of the textbook NKM the functional form of the true social loss function is identical to the objective function under the in(cid:13)ation targeting framework. Inequilibrium, bothin(cid:13)ation andtheoutput gapdependonthe pervious realizationoftheoutput gap. This feature of the optimal commitment policy is not found in the optimal discretion policy. B.3.1 Price level targeting To solve for the equilibrium under discretion we conjecture that the value function of the policymaker is quadratic and depends on the price level of the previous period under price level targeting. We start with theassumptionthatu p;t =(cid:26) u u p;t(cid:0)1 +(cid:27) u " u;t . Whencomparingthesolutionunderthepriceleveltargeting framework to the solution under the optimal commitment policy, however, we will set (cid:26) =0. u The value function of the policymaker satis(cid:12)es ( ) 1 V(p^ t(cid:0)1 ;u p;t )= p m ^t; i x n t 2 p^2 t +(cid:21)PLT (x t )2 +(cid:12)E t V(p^ t ;u p;t+1 ) (180) s:t: (p^ t (cid:0)p^ t(cid:0)1 )=(cid:20)~ p x t +(cid:12)E t private(p^ t+1 (cid:0)p^ t )+u p;t : (181) 72

Online Appendix We conjecture that the value function is quadratic of the form 1 1 V p (p^ t(cid:0)1 ;u p;t )=a 1 u p;t + 2 a 2 (u p;t )2+a 3 u p;t p^ t(cid:0)1 +a 4 p^ t(cid:0)1 + 2 a 5 (p^ t(cid:0)1 )2 (182) implying the derivative with respect to p^ t(cid:0)1 to be of the form V p (p^ t(cid:0)1 ;u p;t )=a 3 u p;t +a 4 +a 5 p^ t(cid:0)1 (183) and that in equilibrium the price level evolves according to p^ t =(cid:2) p p^ t(cid:0)1 +(cid:2) u u p;t : (184) (p^ t (cid:0)p^ t(cid:0)1 )=(cid:20)~ p x t +(cid:12)E t private(p^ t+1 (cid:0)p^ t )+u p;t (185) (p^ t (cid:0)p^ t(cid:0)1 )=(cid:20)~ p x t +(cid:12)((cid:2) p (cid:0)1)p^ t +((cid:12)(cid:2) u (cid:26) u +1)u p;t (186) Combiningequations(181)and(184)toeliminatep^ andimposingexpectationstoberationaldelivers t+1 ! 1 1+(cid:12)(cid:26) (cid:2) x t = (cid:20)~ p^ t (cid:0) (cid:20)~ p^ t(cid:0)1 (cid:0) (cid:20)~ u uu p;t : (187) p p p with ! =1+(cid:12)(1(cid:0)(cid:2) ). Replacing the term x in the value function by the expression in equation (187), p t the Envelop condition associated with the policymaker’s optimization problem implies (cid:21)PLT a 3 u p;t +a 4 +a 5 p^ t(cid:0)1 =(cid:0) (cid:20)~ x t (188) p and the (cid:12)rst order condition with respect to p^ delivers t (cid:21)PLT p^ +! x +(cid:12)[a +a p^]=0: (189) t (cid:20)~ t 4 5 t p Combining the last two equations to eliminate x and applying equation (184) delivers the parameter t restrictions !a (cid:2) = 5 (190) p 1+(cid:12)a 5 !(cid:0)(cid:12)(cid:26) (cid:2) = ua (191) u 1+(cid:12)a 3 5 and a =0. Using this information in the Envelop condition 4 (cid:21)PLT 1 (cid:21)PLT 1 a 3 u p;t +a 5 p^ t(cid:0)1 = (cid:20)~ (cid:20)~ (1(cid:0)!(cid:2) p )p^ t(cid:0)1 + (cid:20)~ (cid:20)~ (1+(cid:12)(cid:26) u (cid:2) u (cid:0)!(cid:2) u )u p;t (192) p p p p we obtain the remaining two conditions (cid:21)PLT a = (1+(cid:12)(cid:26) (cid:2) (cid:0)!(cid:2) ) (193) 3 ((cid:20)~ )2 u u u p 73

