NOTES D’ÉTUDES

ET DE RECHERCHE

DIRECTION GÉNÉRALE DES ÉTUDES ET DES RELATIONS INTERNATIONALES

CENTRAL BANK REPUTATION IN

A FORWARD-LOOKING MODEL

Olivier Loisel

June 2005

NER - R # 127

DIRECTION GÉNÉRALE DES ÉTUDES ET DES RELATIONS INTERNATIONALES DIRECTION DES ÉTUDES ÉCONOMIQUES ET DE LA RECHERCHE

CENTRAL BANK REPUTATION IN

A FORWARD-LOOKING MODEL

Olivier Loisel

June 2005

NER - R # 127

Les Notes d'Études et de Recherche reflètent les idées personnelles de leurs auteurs et n'expriment pas nécessairement la position de la Banque de France. Ce document est disponible sur le site internet de la Banque de France « www.banque-France.fr ». The Working Paper Series reflect the opinions of the authors and do not necessarily express the views of the Banque de France. This document is available on the Banque de France Website “www.banque-France.fr”.

Central bank reputation in a forward-looking modelOlivier Loisel�

April 2005

�Banque de France and CEPREMAP, Paris, France. Address: Banque de France, DGEI-DEER-SEPMF-41-1422,39 rue Croix-des-Petits-Champs, 75049 Paris Cédex 01, France. Phone number: +33 1 42 92 49 42. Fax number:+33 1 42 92 62 92. Email address: olivier.loisel@banque-france.fr. The views expressed in this paper are those ofthe author and do not necessarily re�ect the position of the Banque de France. I thank Laurent Clerc, CarolineJardet, Julien Matheron, Henri Pagès and Marc-Olivier Strauss-Kahn for helpful comments and suggestions. Allremaining errors would be mine.

1

Résumé:Ce papier examine si des préoccupations de réputation peuvent amener la banque centrale àmettre en �uvre la politique monétaire optimale temporellement incohérente dans un modèlenéo-keynésien standard. La nature prospective de ce modèle est à cet égard intéressante à doubletitre: d�une part, elle accentue l�incohérence temporelle de la politique monétaire optimale enajoutant un biais de stabilisation à l�éventuel biais d�in�ation; d�autre part, elle permet de modél-iser la réputation de la banque centrale de manière plus satisfaisante en expliquant la coordinationdes agents privés sur la durée de punition. Nos résultats suggèrent que les biais d�in�ation et destabilisation peuvent être surmontés pour toutes les calibrations utilisées dans la littérature. Cesrésultats permettent d�endogénéiser la perspective atemporelle de Woodford et tendent à mettreen doute le bien-fondé de récentes propositions de délégation de politique monétaire.

Mots-clefs: biais de stabilisation, biais d�in�ation, discrétion, engagement, perspective atem-porelle, réputation.

Abstract:This paper examines whether reputation concerns can induce the central bank to implement thetime-inconsistent optimal monetary policy in a standard New Keynesian model. The forward-looking nature of this model is in this respect interesting on two accounts: �rst, it worsens thetime-inconsistency problem of optimal monetary policy by adding a stabilization bias to the possiblein�ation bias; second, it enables us to model more satisfactorily the reputation of the central bankby accounting for the coordination of the private agents on the punishment length. Our resultssuggest that the in�ation bias and the stabilization bias can be overcome for the calibrations used inthe literature. These results enable us to endogenize Woodford�s timeless perspective and weakenthe case for monetary policy delegation.

Keywords: commitment, discretion, in�ation bias, reputation, stabilization bias, timeless per-spective.

JEL codes: E52, E58, E61.

2

Résumé non technique:

La politique monétaire optimale est connue pour être "temporellement incohérente" (time-inconsistent)sous certaines hypothèses depuis l�article pionnier de Kydland et Prescott (1977). La plus célèbreconséquence de ce problème d�incohérence temporelle est l�existence d�un biais d�in�ation, mis enévidence par Barro et Gordon (1983a), lorsque la banque centrale cherche à stabiliser la productionau-dessus de son niveau potentiel. Barro et Gordon (1983b) montrent cependant que des considéra-tions de réputation peuvent inciter la banque centrale à mettre en �uvre la politique monétaireoptimale sous une hypothèse simple de "mécanisme de punition" (grim-trigger mechanism). Unesolution alternative pour surmonter ce biais d�in�ation, proposée par Rogo¤ (1985), consiste àdéléguer la politique monétaire à un banquier central conservateur. Quelle que soit la solutionretenue, le biais d�in�ation ne semble plus poser problème dans l�environnement actuel d�in�ationfaible.

Le problème d�incohérence temporelle de la politique monétaire optimale a cependant suscité unnouvel intérêt ces dernières années avec le développement des modèles néo-keynésiens. Commel�ont montré Clarida, Galí et Gertler (1999) et Woodford (1999), la nature "prospective" (forward-looking) de ces modèles donne en e¤et naissance à un autre biais, dit de stabilisation, en rendantla politique monétaire optimale temporellement incohérente même lorsque la banque centrale necherche pas à stabiliser la production au-dessus de son niveau potentiel. Ce biais de stabilisationprovient du fait que la politique monétaire optimale à une date donnée consiste à in�uencer lesanticipations des agents privés concernant la politique monétaire future de façon à faciliter lastabilisation du taux d�in�ation et de l�écart de production à la date courante �or cette politiquemonétaire future anticipée, qui permet la mise en �uvre de la politique monétaire optimale à ladate courante, ne coïncide pas avec la politique monétaire qui sera optimale aux dates futures(même en l�absence de nouveaux développements).

Un certain nombre de projets de délégation de la politique monétaire sont déjà apparus dans lalittérature académique pour remédier à l�apparition de ce nouveau biais. Il a ainsi été proposéd�introduire dans la fonction de perte assignée à la banque centrale un objectif de stabilisation duniveau des prix, un objectif de stabilisation de la croissance de la masse monétaire, un objectif destabilisation de la croissance de la production nominale ou encore un objectif de stabilisation dela variation d�écart de production. Mais à notre connaissance il n�existe encore à présent aucuneétude portant sur la réputation de la banque centrale dans les modèles néo-keynésiens. Une telleétude serait pourtant la bienvenue, puisque la délégation de la politique monétaire pourrait bienêtre inutile si le seul souci de sa réputation su¢ sait à ce que la banque centrale mette en �uvre lapolitique monétaire optimale.

Notre papier cherche à combler cette lacune en étudiant la réputation de la banque centrale dansun modèle prospectif, plus précisément dans un modèle néo-keynésien standard, choisi pour sapopularité et sa simplicité. Nous dé�nissons la réputation de la banque centrale comme sa capacitéà in�uencer les anticipations des agents privés, capacité qui dépend de la politique monétaire passéesous une hypothèse simple de mécanisme de punition (i.e. la crédibilité se gagne en joignant le gesteà la parole). Nous montrons notamment que le biais de stabilisation réduit le bien-être d�une façonnon négligeable (puisqu�il équivaut à une augmentation permanente du taux d�in�ation allantde 0,26 à 1,23 points de pourcentage), mais qu�il peut être surmonté par des considérations deréputation pour toutes les calibrations considérées dans la littérature. Ce résultat nous amène enparticulier à mettre en doute le bien-fondé des récentes propositions de délégation de politiquemonétaire évoquées ci-dessus.

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Non-technical summary:

Optimal monetary policy is known to be time-inconsistent under certain assumptions since the sem-inal work of Kydland and Prescott (1977). The best known consequence of this time-inconsistencyproblem is Barro and Gordon�s (1983a) in�ation bias which arises when the central bank seeks tostabilize output above its potential level. Barro and Gordon (1983b) show however that reputationconsiderations can then make the optimal monetary policy sustainable under a simple grim-triggermechanism assumption. An alternative way to overcome this in�ation bias, proposed by Rogo¤(1985), is to delegate monetary policy to a conservative central banker. Whatever the solutionimplemented, the in�ation bias is arguably no longer a relevant issue in the current low in�ationenvironment.

The time-inconsistency problem of optimal monetary policy has however aroused renewed interestin recent years with the development of New Keynesian models. As shown by Clarida, Galí andGertler (1999) and Woodford (1999), the forward-looking nature of these models gives indeed riseto a new bias, called the stabilization bias, by making optimal monetary policy time-inconsistenteven when the central bank does not seek to stabilize output above its potential level. Moreprecisely, in these models the optimal current monetary policy requires to raise some expectationsabout the future monetary policy (in order to facilitate the stabilization of the economy in thepresent) which the central bank will however have no incentive to validate subsequently � even inthe absence of new developments in the meantime.