Online Appendix (cid:21)PLT a = (1(cid:0)!(cid:2) ): (194) 5 ((cid:20)~ )2 p p By combining equations (190), (191), (193), and (194), we obtain the implicit de(cid:12)nition of (cid:2) and (cid:2) p u (cid:21)PLT! (cid:2) = (195) p ((cid:20)~ )2+(cid:12)(cid:21)PLT (1(cid:0)!(cid:2) )+(cid:21)PLT!2 p p (cid:21)PLT (!(cid:0)(cid:12)(cid:26) ) (cid:2) = u (196) u ((cid:20)~ )2+(cid:12)(cid:21)PLT (1(cid:0)!(cid:2) )+(cid:21)PLT (!(cid:0)(cid:12)(cid:26) )2 p p u noting that ! is a function of (cid:2) . p Equippedwiththelawofmotionforpricesandtheoutputgapunderpriceleveltargetingwithdiscretion, we can show that in(cid:13)ation and the output gap follow the same path as under the optimal commitment policy, when (cid:26) =0 and therefore (cid:2) =(cid:2) . Note that equation (187) can be rewritten as u u p 1+(cid:12)(1(cid:0)(cid:2) ) 1 1 !(cid:0) 1 x t = (cid:20)~ p p^ t (cid:0) (cid:20)~ p^ t(cid:0)1 (cid:0) (cid:20)~ u p;t = (cid:20)~ (cid:2)pp^ t (197) p p p p and therefore p^ = (cid:20)~p x . The price level is proportional to the output gap, just as it is the case under t !(cid:0) 1 t (cid:2)p the optimal commitment policy (compare to the optimal targeting rule expressed as p^ =(cid:0) (cid:21) x ). Hence, t (cid:20)~p t we obtain the law of motion for x as t (cid:2) !(cid:0)1 x t = (cid:2) p x t(cid:0)1 + p (cid:20)~ u p;t : (198) p Thus, for the price level targeting framework to implement the optimal commitment policy, (cid:21)PLT must be chosen to satisfy (cid:21)PLT! (cid:7) = (199) x ((cid:20)~ )2+(cid:12)(cid:21)PLT (1(cid:0)!(cid:7) )+(cid:21)PLT!2 p x with(cid:7) beingthesolutiontoequation(178)and! =1+(cid:12)(1(cid:0)(cid:7) ). Furthermore,itis (cid:2)p!(cid:0)1 =(cid:7) given x x (cid:20)~p u the conditions imposed on (cid:7) . x B.3.2 Speed limit policy To solve for the equilibrium under discretion in the speed limit policy framework we conjecture that the value function of the policymaker is quadratic and depends on the output gap of the previous period. The value function of the policymaker satis(cid:12)es ( ) 1 V(x t(cid:0)1 ;u p;t )= (cid:25) m p;t i ; n xt 2 (cid:25)2 p;t +(cid:21)SLP (x t (cid:0)x t(cid:0)1 )2 +(cid:12)E t V(x t ;u p;t+1 ) (200) s:t: (cid:25) =(cid:20)~ x +(cid:12)Eprivate(cid:25) +u : (201) p;t p t t p;t+1 p;t We conjecture that the value function is quadratic implying the derivative with respect to x t(cid:0)1 to be of the form V x (x t(cid:0)1 ;u p;t )=a 3 u p;t +a 4 +a 5 x t(cid:0)1 (202) 74

Online Appendix and that in equilibrium in(cid:13)ation evolves according to (cid:25) p;t =Ω (cid:25) x t(cid:0)1 +Ω u u p;t : (203) Combining equations (201) and (203) and imposing rational expectations delivers (cid:25) =((cid:20)~ +(cid:12)Ω )x +((cid:12)Ω (cid:26) +1)u : (204) p;t p (cid:25) t u u p;t Thus, the Envelop condition associated with the policymaker’s optimization problem implies V x (x t(cid:0)1 ;u p;t )=a 3 u p;t +a 4 +a 5 x t(cid:0)1 =(cid:0)(cid:21)SLP(x t (cid:0)x t(cid:0)1 ): (205) If in equilibrium x evolves according to t x t =(cid:2) x x t(cid:0)1 +(cid:2) u u p;t (206) we obtain the conditions a =(cid:0)(cid:21)SLP(cid:2) , a =0, and a =(cid:21)SLP (1(cid:0)(cid:2) ). 3 u 4 5 x From the (cid:12)rst order condition of the value function, we obtain (cid:25) p;t ((cid:20)~ p +(cid:12)Ω (cid:25) )+(cid:21)SLP (x t (cid:0)x t(cid:0)1 )+(cid:12)a 3 (cid:26) u u p;t +(cid:12)a 5 x t =0 (207) or after substituting out for (cid:25) and x p;t t [( ) ] ((cid:20)~ p +(cid:12)Ω (cid:25) )2+(cid:12)a 5 (cid:2) x +(cid:21)SLP ((cid:2) x (cid:0)1) x t(cid:0)1 [( ) ( )] + ((cid:20)~ +(cid:12)Ω )2+(cid:12)a (cid:2) + ((cid:12)Ω (cid:26) +1)((cid:20)~ +(cid:12)Ω )+(cid:21)SLP(cid:2) +(cid:12)a (cid:26) u =0: (208) p (cid:25) 5 u u u p (cid:25) u 3 u p;t Using the fact that a =(cid:21)SLP (1(cid:0)(cid:2) ) and Ω = (cid:20)~p(cid:2)x 5 x (cid:25) 1(cid:0)(cid:12)(cid:2)x ( ) 1 2 ((cid:20)~ )2 (cid:2) (cid:0)(cid:21)SLP (1(cid:0)(cid:2) )(1(cid:0)(cid:12)(cid:2) )=0 (209) p 1(cid:0)(cid:12)(cid:2) x x x x and (cid:12)nally we obtain a relationship between (cid:2) and (cid:21)SLP x ((cid:20)~ )2 (1(cid:0)(cid:12)(cid:2) )3 p =(1(cid:0)(cid:2) ) x : (210) (cid:21)SLP x (cid:2) x Similarly, we have ( ( ) ) ( ) 1 2 (cid:20)~ ((cid:20)~ )2 +(cid:12)(cid:21)SLP (1(cid:0)(cid:2) ) (cid:2) + (1+(cid:12)Ω (cid:26) ) p +(cid:21)SLP(cid:2) (1(cid:0)(cid:12)(cid:26) ) =0 (211) p 1(cid:0)(cid:12)(cid:2) x u u u 1(cid:0)(cid:12)(cid:2) u u x x where ((cid:20)~ +(cid:12)Ω )(cid:2) +1 Ω = p (cid:25) u (212) u 1(cid:0)(cid:12)(cid:26) u to determine (cid:2) . u We are now in a position to compare the solution under the discretionary speed limit policy to the 75

Online Appendix optimal commitment policy. Rewrite condition (178) as ((cid:20)~ )2 (1(cid:0)(cid:12)(cid:7) ) p =(1(cid:0)(cid:7) ) x : (213) (cid:21) x (cid:7) x and compare to condition (210). For (cid:21)SLP = (cid:21), the speed limit policy imparts some, but less persistence to the output gap than the optimal commitment policy. If (cid:21)SLP =(cid:21)=(1(cid:0)(cid:12)(cid:7) )2, the speed limit policy impart the same persistence on the output gap. Howx ever,theoptimalcommitmentpolicyisnotreplicatedforthisvalueof(cid:21)SLP since(cid:2) ̸=(cid:7) : setting(cid:26) =0 u u u for simplicity condition (211) reduces to (cid:0)(cid:20)~ (cid:2) =(1(cid:0)(cid:12)(cid:7) ) p =(1(cid:0)(cid:12)(cid:7) )(cid:7) : (214) u x (cid:21)f1+(cid:12)(1(cid:0)(cid:7) )g+((cid:20)~ )2 x u x p C Model in Walsh (2003) Walsh (2003) uses the following linear model which resembles our NK model with price indexation and consumption habits. Backward-looking behavior in the hybrid New Keynesian Phillips curve is measured by the parameter ϕ (cid:25) p;t =(1(cid:0)ϕ)(cid:12)E t (cid:25) p;t+1 +ϕ(cid:25) p;t(cid:0)1 +(cid:20)x t +e t (215) where (cid:25) denotes in(cid:13)ation, x the output gap and e a markup shock. The aggregate demand curve p;t t t includes a lagged term of the output gap x t =(cid:18)x t(cid:0)1 +(1(cid:0)(cid:18))E t x t+1 (cid:0)(cid:27)(R t (cid:0)E t (cid:25) p;t+1 )+(cid:22) t (216) where R is the nominal interest rate. The variable (cid:22) summarizes shocks to the natural rate of interest t t (cid:22) t =u t (cid:0)[1(cid:0)(1(cid:0)(cid:18))(cid:13)(cid:22)]y(cid:22) t +(cid:18)y(cid:22) t(cid:0)1 (217) where potential output y(cid:22) and the demand shock u follow AR(1) processes t t y(cid:22) t = (cid:13)(cid:22)y(cid:22) t(cid:0)1 +(cid:24) t (218) u t = (cid:13) u u t(cid:0)1 +(cid:17) t : (219) Finally, the markup shock is given by e t =(cid:13) e e t(cid:0)1 +" t : (220) The welfarecriterion in Walsh(2003) is not derivedfrom the preferences of households, but it is simply stated to be of the form (cid:25)2 +(cid:21)x2: (221) p;t t The parameterization of the model is summarized in Table 3. 76