The literature has already come up with a number of monetary policy delegation schemes asremedies for this stabilization bias. It has thus been proposed to introduce into the loss functionassigned to the central bank a price level stabilization objective, a money growth stabilizationobjective, a nominal income growth stabilization objective or an output gap change stabilizationobjective. But to our knowledge there is so far no study on central bank reputation in NewKeynesian models. Such a study would however be welcome, since monetary policy delegationmay well be useless if reputation concerns alone can induce the central bank to implement thetime-inconsistent optimal monetary policy.

This paper aims at �lling this gap in the literature by considering the issue of central bank reputa-tion in a forward-looking model, namely a standard New Keynesian model chosen for its popularityand analytical tractability. We de�ne the reputation of the central bank as its ability to in�uencethe private agents� expectations, which depends on its monetary policy record under a simplegrim-trigger mechanism assumption (i.e. credibility is gained by matching deeds with words). Wenotably show that the stabilization bias reduces social welfare in a non-negligible way (as muchas a permanent increase in the in�ation rate of 0,26 to 1,23 percentage points), but that it can beovercome by reputation considerations for all the calibrations considered in the literature. Thisresult weakens the case for monetary policy delegation.

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1 Introduction

Optimal monetary policy is known to be time-inconsistent under certain assumptions since the

seminal work of Kydland and Prescott (1977). Under these assumptions the central bank is there-

fore doomed to implement the suboptimal discretionary equilibrium if it cannot credibly commit

to implementing the optimal monetary policy when the private agents have rational expectations.

The best known consequence of this time-inconsistency problem is Barro and Gordon�s (1983a)

in�ation bias which arises when the central bank seeks to stabilize output above its potential level.

Barro and Gordon (1983b) show however that reputation considerations can then make the optimal

monetary policy sustainable under a simple grim-trigger mechanism assumption. An alternative

way to overcome this in�ation bias, proposed by Rogo¤ (1985), is to delegate monetary policy to

a conservative central banker. Now whether because they are conservative or concerned for their

reputation, nowadays central bankers do not seem in practice to aim at stabilizing output above

its potential level, as observed by Blinder (1997), so that the in�ation bias is arguably no longer a

relevant issue in the current low in�ation environment.

The time-inconsistency problem of optimal monetary policy has however aroused renewed in-

terest in recent years with the development of New Keynesian models. As shown by Clarida,

Galí and Gertler (1999) and Woodford (1999), the forward-looking nature of these models gives

indeed rise to a new bias, called the stabilization bias1 , by making optimal monetary policy time-

inconsistent even when the central bank does not seek to stabilize output above its potential level.

More precisely, in these models the optimal current monetary policy requires to raise some expect-

ations about the future monetary policy (in order to facilitate the stabilization of the economy in

the present) which the central bank will however have no incentive to validate subsequently � even

in the absence of new developments in the meantime.

The literature has already come up with a number of monetary policy delegation schemes as

remedies for this stabilization bias: Vestin (2000) proposes to introduce a price level stabilization

objective, Söderström (2001) a money growth stabilization objective, Jensen (2002b) a nominal

income growth stabilization objective, Walsh (2003b) an output gap change stabilization objective

and Svensson and Woodford (2005) a state-contingent linear in�ation contract, into the loss func-

tion assigned to the central bank. But to our knowledge there is so far no study on central bank

reputation in New Keynesian models. Such a study would however be welcome, since monetary

policy delegation may well be useless if reputation concerns alone can induce the central bank to

implement the time-inconsistent optimal monetary policy2 .

1The stabilization bias is presented by Dennis (2003) in a non-technical way and by Walsh (2003a, chapter 11)and Woodford (2003a, chapter 7) in a more technical but very pedagogical way.

2As acknowledged by Woodford (2003b, p. 885), monetary policy delegation can even be counterproductive fora similar reason: �Of course, the assignment to the central bank of an objective di¤erent from the true social loss

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This paper aims at �lling this gap in the literature by considering the issue of central bank

reputation in a forward-looking model, namely a standard New Keynesian model chosen for its

popularity and analytical tractability. We de�ne the reputation of the central bank as its ability

to in�uence the private agents�expectations, which depends on its monetary policy record under

a simple grim-trigger mechanism assumption (i.e. credibility is gained by matching deeds with

words). More precisely, we generalize Currie and Levine�s (1993, chapter 5) framework � itself an

extension of Barro and Gordon�s (1983b) framework to a dynamic stochastic model � to a �nite

punishment length. For simplicity, we assume that monetary policy is fully transparent so that

the private agents understand the central bank�s incentive to deviate from the optimal monetary

policy3 and can immediately detect such a deviation4 .

The consideration of grim-trigger mechanisms is particularly interesting in our standard New

Keynesian model for two reasons. First, numerical calibrations of this model enable us to conclude

unambiguously about the sustainability of the optimal monetary policy for a given punishment

length. By contrast, �qualitative models� usually lead to inconclusive results since the optimal

monetary policy (and more generally any time-inconsistent monetary policy superior to the discre-

tionary monetary policy) is necessarily non-sustainable when the discount factor is close enough

to zero and sustainable when the discount factor is close enough to one and the punishment is

long enough, as implied by the folk theorem. Second, the forward-looking nature of this model

is shown to facilitate greatly the coordination of the atomistic private sector on the punishment

length � except in the particular case of serially uncorrelated cost-push shocks. By contrast, this

coordination is usually left unexplained in non-forward-looking models.

The remaining of the article is organized as follows. Section 1 gives an overview of our framework

and methodology. Section 2 focuses on central bank reputation as a means to overcome the

in�ation bias. Though apparently no longer (if ever5) a cause for concern, the in�ation bias is

function, in the expectation that it will pursue that objective with discretion, is not the only possible approach tothe achievement of a desirable pattern of responses to disturbances. One defect of the �optimal delegation�approachconsidered here is that it presumes that the stationary Markov equilibrium associated with a particular distortedobjective will be realized. Yet there may well be other possible rational expectations equilibria consistent withdiscretionary optimization by the central bank, �reputational� equilibria in which the bank may do a better job ofminimizing the objective it has been assigned, but as a consequence bring about a pattern of responses that is lessdesirable from the point of view of the true social objective.� (Woodford�s emphasis.)

3The publication by the central bank of an explicit loss function with a numerical relative weight for the outputgap stabilization objective, along the lines set by Svensson (2003), would undoubtedly help the private agents tounderstand the central bank�s incentive to deviate from the optimal monetary policy.

4This assumption is more easily justi�ed in a rule-based policy-making framework along the lines set by Kydlandand Prescott (1977, p. 487): �In a democratic society, it is probably preferable that selected rules be simpleand easily understood, so it is obvious when a policymaker deviates from the policy�, and by Woodford (1999,p. 292): �A simple feedback rule would make it easy to describe the central bank�s likely future conduct withconsiderable precision, and veri�cation by the private sector of whether such a rule is actually being followedshould be straightforward as well.� In the real world, the monetary policy frameworks closest to transparentrule-based policy-making are currently those of the Reserve Bank of New Zealand and the Bank of Canada, whichpublish macroeconomic projections conditional on a future nominal interest rate path derived from a pre-determinedmonetary policy reaction function.

5This question is notably addressed by Ireland (1999), who argues that the in�ation bias can explain the behaviour

6

worth considering precisely as a way to test the relevance of our analysis by checking whether

reputation considerations can indeed overcome this bias in our framework. Section 3 focuses on

the main issue at stake, namely central bank reputation as a means to overcome the stabilization

bias. We then shortly conclude and provide a technical appendix.

2 Central bank reputation in a standard New Keynesianmodel

This section gives an overview of our framework and methodology.

2.1 A standard New Keynesian model

We consider a standard New Keynesian model with structural in�ation inertia developed by Wood-

ford (2003a)6 , whose reduced form (log-linearized around the steady state) is isomorphic to, and

includes as a particular case, that of the canonical New Keynesian model without structural in-

�ation inertia used notably by Clarida, Galí and Gertler (1999), Walsh (2003a) and Woodford

(2003a). For our purpose, this reduced form can be limited to a Phillips curve and a social loss

function.