Online Appendix Table 3: Parameter Values for Walsh(2003) Parameter Description Value (cid:12) discountfactor 0.99 (cid:20) slopeofNKPC 0.05 (cid:21) weightonoutputgap 0.25 (cid:27) inverseofelast. subs. 1.5 ϕ laggedin(cid:13)ationinNKPC 0.5 (cid:18) laggedconsumptioninAD 0.5 Shock Description Value (cid:27)" autocorr. markup 0.015 (cid:13) std. markup 0 e (cid:27)u autocorr. demandshock 0.015 (cid:13) std. demandshock 0.3 u (cid:27) autocorr. naturaloutput 0.005 (cid:24) (cid:13)(cid:22) std. naturaloutput 0.97 We solve the model for in(cid:13)ation targeting, price level targeting, speed limit policy, and the second nominal income targeting framework under commitment and discretion. Walsh (2003) only reports results under discretion. Figure (14) shows the welfare outcomes for each framework relative to the IT framework as a function of the degree of price indexation. The top two panels report welfare differences in percent deviations from the IT framework as in Walsh (2003) while the bottom two panels report the welfare differences in terms of consumption equivalent variation (CEV). Similar to our (cid:12)ndings, the price level targeting and speed limit policy frameworks perform worse than the IT framework when policymakers act under commitment. Given that Walsh (2003) evaluates welfare usingalossfunctionthathasthesamefunctionalformastheobjectivefunctionunderIT,thisresultholds by assumption. Under discretion, the price level targeting and the speed limit policy perform much better thantheITframeworkformoderatedegreesofpriceindexation. Thiscontrastswithour(cid:12)ndingthatprice level targeting and the speed limit policy outperform the IT framework for all degrees of price indexation in the model with moderate consumption habits. 77

Online Appendix Figure 14: Welfare evaluation of targeting frameworks in Walsh (2003) 0 0.2 0.4 0.6 0.8 1 price indexation φ tnecreP Discretion 15 PLT SLP 10 NITII - IT 5 0 -5 -10 -15 -20 -25 -30 0 0.2 0.4 0.6 0.8 1 price indaxation φ tnecreP Commitment 0 -5 -10 -15 -20 -25 -30 0 0.2 0.4 0.6 0.8 1 price indexation φ VEC∆*001 Discretion 1 0 -1 -2 -3 -4 -5 -6 -7 0 0.2 0.4 0.6 0.8 1 price indaxation φ VEC∆*001 Commitment 0 -1 -2 -3 -4 -5 -6 -7 Note: The (cid:12)gure shows the welfare performance of price level targeting (PLT), speed limit policy (SLP), and nominal income targeting (NIT-II)relativetoin(cid:13)ationtargeting(IT)underdiscretionandcommitmentinthemodelofWalsh(2003). Intheupperpanels,weexpress the welfare differences of each targeting framework as the percent deviation from the in(cid:13)ation targeting framework as in Walsh (2003), whileinthelowerpanelsweexpressthedifferencesbetweenframeworksintermsoftheconsumptionequivalentvariation(CEV). 78

Online Appendix D The CEE/SW model This section lays out the nonlinear version of the CEE/SW model as implemented in our paper following Smets and Wouters (2007). D.1 Households D.1.1 Household Agent Each period t, household j chooses consumption C (j), labor supply N (j), investment I (j), the capital t t t stockK (j),capitalutilizationZ (j),anddomesticbondholdingstomaximizeexpecteddiscountedlifetime t t utility. In doing so the household takes prices, wages and transfers as given. Household j’s preferences over consumption and leisure are given by { ( )} E t0 ∑1 (cid:12)t(cid:0)t0 1(cid:0) 1 (cid:27) ( C t (j)(cid:0)hC t A (cid:0)1 ) 1(cid:0)(cid:27)Cexp (cid:27) 1 C + (cid:0) (cid:27) 1 N t (j)(1+(cid:27)L) (222) C L t=t0 CA refers to the level of aggregate consumption in the previous period; the parameter h captures the t(cid:0)1 degree of external consumption habits. The budget constraint of the household is given by B (j) P t C t (j)+P t I t (j)+ (cid:24)R t R =W t fN t (j)+R t kK t(cid:0)1 (j)Z