The Phillips curve, derived from the �rms�pro�ts maximization programme, is written:

zt = � eEt fzt+1g+ �xt + ��t with zt � �t � �t�1,where �t denotes the in�ation rate and xt the output gap at date t, while eEt f:g stands for theprivate agents�expectation operator conditionally on the information available at date t, which

includes the past and present variables and shocks. Parameters �, , � and � are such that

0 < � < 1, 0 � � 1, � 2 f0; 1g and � > 0. This Phillips curve is forward-looking (both in

terms of the in�ation rate � and in terms of the quasi-di¤erenced in�ation rate z) because of the

underlying Calvo-type price-setting assumption, as �rms know that the price they choose today

will remain e¤ective for more than one period on average. When > 0 it is also backward-looking

(in terms of the in�ation rate �) because of the underlying assumption that non-optimally reset

prices are indexed to past in�ation, with parameter measuring the degree of indexation. The

exogenous cost-push disturbance �, which captures any factor altering the relationship between

real marginal costs and the output gap, is assumed to follow an autoregressive process of order

one: �t = ��t�1 + "t with 0 � � < 1, where " is a white noise of variance V" > 0. We assume that

of in�ation in the United States if one allows for time variation in the natural rate of unemployment.6Walsh (2005) uses a very close model, whose reduced form is identical to that of our model except for the

speci�cation of the cost-push shock stochastic process.

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the distribution of " is continuous, symmetric and bounded7 , that is to say that the set of possible

values for this shock is the real interval [�"; "] where " > 0.

The social loss function at date t, derived by Woodford (2003a, chapter 6) as negatively related

to the second-order approximation of the representative household�s utility function taken in the

neighbourhood of the steady state, is written:

Lt = Et

nX+1

k=0�kh(zt+k)

2+ � (xt+k � x�)2

io,

where Et f:g stands for the rational expectation operator conditionally on the information available

at date t, which includes the past and present variables and shocks, while parameters � and

x� are such that � > 0 and x� � 0. The presence of a nominal variable (namely the quasi-

di¤erenced in�ation rate) in this loss function comes from the fact that the absence of price-setting

synchronization entails relative price distortions which lead to an ine¢ cient sectoral allocation of

labour, even when the output gap is equal to zero. The case x� = 0 can be justi�ed by the existence

of structural policies o¤setting �rst-order distortions.

In what follows, we proceed as if the central bank controlled directly variables z and x to

minimize the social loss function subject to the Phillips curve8 . The central bank may minimize

the loss function under discretion (i.e. at each date) or under commitment (i.e. once and for

all). We note�zc;Tt ; xc;Tt ; Lc;Tt

�for t � T the value of (zt; xt; Lt) when the central bank optimizes

once and for all at a given date T ,�zdt ; x

dt ; L

dt

�for t 2 Z the value of (zt; xt; Lt) when the central

bank re-optimizes at each date and�ztpt ; x

tpt ; L

tpt

�for t 2 Z the value of (zt; xt; Lt) when the

central bank optimizes at each date from Woodford�s (1999) timeless perspective, that is to say

equivalently the value limT�!�1

�zc;Tt ; xc;Tt ; Lc;Tt

�, where �c; T�, �d�and �tp�stand respectively for

�commitment at date T�, �discretion�and �timeless perspective�. We also note�Lc; Ld; Ltp

�the

unconditional mean of�Lc;tt ; L

dt ; L

tpt

�for any date t 2 Z. Finally, when 6= 1 we measure the gain

from commitment by Jensen�s (2002b) �in�ation equivalent�, which corresponds to the value of

the permanent increase in the in�ation rate (expressed in percentage points) leading to an increase

in the unconditional mean of the loss function equal to Ld � Lc or Ld � Ltp, that is to say in our

context:

�c � 100p(1� �) (Ld � Lc)

1� and (if Ltp < Ld) �tp � 100p(1� �) (Ld � Ltp)

1� .

Whatever t 2 Z, Lc;tt is by construction the minimal value which the loss function Lt can take

(and in particular Lc;tt < Ldt ). This value may however not be achievable when a commitment

7The �nite distribution assumption, necessary for reputation concerns to overcome the stabilization bias in allsituations (as clear from subsection 3.2), is actually unavoidable in our log-linearized framework.

8 Interest rate rules implementing the discretionary equilibrium or the precommitment equilibrium can be easilydesigned from the missing IS equation to justify this procedure.

8

technology is lacking and when the private agents have rational expectations because the precom-

mitment equilibrium is time-inconsistent, so that the central bank would choose to (re-)commit

at each date if the private agents did wrongly trust each commitment. The next subsection ex-

amines how this time-inconsistency problem can be overcome when the central bank may credibly

re-commit at the prior cost of a temporary loss of reputation. Note �nally that the precommitment

equilibrium proves time-inconsistent when x� 6= 0 because of an in�ation bias à la Barro and Gor-

don (1983a) and when � = 1 because of a stabilization bias à la Clarida, Galí and Gertler (1999)

and Woodford (1999). We choose to consider these two biases alternatively since if it exists the

in�ation bias is likely to overshadow the stabilization bias. Thus section 2 focuses on the in�ation

bias by assuming � = 0 and x� 6= 0, while section 3 focuses on the stabilization bias by assuming

� = 1 and x� = 0.

2.2 Central bank reputation

In this subsection the private agents are assumed to behave according to a grim-trigger mechanism

such that once lost, the central bank�s reputation takesD periods to be restored, whereD 2 N� is an

exogenous parameter. More precisely, at each date t � T0 the private agents form their expectationeEt fzt+1g in accordance with the T0-commitment equilibrium if and only if this equilibrium has

been implemented from date T0 to date t included. Let T 00 denote the �rst date when the central

bank deviates from this equilibrium (if it does). By recurrence on j 2 N, we assume that if T 0jexists then: i) if D � 2 then at each date t 2

�T 0j ; :::; T

0j +D � 2

the private agents expect the

central bank to act in a discretionary way from date t + 1 to date T 0j + D � 1 included; ii) at

each date t 2�T 0j ; :::; T

0j +D � 1

the private agents expect the central bank to implement the

Tj+1-commitment equilibrium at date Tj+1 � T 0j +D; and iii) at each date t � Tj+1 the private

agents form their expectation eEt fzt+1g in accordance with the Tj+1-commitment equilibrium if

and only if this equilibrium has been implemented from date Tj+1 to date t included. Finally, T 0j+1

is then de�ned as the �rst date when the central bank deviates from this equilibrium (if it does).

Four points are worth noting at this stage. First, we need to resort to a recurrence on j 2 N to

de�ne this grim-trigger mechanism because in our dynamic model with a �nite punishment length,

the situation prevailing immediately after a punishment interval (i.e. at date Tj for a given j 2 N�)

di¤ers from the situation which would have prevailed at the same date had the central bank not

deviated from the Tj�1-commitment equilibrium. More precisely, we assume that at each date Tj

(if this date exists) for j 2 N� the meter is reset to zero in the sense that the central bank can

start to implement the Tj-commitment equilibrium, but cannot revert to the implementation of

the T0-commitment equilibrium9 . Second, the punishment equilibrium implemented from date T 0j9Alternative assumptions to examine the sustainability of the timeless perspective equilibrium (rather than that

9

to date T 0j+D�1 included (if these dates exist) for j 2 N will take into account the private agents�

expectations of the central bank�s re-commitment at date Tj+1 and will therefore not correspond to

the (permanent) discretionary equilibrium displayed in subsections 2.1 and 3.1. Third, the private

agents� �cynical� expectations during the punishment intervals (as opposed to their �trustful�

expectations outside these intervals) are rational, so that the central bank has no incentive to

surprise the private agents during these intervals. Fourth, because the private agents are assumed

not to be unionized, each atomistic private agent is �expectations-taker�so that this grim-trigger

mechanism is not strategically chosen but should rather be considered as a postulate about the

private sector�s behaviour, which explains why D is exogenous10 and why there is no possibility

of renegotiation during the punishment intervals between the private agents on the one hand and

the central bank on the other hand.

The timing of the model at each date t can be presented as follows: 1) shock "t occurs, which

is observed by the central bank and the private agents; 2) if 9j 2 N such that t = Tj or such that

the Tj-commitment equilibrium has been implemented from date Tj to date t � 1 included, then

the central bank decides whether to implement the Tj-commitment equilibrium or to deviate from

this equilibrium, and this decision is observed by the private agents; 3) the private agents form

their expectation eEt fzt+1g; 4) the central bank chooses zt and xt. As clear from this timing, we

assume for simplicity that the central bank and the private agents have an information set which

goes beyond the perfect knowledge of the structure of the model and the value of its parameters.