t (j)(cid:0)a(Z t (j))K t(cid:0)1 (j)P t +Profits t +T t t t (223) The household earns income by supplying homogeneous labor services to labor union N (j) and earns the t wagerateW t f. Furthermore,thehouseholdderivesincomefromrentingoutitscapitalstock,R t kK t(cid:0)1(j) Z t (j) netofcapitalutilizationcosta(Z t (j))K t(cid:0)1 (j)P t . Finally, thehouseholdreceivespaymentsfromholding(cid:12)nancialassets,B (j),Profits andgovernmenttransfersT . Thisincomeisspentonconsumption,P C (j), t t t t t investment, P I (j), and (cid:12)nancial assets. t t Capital accumulates following [ ( )] I (j) K t (j)=(1(cid:0)(cid:14))K t(cid:0)1 (j)+(cid:24)I t I t (j) 1(cid:0)S I t(cid:0) t 1 (j) (224) with the investment adjustment cost function ( ) ( ) I (j) (cid:20) I (j) 2 S t = t (cid:0)(cid:13) (225) I t(cid:0)1 (j) 2 I t(cid:0)1 (j) where S((cid:13))=0, S′((cid:13))=0, S′′(:)=(cid:20)>0. Capital utilization costs are governed by [ ( ) ] (Rk)2 z a(Z (j))= exp (Z (j)(cid:0)1) (cid:0)1 (226) t z Rk t (cid:14) is the depreciation rate. The utilization function satis(cid:12)es a(1)=0, a ′ (1)=Rk, and a ′′ (1)=z. 79

Online Appendix D.1.2 Labor unions and bundlers Households supply their homogeneous labor to intermediate labor unions. These unions differentiate the laborservices,andresellthemtolaborbundlers. Theunionactsundermonopolisticcompetitionandsetsits wagerateusingstaggeredcontractsasinCalvo(1983). Thelaborbundlerscombinethedifferentiatedlabor services into an aggregate labor service that is sold to the intermediate goods producers in a competitive market. Labor bundling takes the form ∫ ( ) 1 L (j) G t di=1 (227) L 0 t following Kimball (1995). G is assumed to be a strictly concave and increasing function ( ) ( ) [( ) ] L (j) 1+(cid:18) 1+(cid:18) (cid:0)(cid:18) ϵ L (j) (cid:18) ϵ 1(cid:0)(cid:18)wϵw (cid:18) +(cid:18) ϵ G t = w w w w t + w w 1+(cid:18)w(cid:0)(cid:18)wϵw (cid:0) w w w L 1(cid:0)(cid:18) ϵ 1+(cid:18) L 1+(cid:18) 1(cid:0)(cid:18) ϵ t w w w t w w w (228) where 1+ (cid:18)w (cid:18)w referstotheelasticityofsubstitutionamonglaborvarieties,andϵ w isreferredtoastheKimball elasticity. For ϵ = 0, the function G reduces to the standard Dixit-Stiglitz aggregator with a constant w elasticity of substitution between varieties. Each labor bundler buys differentiated labor services from all unions and packages the differentiated services into an aggregate labor service L . In doing so, a bundler solves the pro(cid:12)t maximization problem t ∫ 1 max W L (cid:0) W (j)L (j)dj (229) t t t t Lt(i) ∫ ;Lt ( 0) 1 L (j) s:t: G t djL =L ((cid:21)L): (230) L t t t 0 t The (cid:12)rst order conditions imply the bundlers’ demand function for labor of type j ( ) L (j) 1+(cid:18) W (j)W (cid:0)1+(cid:18)w(cid:0)(cid:18)wϵw (cid:18) ϵ t = w t t (cid:18)w (cid:0) w w (231) L t 1+(cid:18) w (cid:0)(cid:18) w ϵ w W t (cid:21)L t 1+(cid:18) w (cid:0)(cid:18) w ϵ w and wage costs charged to an intermediate goods produced satis(cid:12)es [∫ ( ) ] (cid:21)L 1 W (j)W (cid:0)1(cid:0)(cid:18)wϵw (cid:0) (cid:18)w t = t t (cid:18)w dj 1(cid:0)(cid:18)wϵw: (232) W t 0 W t (cid:21)L t Each labor union measures the costs of the labor services it differentiates in terms of the marginal rate of substitution of the supplying households. The unions are subject to nominal rigidities as in Calvo (1983). A union can readjust its nominal wage with probability 1(cid:0)(cid:24) in each period. For those that w cannot adjust wages optimally in the current period, wages increase as the weighted average of in(cid:13)ation in the previous period (cid:5) t = P P t(cid:0) t 1 and in(cid:13)ation rate along the balance growth path (cid:5)(cid:22) taking into account the labor-augment technological progress (cid:13), i.e., ( ) W (j)=W~ (j) (cid:5)(cid:19)w(cid:5)(cid:22)(1(cid:0)(cid:19)w)(cid:13) (233) t+1 t t For those that can adjust, the problem is to choose a wage W~ (j) that maximizes the wage income in all t 80

Online Appendix states of nature where union has to maintain that wage in the future: Wage setting behavior for labor variety j ∑1 (cid:12)s(cid:21) [ ] maxE ((cid:24) )s t+s W (j)(cid:0)Wh L (j) W~ t(j) t s=0 w (cid:21) t t+s t+s t+s ( ) L (j) 1+(cid:18) W (j)W (cid:0)1+(cid:18)w(cid:0)(cid:18)wϵw (cid:18) ϵ s:t: t+s = w t+s t+s (cid:18)w (cid:0) w w L 1+(cid:18) (cid:0)(cid:18) ϵ W (cid:21) 1+(cid:18) (cid:0)(cid:18) ϵ t+s w w w t+s t+s w w w W (j)=W~ (j)XW 8t+s t t;s >>< 1 for s=0 X t W ;s = >>: ∏s ((cid:5)(cid:19) t w +l(cid:0)1 (cid:5)(cid:22)1(cid:0)(cid:19)w(cid:13)) for s=1;:::;1 (234) l=1 A wage markup shock is modeled by allowing (cid:18) to vary over time. This shock is assumed to follow an w ARMA(1,1) process. Accordingly, (cid:18) is replaced by (cid:18) with w w;t log((cid:18) w;t )=(1(cid:0)(cid:26) w )log((cid:18) w )+(cid:26) w log((cid:18) w;t(cid:0)1 )+" w;t (cid:0)(cid:26) w;ϵ ϵ w;t(cid:0)1 (235) " is white noise following N(0;(cid:27)2). w;t w D.2 Firms There are two types of (cid:12)rms: intermediate goods producers and (cid:12)nal good producers. D.2.1 Intermediate Goods Producer Intermediate goods producers choose capital and labor to minimize the cost of producing an intermediate goods variety using a Cobb-Douglas technology. In doing so they take the capital rental rate Rk and the t aggregate wage rate W as given. The cost minimization problem is then given by t min RkK (i)+W L (i) t t t t Kt(i);Lt(i) s:t: Y (i)=(cid:24) K (i)!k ((cid:13)tL (i))!l (cid:0)(cid:13)t(cid:8) (236) t A;t t t where(cid:8)isa(cid:12)xedcostthatischosentosettheproducer’spro(cid:12)tsequaltozerointhesteadystate. Marginal costsareequalizedacross(cid:12)rmsas(cid:12)rmssharethesametechnologyandfactormarketsarefrictionless. ^(cid:24) A;t denotes a shock to total factor productivity ( ) ( ) log (cid:24) =(1(cid:0)(cid:26) )log((cid:13))+(cid:26) log (cid:24) +" (237) A;t A A A;t A;t " is white noise following N(0;(cid:27)2). (cid:13) refers to steady sate labor-augment technology progress. A;t A Intermediate goods producers set prices using staggered contracts as in Calvo (1983). Each period, a (cid:12)rmcanresetitspriceoptimallywithaconstantprobability1(cid:0)(cid:24) . Thisprobabilityisindependentacross p producers and time. Producers that cannot optimally adjust their price in the current period adjust by a weighted average of (cid:5) the nominal price in(cid:13)ation in the previous period and (cid:5)(cid:22) the steady state in(cid:13)ation t 81