In particular, the central bank observes the realization of the current shock before deciding which

equilibrium to implement11 . Moreover, the private agents can adjust their expectation of the future

quasi-di¤erenced in�ation rate to the central bank�s decision12 , which implies that the central

bank may surprise the private agents only by disappointing their previous expectations (and not

by disappointing their current expectations), so that the cost to renege may be biased upwards.

Finally, the fact that zt and xt are chosen after the private agents form their expectation is a

of the precommitment equilibrium) could be formulated in exactly the same way except that the expressions �the Tj -commitment equilibrium�for j 2 N would be replaced by �the timeless perspective equilibrium�. In this framework,the necessary and su¢ cient condition for the in�ation bias to be overcome is found to be independent of parameterD and to be satis�ed for calibration 1, while the necessary and su¢ cient condition for the stabilization bias to beovercome proves not analytically determinable except in the particular case � = 0.10Even in appendix F where the private agents manage to coordinate on an endogenized value of D, this value

of D is not strategically chosen by the private agents.11Were steps 1 and 2 inverted, the condition for the stabilization bias to be overcome would no longer be analyt-

ically determinable when 0 < � � 12.

12This assumption may be better understood in a framework with an explicit IS equation. Indeed, the corres-ponding timing would then be the following one: 1) shock "t occurs, which is observed by the central bank and theprivate agents; 2) the central bank sets the nominal interest rate, which is observed by the private agents; 3) theprivate agents form their expectations about the future situation, set their prices, produce and consume. Jensen(2002a) also assumes that the private agents observe the central bank�s action before forming their expectation offuture in�ation. As argued by Walsh (2000, p. 249): �With most major central banks using a short-term interestrate to implement monetary policy, policy changes are immediately and widely noted in the press. Expectationsabout future in�ation can respond immediately to any change in policy, a¤ecting both current and future equilibria.This response has the potential to discipline an opportunistic central bank�.

10

natural consequence of the Calvo-type price-setting assumption, which implies that the current

quasi-di¤erenced in�ation rate depends on the private agents� expectation of the future quasi-

di¤erenced in�ation rate and not the other way round.

Now let us set T0 equal to 0 for simplicity, consider a given n 2 N and assume that the 0-

commitment equilibrium has been implemented from date 0 to date n � 1 included if n � 1.

At date n, the central bank either implements this equilibrium (option A) or deviates from this

equilibrium (option B). Let�z�n+k; x

�n+k

�for k 2 N denote the value of (zn+k; xn+k) with option

� 2 fA;Bg, so that the value taken by the loss function at date n with option � 2 fA;Bg is

L�n = En

�X+1

k=0�k��z�n+k

�2+ �

�x�n+k � x

��2��

. (1)

Let �nally�zd;Tt ; xd;Tt

�for t < T denote the value of (zt; xt) when from date t to date T � 1

included the private agents expect the central bank to act in a discretionary way from date t to

date T � 1 included and to implement the T -commitment equilibrium at date T . We then have in

particular:

�zAn ; x

An

�=�zc;0n ; xc;0n

�and�

zBn+k; xBn+k

�=�zd;n+Dn+k ; xd;n+Dn+k

�for 0 � k � D � 1.

In the absence of any commitment technology, the central bank seeks to minimize Lt at each date

t � 0. At date n in particular, it chooses the option � 2 fA;Bg which minimizes Ln. A necessary

and su¢ cient condition for the central bank never to deviate from the 0-commitment equilibrium

is that it chooses not to deviate in the most tempting situation. Given that LAn and LBn depend

only on ��1 and "i for 0 � i � n13 , this necessary and su¢ cient condition can be written in the

following way:

S � Supn, ��1 and "ifor 0 � i � n

�LAn � LBn

�� 0 and, if it exists, M � Max

n, ��1 and "ifor 0 � i � n

�LAn � LBn

�< 0.

Now let us de�ne bLAn as the value taken by LAn when zAn+k and xAn+k are arbitrarily replaced byzc;0n+k and x

c;0n+k respectively for k � 1 in equation (1), and bLBn as the value taken by LBn when zBn+k

13 Indeed, given the grim-trigger mechanism considered, at each date t � n the central bank implements eithera T -commitment equilibrium (where T � 0) or a punishment equilibrium. As shown by appendices B and C,the corresponding quasi-di¤erenced in�ation rates and output gaps depend only on ��1 and "i for 0 � i � t. As a

consequence,�zAn+k; x

An+k

�k�0

and�zBn+k; x

Bn+k

�k�0

depend only on ��1 and "i for 0 � i � n + k, so that LAn

and LBn depend only on ��1 and "i for 0 � i � n.

11

and xBn+k are arbitrarily replaced by zc;n+Dn+k and xc;n+Dn+k respectively for k � D in equation (1).

Let us further de�ne

bS � Supn, ��1 and "ifor 0 � i � n

�bLAn � bLBn � and, if it exists, cM � Maxn, ��1 and "ifor 0 � i � n

�bLAn � bLBn � .Appendix D shows the following equivalence:

(S � 0 and, if it exists, M < 0)()�bS � 0 and, if it exists, cM < 0

�, (2)

so that�bS � 0 and, if it exists, cM < 0

�is a necessary and su¢ cient condition for the central bank

never to deviate from the 0-commitment equilibrium, that is to say a necessary and su¢ cient

condition for the precommitment equilibrium to be a reputational equilibrium. This result: i)

greatly facilitates our analysis, since bS and cM are much easier to compute than S and M ; ii)

can easily be generalized to other dynamic stochastic models (whether forward-looking or not), as

clear from appendix D; and iii) improves on the existing literature about central bank reputation

in dynamic stochastic models, like Currie and Levine (1993, chapter 5) for instance, since this

literature typically avoids handling�zBn+k; x

Bn+k

�k�D by considering an in�nite punishment length.

Appendix E shows that �bS�D � 0 when � = 0 (as in section 2) or x� = 0 (as in section

3). As a consequence, the longer the punishment length D, the more deterred the central bank

from deviating from the precommitment equilibrium and thus the more likely the precommitment

equilibrium to be a reputational equilibrium. This conventional result ensures the unicity (but not

the existence) of D 2 N� such that the precommitment equilibrium is a reputational equilibrium

if and only if D � D, that is to say such that�bS � 0 and, if it exists, cM < 0

�() (D � D).

Now calibrating parameter D is a challenging task, which we wisely choose to circumvent. We

agree with Rogo¤ (1987, p. 151) that �it is not intuitively appealing to have a long or in�nite

punishment interval�. We argue moreover that the most plausible values for D are of the same

order as monetary policy committees�terms of o¢ ce, which are typically a few years long. As a

rule of thumb, we therefore decide that the precommitment equilibrium quali�es as a reputational

equilibrium if and only if D exists and is of the order of a few years.

3 Central bank reputation and the in�ation bias

This section focuses on the in�ation bias, that is to say on the case � = 0 and x� 6= 0.

12

3.1 The in�ation bias

In this subsection the private agents are assumed to have rational expectations, so that they form

the same expectations as the central bank: eEt f:g = Et f:g at all dates t 2 Z.We �rst determine the discretionary equilibrium, that is to say the equilibrium obtained when

at each date t 2 Z the central bank chooses zt and xt so as to minimize Lt subject to the Phillips

curve taken at date t. As shown in appendix A, the central bank then chooses the same trade-o¤

between a quasi-di¤erenced in�ation rate higher than 0 and an output gap lower than x� at each

date t:

zdt =��x�

�2 + � (1� �) and xdt =

� (1� �)x��2 + � (1� �) .

We then determine the precommitment equilibrium or more precisely (without any loss in

generality) the 0-commitment equilibrium, that is to say the equilibrium obtained when at date

0 the central bank chooses zt and xt for all t � 0 so as to minimize the loss function L0 subject

to the Phillips curve taken at all dates t � 0. As shown in appendix B, we obtain the following

results for t � 0:

zc;0t =� (1� !)!tx�

�and xc;0t = !t+1x�,

where ! 2 ]0; 1[ is a parameter depending on �, �, � given in appendix B.

Unlike the discretionary equilibrium, the precommitment equilibrium makes therefore the quasi-

di¤erenced in�ation rate and the output gap vary over time, and more precisely decrease exponen-

tially towards zero. This comes from the fact that the commitment technology enables the central

bank to trade o¤ not only between a quasi-di¤erenced in�ation rate higher than 0 and an output

gap lower than x� at date t, but also between the situation at date t and the future situations. By

lowering the private agents�expectations eEt fzt+1g = Et fzt+1g in the Phillips curve taken at datet, the central bank indeed improves the trade-o¤ between a value of zt higher than 0 and a value

of xt lower than x�. Of course, the precommitment equilibrium proves time-inconsistent since at

date t+ 1 the central bank faces the same optimization problem as at date t and has therefore no

incentive to choose a quasi-di¤erenced in�ation rate di¤erent from the quasi-di¤erenced in�ation

rate chosen at date t. This in�ation bias implies that the quasi-di¤erenced in�ation rate is higher

in the discretionary equilibrium than in the precommitment equilibrium:

zdt � zc;0t � ��2 (1� !)2 x�

� [�2 + � (1� �)] > 0 for t � 0.

The social loss function is easily shown to take the following values:

13

Ldt =�2�

��2 + �

�x�2

(1� �) [�2 + � (1� �)]2= Ld for t 2 Z,

Lc;00 =� (1� !)2

h�2 + � (1� �!)2

ix�2

(1� �)�2 (1� �!) (1� �!2) = Lc,

Lc;0t =

�1

1� � �!t+1 (2� !t)1� �!

��x�2 for t � 0,

Ltpt =�x�2

1� � = Ltp for t 2 Z.

Note �nally that Lc;0t is a strictly increasing function of t 2 N, so that Lc < Lc;0t < Ltp for

all t > 0. We naturally also have Lc < Ld and we �nd moreover that Ltp < Ld if and only if

�2 > 2�2 (1� �)+� (1� �)2, which implies that Ltp > Ld is obtained only for very unlikely values

of parameters �, � and �.

3.2 Central bank reputation

In this subsection the private agents are assumed to behave according to the grim-trigger mechan-

ism described in subsection 1.2. As shown in appendix C, we have for 0 � k � D � 1:

zd;n+Dn+k =

"�2 � �� (1� !)2

���

�2 + �

�D�k#�x�

� [�2 + � (1� �)] ,

xd;n+Dn+k =

"(1� �) + � (1� !)2

���

�2 + �

�D�k#�x�

�2 + � (1� �) . (3)

This result shows in particular the existence of a bijective relationship, for a given value of

(�; �; �; x�) and at each date n + k 2 fn; :::; n+D � 1g of the punishment interval, between

the quasi-di¤erenced in�ation rate zn+k = zd;n+Dn+k or the output gap xn+k = xd;n+Dn+k on the

one hand and the punishment length D on the other hand. The existence of this bijective re-

lationship, which is a consequence of the forward-looking nature of the model and the fact that

En�zBn+D

= En

nzc;n+Dn+D

o6= 0 (since x� 6= 0), would greatly facilitate the coordination of the

private agents on a given punishment length, should they know the value of all the model�s para-

meters except D. Indeed, they could then coordinate on a given value of D as soon as date n+ 1,

as shown by appendix F, because the observation of zn and xn would reveal to each private

agent some information about the beliefs of other private agents about the punishment length.

By contrast, in non-forward-looking models the observation of aggregate variables by each private

agent during the punishment interval cannot reveal anything about the beliefs of other private

agents about the punishment length, so that the coordination problem is more serious.

14

Appendix G shows that the precommitment equilibrium is a reputational equilibrium if and

only if

�2

(1� �)� +2�D+2�2 (1� !)2

[�2 + � (1� �)]2

"1�

��

�2 + �

�D#�

�1� �D

��2h�2 + � (1� �)2

i(1� �) [�2 + � (1� �)]2

+

�D (1� !)2h�2 + � (1� �!)2

i(1� �)� (1� �!) (1� �!2) +

�2��2 + �

�� (1� !)4

[�2 + � (1� �)]2�

��

�2 + �

�2D 2641� (�2+�)

2D

�D�2D

1� (�2+�)2

��2

375 . (4)

Two points are worth noting about this proposition. First, whether the precommitment equilib-

rium is a reputational equilibrium does not depend on the value of x�. This result comes from

the fact that the values taken by the loss function under commitment and under discretion are

both proportional to x�2. Second, as shown in appendix G, deviating from the precommitment

equilibrium becomes more tempting with time for the central bank, so that bS is attained onlyasymptotically (i.e. for n ! +1, which implies that cM does not exist). This result implies that

the precommitment equilibrium is a reputational equilibrium if and only if the timeless perspective

equilibrium is a reputational equilibrium.

The consideration of one calibration of (�; �; �) found in the literature (detailed in appendix

J) leads to the numerical results reported in table 1, where the notation NA stands for �non-

available�.

Table 1: numerical results for calibration 1.

No. ! �c �tp D

1a 0; 64 9; 44 9; 43 31b 0; 64 18; 88 18; 86 3

Three points are worth noting about these results. First, �c and �tp are very close to each other, so

that the timeless perspective equilibrium hardly reduces welfare compared to the precommitment

equilibrium. Second, �c and �tp are as high as 9,5% or even 19%, that is to say that the welfare

gain from commitment is very large. Third, D exists and is very small. More precisely, the

precommitment equilibrium proves a reputational equilibrium provided that the private agents

�punish� the central bank during at least three quarters only. This result suggests that today�s

apparent absence of in�ation bias can be explained by reputation considerations in our standard

New Keynesian model.

4 Central bank reputation and the stabilization bias

This section focuses on the stabilization bias, that is to say on the case � = 1 and x� = 0.

15

4.1 The stabilization bias

In this subsection the private agents are assumed to have rational expectations, so that they form

the same expectations as the central bank: eEt f:g = Et f:g at all dates t 2 Z.We �rst determine the discretionary equilibrium, that is to say the equilibrium obtained when

at each date t 2 Z the central bank chooses zt and xt so as to minimize Lt subject to the Phillips

curve taken at date t. As shown in appendix A, the central bank then faces a trade-o¤ between

a positive quasi-di¤erenced in�ation rate and a negative output gap following a positive cost-push

shock, and makes the current variables zt and xt depend only on the current disturbance �t:

zdt =�

�2 + � (1� ��)�t and xdt =

���2 + � (1� ��)�t.

We then determine the precommitment equilibrium or more precisely (without any loss in

generality) the 0-commitment equilibrium, that is to say the equilibrium obtained when at date 0

the central bank chooses the state-contingent values of zt and xt for all t � 0 so as to minimize

the loss function L0 subject to the Phillips curve taken at all dates t � 0. To that aim, we

follow the undetermined coe¢ cients method and specify the variables prior to optimization as the

following linear combinations of shocks14 : zt =P+1

j=0 aj;t"t�j and xt =P+1

j=0 bj;t"t�j for t � 0,

with (aj;t; bj;t) 2 R2 for (j; t) 2 N2. As shown in appendix B, we obtain the following results15

for t � 0:

zc;0t =! [(1� �) �t � (1� !) �t]

(1� ��!) (! � �) and xc;0t =�! (��t � !�t)

� (1� ��!) (! � �) ,

where �t �Pt

j=0 !j"t�j + !

t���1. Thus written, these results hold only in the case ! 6= �, but

they can easily be extended by continuity to the case ! = �.

Unlike the discretionary equilibrium, the precommitment equilibrium makes therefore the cur-

rent variables zt and xt depend not only on the current disturbance �t, but also on a linear combin-

ation of present and past shocks �t which di¤ers from �t when ! 6= �. This result is best understood

in the particular case = � = 0. In that case indeed, the discretionary equilibrium makes the

current variables �t and xt depend only on the current shock "t, while the precommitment equilib-

rium makes the current variables �t and xt depend not only on the current shock "t, but also on

past shocks "t�j for j � 1. This inertia or �history-dependence�of the precommitment equilibrium14Note that we allow the opportunist central bank to make the quasi-di¤erenced in�ation rate and the out-

put gap depend on the shocks occurred before the commitment date by considering (possibly time-variant) linearcombinations of the entire history of shocks.15These results imply that the response of the in�ation rate to a positive cost-push shock is hump-shaped if and

only if + � + ! > 2 (which requires in particular > 0 and � > 0), while the response of the output gap to anegative cost-push shock is hump-shaped if and only if �+ ! > 1 (which requires in particular � > 0). This makesour New Keynesian model less vulnerable to the lack-of-empirical-validity criticism which has been addressed tothe canonical New Keynesian model (corresponding to the particular case = � = 0) for its inability to match thehump-shaped responses of variables to shocks observed in the data.