Online Appendix rate. ( ) P (i)=P~(i) (cid:5)(cid:19)p(cid:5)(cid:22)1(cid:0)(cid:19)p : (238) t+1 t t The intermediate goods producer i solves the pro(cid:12)t maximization problem ∑1 maxE ((cid:24) )s [(P (i)(cid:0)MC )]Y (i) t p t;t+s t+s t+s t+s P~ t(i) s=0 s:t: ( ) (cid:0)1+(cid:18)p(cid:0)(cid:18)pϵp Y (i) 1+(cid:18) P (i)P (cid:18)p (cid:18) ϵ t+s = p t+s t+s (cid:0) p p Y t+s 1+(cid:18) p (cid:0)(cid:18) p ϵ p P t+s (cid:21)Y t+s 1+(cid:18) p (cid:0)(cid:18) p ϵ p P (i)=P~(i)XP 8t+s t t;s >>< 1 for s=0 X t P ;s = >>: ∏s ((cid:5)(cid:19) t p +l(cid:0)1 (cid:5)(cid:22)1(cid:0)(cid:19)p) for s=1;:::;1 (239) l=1 by (cid:12)xing the price in the current period. D.2.2 Final Good Producer Differentiatedintermediatedproductsarecombinedtoformthecompositegoodbyacontinuumofbundlers in a perfectly competitive environment. Using a technology of the form in Kimball (1990), it is ∫ ( ) 1 Y (i) G t di=1 (240) Y 0 t and G ( Y t (i) ) = 1+(cid:18) p [( 1+(cid:18) p (cid:0)(cid:18) p ϵ p ) Y t (i) + (cid:18) p ϵ p ]( 1+ 1 (cid:18) (cid:0) p (cid:18) (cid:0) p (cid:18) ϵ p p ϵp ) (cid:0) (cid:18) p +(cid:18) p ϵ p Y 1(cid:0)(cid:18) ϵ 1+(cid:18) Y 1+(cid:18) 1(cid:0)(cid:18) ϵ t p p p t p p p (241) where 1+(cid:18)p refers to the elasticity of substitution between intermediate varieties, and ϵ stands for the (cid:18)p p Kimballelasticity. Ifϵ =0, theKimballaggregatorreducestothestandardDixit-Stiglitzaggregatorwith p [∫ ] 1 1+(cid:18)p 1 Y t = Y t (i)1+(cid:18)pdi (242) 0 Pro(cid:12)t maximization for intermediate producer i is de(cid:12)ned as: ∫ 1 max P Y (cid:0) P (i)Y (i)di t t t t Yt(i);Yt 0 s:t: ∫ ( ) 1 Y (i) G t diY =Y ((cid:21)Y): (243) Y t t t 0 t 82

Online Appendix The (cid:12)rst order conditions deliver the demand function for each intermediate good and an expression for the aggregate price index ( ) (cid:0)1+(cid:18)p(cid:0)(cid:18)pϵp Y t (i) = 1+(cid:18) p P t (i) P t (cid:18)p (cid:0) (cid:18) p ϵ p (244) Y t 1+(cid:18) p (cid:0)(cid:18) p ϵ p P t (cid:21)Y t 1+(cid:18) p (cid:0)(cid:18) p ϵ p and 2 3 (cid:21)Y t = 4 ∫ 1 ( P t (i) P t ) (cid:0)1(cid:0) (cid:18) (cid:18) p pϵp di 5 (cid:0) 1(cid:0) (cid:18) (cid:18) p pϵp : (245) P t 0 P t (cid:21)Y t Again, if ϵ = 0, the demand of each differentiate becomes p ( ) (cid:0)1+(cid:18)p Y (i)= P t (i) (cid:18)p Y (246) t P t t and the aggregate price index is [∫ ] 1 (cid:0)(cid:18)p (cid:0) 1 P t = P t (i) (cid:18)pdi : (247) 0 Timevariationinthemarkupcanbeintroducedbyreplacing(cid:18) with(cid:18) ,where(cid:18) followsanARMA(1,1) p p;t p;t process ( ) log((cid:18) p;t )= 1(cid:0)(cid:26) p log((cid:18) p )+(cid:26) p log((cid:18) p;t(cid:0)1 )+" p;t (cid:0)(cid:26) p " p;t(cid:0)1 (248) " is white noise following N(0;(cid:27)2). p;t p D.2.3 Fiscal and Monetary Policy Government budget is balanced with B P t G t +B t(cid:0)1 =T t + R t (249) t and G =(cid:24) Y (250) t G;t ss The government spending shock (cid:24) follows the stochastic process G;t ( ) ( ) log (cid:24) =(1(cid:0)(cid:26) )log(g )+(cid:26) log (cid:24) +(cid:26) log((cid:24) )(cid:0)(cid:26) log((cid:24) )+" : (251) G;t G y G G;t(cid:0)1 AG A;t AG A;t(cid:0)1 G;t " is white noise following N(0;(cid:27)2). where g is the government spending to GDP ratio in the steady G;t G y state. 83