16

comes from the fact that the commitment technology enables the central bank to spread the burden

of the adjustment to shocks over time: following a positive cost-push shock "t, the central bank

can trade o¤ not only between a higher in�ation rate and a lower output gap at date t, but also

between the situation at date t and the future situations. As shown by Clarida, Galí and Gertler

(1999), this trade-o¤ between the present and the future improves the trade-o¤ between output

and in�ation in the present, as an expected future in�ation term eEt f�t+1g = Et f�t+1g in-between� "t� and 0 o¤sets part of the e¤ect of the cost-push shock "t in the Phillips curve taken at date t.

Of course, the precommitment equilibrium proves time-inconsistent since at date t+ 1 the central

bank has no incentive to go on reacting to the bygone shock "t in this purely forward-looking

framework, and this time-inconsistency gives rise to the so-called stabilization bias.

The social loss function is easily shown to take the following values:

Ldt =���2 + �

� �(1� �) �2t + �V"

�(1� �) (1� ��2) [�2 + � (1� ��)]2

for t 2 Z,

Ld =���2 + �

�V"

(1� �) (1� �2) [�2 + � (1� ��)]2,

Lc;00 =!�(1� �) �20 + �V"

�(1� �) (1� ��2) (1� ��!)2

,

Lc =!V"

(1� �) (1� �2) (1� ��!)2,

Ltp =

��2 (1 + �!) + 2� (1� �) (1� !)

�!2V"

(1� �)� (1� �2) (1� !2) (1� �!) (1� ��!)2.

Finally, the unconditional mean of Lc;0t is easily shown to be a strictly increasing function of t 2 N,

so that we have Lc < EnLc;0t

o< Ltp for all t > 0, where E f:g represents the unconditional mean

operator. We naturally also have Lc < Ld and we �nd moreover that Ltp < Ld is obtained for all

calibrations reported in appendix J16 .

4.2 Central bank reputation

In this subsection the private agents are assumed to behave according to the grim-trigger mechan-

ism described in subsection 1.2. As shown in appendix C, we have for 0 � k � D � 1:

zd;n+Dn+k =

"1� �! (1� !)

1� ��!

����

�2 + �

�D�k# ��n+k�2 + � (1� ��) ,

xd;n+Dn+k = �"1� �! (1� !)

1� ��!

����

�2 + �

�D�k# ��n+k�2 + � (1� ��) .

16We also �nd that Ltp > Ld is theoretically possible, at least for some unlikely values of the parameters, inaccordance with Blake�s (2001) results.

17

Like in subsection 2.2, this result shows in particular the existence of a bijective relationship,

for a given value of (�; �; �; �) such that � 6= 0 and at each date n + k 2 fn; :::; n+D � 1g

of the punishment interval provided that �n+k 6= 0, between the quasi-di¤erenced in�ation rate

zn+k = zd;n+Dn+k or the output gap xn+k = xd;n+Dn+k on the one hand and the punishment length

D on the other hand. The existence of this bijective relationship, which is a consequence of the

forward-looking nature of the model and the fact that En�zBn+D

= En

nzc;n+Dn+D

o6= 0 (when

� 6= 0)17 , would greatly facilitate the coordination of the private agents on a given punishment

length, as shown by appendix F, should they initially ignore the value of D.

Appendix H shows that a necessary and su¢ cient condition when 0 � � � 12 and a su¢ cient

condition when 12 < � < 1 for the precommitment equilibrium to be a reputational equilibrium is

!"2

� (1� �) (1� !)V"+

�1� �D�1

�(1� �)

(1� �) (1� ��2) ����2 + �

�(1� ��!)2

(1 + �) [�2 + � (1� ��)]2 !

" 1� �D�1

1� � �

�21� �D�1�2D�2

1� ��2

!� 2�! (1� !)

1� ��!

����

�2 + �

�D�10B@1� (�2+�)

D�1

�D�1�D�1

1� �2+���

� �21� (�

2+�)D�1

�D�1

�D�1

1� (�2+�)��

1CA+�2!2 (1� !)2

(1� ��!)2����

�2 + �

�2D�20B@1� (�2+�)2D�2

�D�1�2D�2�2D�2

1� (�2+�)2

��2�2

� �21� (�

2+�)2D�2

�D�1�2D�2

1� (�2+�)2

��2

1CA375 . (5)

Two points are worth noting about this proposition. First, thus written this inequality is de�ned

only for � > 0, but it can easily be extended by continuity to the case � = 0. Second, as shown in

appendix H, if 0 � � � 12 then

bS is attained only asymptotically (i.e. for n! +1, which implies

that cM does not exist) and more precisely in the case of an in�nite sequence of shocks equal to

". Like in subsection 2.2, this result implies that the precommitment equilibrium is a reputational

equilibrium if and only if the timeless perspective equilibrium is a reputational equilibrium.

Now let us de�ne F as the function

]0; 1[� R� � R� � [0; 1[� N��!R

(�; �; �; �;D) 7�! F (�; �; �; �;D)

17The condition � 6= 0 is necessary and su¢ cient for EnnzBn+D

o= En

nzc;n+Dn+D

oto di¤er from zero in our

canonical New Keynesian model with (x�; �) = (0; 1) and under our grim-trigger mechanism assumption. But

EnnzBn+D

o= En

nzc;n+Dn+D

o6= 0 could also be obtained for � = 0 in the same model under an alternative grim-

trigger mechanism assumption (for instance if the equilibrium expected to be implemented immediately after thepunishment interval is the timeless perspective equilibrium instead of the precommitment equilibrium) or under thesame grim-trigger mechanism assumption in an alternative model (for instance the model with the same Phillipscurve and social loss function as ours except that the in�ation rate � would replace the quasi-di¤erenced in�ationrate z in the social loss function).

18

such that inequality (5) is equivalent to

"2

V"� F (�; �; �; �;D) .

In the particular case � = 0, we have

F (�; �; �; 0; D) =�2�1� �D�1

�� (1� !)2

(1� �) (�2 + �)! .

Appendix I determines function F�s variations and limits when � = 0, reported in table 2.

Table 2: function F�s variations and limits when � = 0.

Par. Left limit Variation Right limit

� lim��!0+

F (�; �; �; 0; D) = 0 @F@� > 0 lim

��!1�F (�; �; �; 0; D) = �2(D�1)

�2+�

� lim��!0+

F (�; �; �; 0; D) = 0 @F@� > 0 lim

��!+1F (�; �; �; 0; D) =

�2(1��D�1)1��

� lim��!0+

F (�; �; �; 0; D) =�2(1��D�1)

1��@F@� < 0 lim

��!+1F (�; �; �; 0; D) = 0

D F (�; �; �; 0; 1) = 0 �F�D > 0 lim

D�!+1F (�; �; �; 0; D) = �2�(1�!)2

(1��)(�2+�)!

These results are in accordance with conventional wisdom: the precommitment equilibrium is all

the more likely to be a reputational equilibrium as � is low18 and �, �, D are large, i.e. as the central

bank is patient and conservative, the short-run Phillips curve is steep (for given expectations)

and the punishment interval is long. In particular, the result limD�!+1

lim��!1�

F (�; �; �; 0; D) =

lim��!1�

limD�!+1

F (�; �; �; 0; D) = +1 is a direct consequence of the folk theorem, even though the

discount factor � also appears in the Phillips curve.

Let Dr 2 N� for r � 1 denote the value of D when "2

V"is equal to r. We focus on the following

cases: i) "2

V"= 1, corresponding to the limit case of a Dirac distribution; ii) "2

V"= 3, corresponding

to the uniform distribution; iii) "2

V"= 6, corresponding to an �isoceles triangle distribution�. The

consideration of various calibrations found in the literature (detailed in appendix J) then leads

to the numerical results reported in table 3, where the notation NA stands for �non-available�.

18We proceed here as if � were independent of the other parameters, that is to say as if the central bank sought tominimize an ad hoc loss function instead of the social loss function, in order to be able to de�ne the central bank�sdegree of conservatism and to assess its e¤ect of the central bank�s reputation. Appendix J makes clear howeverthat all the calibrations considered in this paper set � to its model-consistent value, so that the corresponding lossfunctions are the social loss functions.

19

Table 3: numerical results for calibrations 2-6.

No. ! �c �tp limD�!+1

F (�; �; �; �;D) D1 D3 D6

2a 0; 64 NA NA 16; 04 8 22 482b 0; 64 0; 26 0; 26 16; 04 8 22 482c 0; 64 0; 43 0; 43 16; 04 8 22 483 0; 66 1; 23 1; 22 13; 19 10 28 634 0; 51 0; 48 0; 48 24; 55 7 15 305 0; 64 NA NA 11; 66 � 14 � 35 � 776 0; 64 NA NA 10; 05 � 17 � 42 � 97

Three points are worth noting about these results. First, when they exist �c and �tp are very

close to each other, which implies that the timeless perspective equilibrium hardly reduces welfare

compared to the precommitment equilibrium. Second, when they exist �c and �tp can substantially

vary from one calibration to another, ranging from 0; 26% to 1; 23% for �c and from 0; 26% to

1; 22% for �tp, but they are always sizeable (except arguably for calibration 2b) so that the welfare

gain from commitment is never negligible. Third, whatever the calibration considered D1, D3 and

D6 exist and are respectively found in-between 7 and 17 quarters (134 year and 4

14 years), 15 and

42 quarters (3 34 years and 1012 years), 30 and 97 quarters (7

12 years and 24

14 years).

Though some of these �gures are clearly beyond the order of a few years, we argue that the

precommitment equilibrium should nonetheless qualify as a reputational equilibrium for three

reasons. First, if we limit ourselves to the calibrations for which the exact values of D1, D3 and

D6 are known (i.e. the calibrations 2-4, which set � � 12 ), then the upper bound falls from 17 to

10 quarters (from 414 to 212 years) for D1, from 42 to 28 quarters (from 10 12 to 7 years) for D3 and

from 97 to 63 quarters (from 24 14 to 1534 years) for D6. Second, the �isoceles triangle distribution�,

corresponding to D = D6, for which D takes its maximal values, and to a lesser extent the uniform

distribution, corresponding to D = D3, for which D takes intermediate values, might not be the

most relevant distributions to consider in our context. Indeed, for the stabilization bias to be

overcome we require that the central bank should prefer not to deviate even in the most tempting

situation, which corresponds to an in�nite sequence of shocks equal to ". Now the probability

that this situation should occur is equal to zero whatever the distribution considered, and any

situation nearby (such as a long sequence of shocks close to ") is all the more unlikely as the

probability of " being close to " is low. This implies that our condition for the stabilization bias to

be overcome is demanding for the uniform distribution and even more demanding for the �isoceles

triangle distribution�. Third, if in reality the private agents are not initially coordinated on a given

value of D, then our results underestimate the social cost of deviating from the precommitment

equilibrium and hence overestimate the value taken by Dr for any r � 1, as shown in appendix

20

F19 .

Conclusion

This paper examines whether reputation concerns can induce the central bank to implement the

time-inconsistent optimal monetary policy in a standard New Keynesian model. Our analysis

rests on a simple grim-trigger mechanism assumption in an in�nite-horizon repeated game with

complete information. This grim-trigger mechanism assumption is all the more relevant in our

framework as the forward-looking nature of our standard New Keynesian model greatly facilitates

the coordination of the private agents on the punishment length � except in the particular case

of serially uncorrelated cost-push shocks. Our results suggest that the in�ation bias and the

stabilization bias can be overcome for the calibrations used in the literature. These results enable

us to endogenize Woodford�s (1999) timeless perspective and tend to weaken the case for monetary

policy delegation shortly presented in the introduction of this article.

Examining the issue of central bank reputation in a dynamic and possibly stochastic model

with a �nite punishment length raises some practical di¢ culties. In this paper we have overcome

these di¢ culties for a standard New Keynesian model by focusing on the most tempting situation

for the central bank, thus determining a necessary and su¢ cient condition for the central bank

never to deviate from the precommitment equilibrium, rather than a necessary and su¢ cient

condition for the central bank not to deviate from this equilibrium in a given situation. Now

this method can easily be generalized to many other dynamic and possibly stochastic models for

which it should provide a useful indicator of whether to consider the discretionary equilibrium or

the timeless perspective equilibrium in the presence of a sizeable in�ation or stabilization bias. In

particular, to apply our method to the popular New Keynesian model with structural in�ation

inertia �rst considered by Clarida, Galí and Gertler (1999)20 would be interesting for two reasons:

�rst, because the stabilization bias may be even larger in this model than in our model with = 0,

as shown by Dennis and Söderström (2002), even though this bias disappears in the limit case of

a purely backward-looking Phillips curve; second, because the stabilization bias remains sizeable

19 In addition, Dr for any r � 1 is a¤ected by two assumptions about the timing of the model. On the onehand, our assumption that the central bank observes the current shock before deciding whether to deviate fromthe precommitment equilibrium tends to bias Dr for any r � 1 upwards, as clear from appendix H. On the otherhand, our assumption that the private agents can instantly observe and react to a deviation from the precommitmentequilibrium (as mentioned in subsection 1.2) tends to bias Dr for any r � 1 downwards.20We are referring to the model with a Phillips curve of the form �t = a eEt f�t+1g+ b�t�1 + cxt + �t (sometimes

called the �hybrid� New Keynesian Phillips curve for its partly forward-looking, partly backward-looking nature

in terms of the in�ation rate) and a social loss function of the form Lt = EtnP+1

k=0 �kh(�t+k)

2 + d (xt+k)2io,

where a, b, c and d are strictly positive real numbers. This work could be carried out only in the form of numericalcomputations (since no tractable analytical results would then be available) and would notably require the use of asoftware programme maximizing numerically a quadratic function (bLAn � bLBn ) of a large number of bounded variables("i 2 [�"; "] for 0 � i � n with n! +1).

21

when information and/or transmission lags21 are introduced into this model, as shown by Dennis

and Söderström (2002) and Lam and Pelgrin (2004)22 .

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21The presence of information lags in our framework would mean that the central bank and the private agentsobserve variables and shocks with delay, but the private agents could still observe the central bank�s decision (aboutwhether to deviate from the precommitment equilibrium) without delay if the central bank measures variables andshocks and follows its interest rate rule in a transparent way.22Depending on the calibration and on the nature of the lag(s) considered, �c ranges indeed from 0; 24% to 0; 86%

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Ld

�and (if Ltp < Ld)

tp � 100�1� Ltp

Ld

�. But the relevance of this measure of the gain from commitment can be questioned on two

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22

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theory and practice: essays in honour of Charles Goodhart edited by P. Mizen (Edward

Elgar, Cheltenham, UK, Northampton, US), Volume 1, 135-152.

Svensson, L. E. O. and M. Woodford, 2005, Implementing optimal policy through in�ation-forecast

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(University of Chicago Press, Chicago, US), forthcoming.

Vestin, D., 2000, Price-level targeting versus in�ation targeting in a forward-looking model, Sveriges

Riksbank Working Paper No. 106.

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Walsh, C. E., 2003a, Monetary theory and policy, second edition (MIT Press, Cambridge, US).

23

Walsh, C. E., 2003b, Speed limit policies: the output gap and optimal monetary policy, American

Economic Review 93, 265-278.

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Appendix

In this appendix, we adopt the conventionsP�1

j=0 (:) =P0

j=1 (:) = 0 and 00 = 1.

A Determination of the discretionary equilibrium

At each date t considered the central bank chooses zt and xt so as to minimize Lt subject to the

Phillips curve taken at date t. Since zt+k and xt+k for k � 1 will be chosen in the future and since

today�s choice of zt and xt will not in�uence tomorrow�s choice of zt+k and xt+k (as the model

is purely forward-looking in terms of the quasi-di¤erenced in�ation rate z), the private agents�

expectations Etfzt+kg and Etfxt+kg do not depend on the choice of zt and xt, so that the central

bank considers these expectations as given when minimizing Lt subject to the Phillips curve taken

at date t.