Online Appendix D.2.4 Resources Constraint ∫ 1 Capital market clearingThemarketforcapitalclearsifthetotalamountdemandedby(cid:12)rms K (i)di 0 t equals the amount supplied by the households ∫ ∫ 1 1 K t (i)di=Z t K t(cid:0)1 (j)dj: (252) 0 0 LabormarketclearingTherelationshipbetweenlaborsupplyandaggregatelabordemandcanbestated as N =ΩlL : (253) t t t It can be shown that Ωl ⩾ 1 due to the concavity of the Kimball aggregator. Ωl is de(cid:12)ned implicitly by t t the above equation; see also Appendix B.1. Final product market clearing Demand for the (cid:12)nal product is Y t =C t +I t +G t +a(Z t )K t(cid:0)1 : (254) The (cid:12)nal product is purchased by households for consumption and investment and capital utilization, and by the government. Supply of the (cid:12)nal product is given by (cid:18) ΩyY =(cid:24) K(cid:11)((cid:13)tL )1(cid:0)(cid:11)(cid:0) p (cid:13)tK L : (255) t t L;t t t 1+(cid:18) ss ss p It can be shown that Ωy ⩾ 1 due to the concavity of the Kimball aggregator. Ωy is de(cid:12)ned implicitly by t t the above equation; see also Appendix B.1. Table 4 summarizes the parameters estimated for the CEE/SW model. Figure 4 plots the impulse responses of selected variables to a price and a wage markup shock under the optimal commitment policy, and in(cid:13)ation targeting, price level targeting and speed limit policy under discretion. 84

Online Appendix Table 4: Parameter values for CEE/SW model estimated with US data a:CalibratedandEstimatedParameters Parameter Description Value Parameter Description Value (cid:14) depreciationrate 0.025 (cid:20) invest. adjust. cost 5.48 ϵp Kimballelas. goods 10 ϵw Kimballelas. labor 10 gy s.s. G/Y 0.18 (cid:12) discountfactor 0.9984 (cid:13) tech. progress 1.0043 (cid:25)(cid:22) s.s. in(cid:13)ationrate 1.0081 (cid:27)C inversecons. elastic. 1.39 (cid:27)L inverselabor. elastic. 1.92 (cid:18)w s.s. netwagemarkup 0.5 (cid:18)p s.s. netpricemarkup 0.61 h habitpersistence 0.71 capitalutil. cost 0.54 (cid:24) pricestickiness 0.65 (cid:24) wagestickiness 0.73 p w (cid:19)p priceindexation 0.22 (cid:19)w wageindexation 0.59 !k capitalshare 0.19 !l laborshare 0.81 (cid:28)(cid:22)p pricesubsidies 0 (cid:28)(cid:22)w wagesubsidies 0 b:ParametersforShockProcess Shock AR(1) MA(1) Standarddeviation (%) Value technology (cid:26) A 0.95 - - (cid:27)A 0.45 riskpremium (cid:26) R 0.18 - - (cid:27)R 0.24 gov. spending (cid:26) G 0.97 (cid:26) AG 0.52 (cid:27)G 0.52 invest. speci(cid:12)c (cid:26) I 0.71 - - (cid:27)I 0.45 pricemarkup (cid:26) p 0.90 (cid:26) p" 0.74 (cid:27)p 0.14 wagemarkup (cid:26) w 0.97 (cid:26) w" 0.88 (cid:27)w 0.24 85

Online Appendix Table 5: Parameter values for CEE/SW model estimated with euro area data a:CalibratedandEstimatedParameters Parameter Description Value Parameter Description Value (cid:14) depreciationrate 0.025 (cid:20) invest. adjust. cost 5.68 ϵp Kimballelas. goods 10 ϵw Kimballelas. labor 10 gy s.s. G/Y 0.18 (cid:12) discountfactor 0.9977 (cid:13) tech. progress 1.0039 (cid:25)(cid:22) s.s. in(cid:13)ationrate 1.0068 (cid:27)C inversecons. elastic. 1.32 (cid:27)L inverselabor. elastic. 2.60 (cid:18)w s.snetwagemarkup 0.5 (cid:18)p s.s. netpricemarkup 0.77 h habitpersistence 0.72 capitalutil. cost 0.24 (cid:24) pricestickiness 0.64 (cid:24) wagestickiness 0.75 p w (cid:19)p priceindexation 0.128 (cid:19)w wageindexation 0.374 !k capitalshare 0.16 !l laborshare 0.84 (cid:28)(cid:22)p pricesubsidies 0 (cid:28)(cid:22)w wagesubsidies 0 b:ParametersforShockProcess Shock AR(1) MA(1) Standarddeviation (%) Value technology (cid:26) A 0.99 - - (cid:27)A 0.27 riskpremium (cid:26) R 0.69 - - (cid:27)R 0.10 gov. spending (cid:26) G 0.99 (cid:26) AG 0.39 (cid:27)G 0.29 invest. speci(cid:12)c (cid:26) I 0.12 - - (cid:27)I 0.55 pricemarkup (cid:26) p 0.99 (cid:26) p" 0.92 (cid:27)p 0.16 wagemarkup (cid:26) w 0.98 (cid:26) w" 0.84 (cid:27)w 0.15 86