The �rst-order condition of the minimization programme at date t is �zt + �xt = �x�, from

which we derive Et fzt+1g = �2+��� zt � �x�

� � ��t� with the Phillips curve taken at date t. For

k � 1 similarly, the �rst-order condition of the minimization programme at date t + k taken in

expectations Et f:g is �Et fzt+kg+�Et fxt+kg = �x�, from which we derive the recurrence equation

Et fzt+k+1g = �2+��� Et fzt+kg� �x�

� � ��k�t� with the Phillips curve taken in expectations Et f:g at

date t+ k. These two equations lead in turn to

Et fzt+kg =

��2 + �

��

�k �zt �

��x�

�2 + � (1� �) ����t

�2 + � (1� ��)

�+

��x�

�2 + � (1� �) +���k�t

�2 + � (1� ��)

for k � 1. The solution to the optimization programme satis�es therefore

24

zt =��x�

�2 + � (1� �) +���t

�2 + � (1� ��) ,

since Lt would take an in�nite value otherwise. The condition �zt + �xt = �x� then leads to

xt =� (1� �)x��2 + � (1� �) �

���t�2 + � (1� ��) .

B Determination of the 0-commitment equilibrium

We follow the undetermined coe¢ cients method to solve analytically the central bank�s optimiza-

tion problem. Since the central bank has observed shocks "�i for i � 0 at date 023 , the variables

can be rewritten in the following way prior to the minimization of L0:

zk �Xk�1

j=0aj;k"k�j + gk and xk �

Xk�1

j=0bj;k"k�j + hk

for k � 0. We look for the coe¢ cients aj;k, bj;k, gk and hk for k � 0 and 0 � j � k � 1 which

minimize L0 subject to the Phillips curve considered at all dates, i.e. which minimize the following

Lagrangian:

E0

nX+1

k=0�kh(zk)

2+ � (xk � x�)2

io�X+1

k=0�k (zk � �Ek fzk+1g � �xk � ��k) .

The �rst-order conditions of the Lagrangian�s minimization with respect to a0;k for k � 1, aj;k for

k � 2 and j 2 f1; :::; k � 1g, bj;k for k � 1 and j 2 f0; :::; k � 1g, g0, gk for k � 1, hk for k � 0 can

be respectively written in the following way:

2�kV"a0;k � �k"k = 0 for k � 1,

2�kV"aj;k � �k"k�j + ��k�1"k�j = 0 for k � 2 and j 2 f1; :::; k � 1g ,

2�k�V"bj;k + ��k"k�j = 0 for k � 1 and j 2 f0; :::; k � 1g ,

2g0 � �0 = 0,

2�kgk � �k + ��k�1 = 0 for k � 1,

2�k� (hk � x�) + ��k = 0 for k � 0,

and the Phillips curve considered at all dates provides the following two additional equations:

�aj+1;k+1 � aj;k + �bj;k = ���j for k � 1 and j 2 f0; :::; k � 1g ,

�gk+1 � gk + �hk = ���k�0 for k � 0.23Similar computations show that the 0-commitment equilibrium would be unchanged under the alternative

assumption that the central bank has observed shocks "�i for i � 1 but not "0 when committing at date 0.

25

Let us note u � k � j, v � j, Au;v � aj;k and Bu;v � bj;k, so that Au;v and Bu;v characterize

respectively the responses of zu+v and xu+v to "u. Our eight equations are then equivalent to the

following systems of equations:

8<: �g0 + �h0 = �x�

�gk+1 + �hk+1 � �hk = 0�gk+1 � gk + �hk = ���k�0

for k � 0for k � 0

(6)

and

8<: �Au;0 + �Bu;0 = 0�Au;v+1 + �Bu;v+1 � �Bu;v = 0�Au;v+1 �Au;v + �Bu;v = ���v

for u � 1for u � 1 and v � 0for u � 1 and v � 0

. (7)

System (6) implies that the coe¢ cients gk satisfy the following equations:

��g1 ���2 + �

�g0 = ���x� � ���0,

��gk+2 ����+ �2 + �

�gk+1 + �gk = �� (1� �) �k�0 for k � 0.

The latter equation corresponds to a recurrence equation on the gk for k � 0. The corresponding

characteristic polynomial has three positive real roots �, ! and !0 with:

! ����+ �2 + �

��q(��+ �2 + �)

2 � 4��2

2��< 1,

!0 ����+ �2 + �

�+

q(��+ �2 + �)

2 � 4��2

2��> 1.

The general form of the solution to the recurrence equation is therefore gk = p1�k+p2!k+p3!0k for

k � 0, where (p1; p2; p3) 2 R3. Three equations are then needed to determine (p1; p2; p3). Two are

provided by the initial conditions ��g1���2 + �

�g0 = ���x�����0 and ��g2�

���+ �2 + �

�g1+

�g0 = �� (1� �) �0. The third one is simply p3 = 0 and comes from the fact that �!02 � 1, as can

be readily checked, so that no solution with p3 6= 0 would �t the bill as L0 would then be in�nite.

We thus eventually obtain the following solution for system (6):

gk =� (1� !)!kx�

�+�!�(1� �) �k � (1� !)!k

��0

(1� ��!) (! � �) for k � 0,

hk = !k+1x� +��!

��k+1 � !k+1

��0

� (1� ��!) (! � �) for k � 0.

The similarity between systems (6) and (7) enables us to derive the solution of system (7) from

the solution of system (6) in a straightforward way:

26

Au;v =�! [(1� �) �v � (1� !)!v ]

(1� ��!) (! � �) for u � 1 and v � 0,

Bu;v =��!

��v+1 � !v+1

�� (1� ��!) (! � �) for u � 1 and v � 0,

so that we eventually obtain the following results for k � 0:

zk =� (1� !)!kx�

�+�! [(1� �) �k � (1� !) �k]

(1� ��!) (! � �) ,

xk = !k+1x� +��! (��k � !�k)� (1� ��!) (! � �) .

where �k �Pk

j=0 !j"k�j + !

k���1.

C Determination of the punishment equilibrium

The central bank acts in a discretionary way from date n to date n + D � 1 included since it

cannot in�uence the private agents�expectations during this punishment interval. The �rst-order

condition of the minimization programme at date n+k for 0 � k � D�1 is �zn+k+�xn+k = �x�,

from which we derive zn+k =���2+�

eEn+k fzn+k+1g + ��x�

�2+� +���n+k�2+� with the Phillips curve taken

at date n+ k. The latter equation leads to

zn+k =

���

�2 + �

�D�k eEn+k fzn+Dg+ "1� � ��

�2 + �

�D�k#��x�

�2 + � (1� �)

+

"1�

����

�2 + �

�D�k# ���n+k�2 + � (1� ��)

for 0 � k � D � 1, so that by replacing eEn+k fzn+Dg byEn+k

nzc;n+Dn+D

o=� (1� !)x�

�+�!�D�k�n+k1� ��!

we eventually get for 0 � k � D � 1

zn+k =

"�2 � �� (1� !)2

���

�2 + �

�D�k#�x�

� [�2 + � (1� �)]

+

"1� �! (1� !)

1� ��!

����

�2 + �

�D�k# ���n+k�2 + � (1� ��)

and, with the �rst-order condition �zn+k + �xn+k = �x�,

27

xn+k =

"(1� �) + � (1� !)2

���

�2 + �

�D�k#�x�

�2 + � (1� �)

�"1� �! (1� !)

1� ��!

����

�2 + �

�D�k# ���n+k�2 + � (1� ��) .

D Proof of proposition (2)

Suppose that S � 0 and (if it exists) M < 0. The only admissible values for the rational ex-

pectations of�zAn+k; x

An+k

�k�1 and

�zBn+k; x

Bn+k

�k�D are then respectively

�zc;0n+k; x

c;0n+k

�k�1

and�zc;n+Dn+k ; xc;n+Dn+k

�k�D

. Therefore S = bS and (if they exist)M = cM , so that bS � 0 and (if it exists)cM < 0.

Now suppose that bS � 0 and (if it exists) cM < 0. Because Lc;n+Dn+D is the minimal value of

Ln+D, whatever���1, n, "0, ..., "n

�we have �DEn

nLc;n+Dn+D

o� �DEn fLn+Dg and hence

En

�X+1

k=D�k��zc;n+Dn+k

�2+ �

�xc;n+Dn+k � x�

�2��� En

nX+1

k=D�kh�zBn+k

�2+ �

�xBn+k � x�

�2io,

so that bLBn � LBn whatever�n, ��1, "0, ..., "n

�. The inequalities bS � 0 and (if it exists) cM < 0

imply therefore that

Supn, ��1 and "ifor 0 � i � n

�bLAn � LBn � � 0 and, if it exists, Maxn, ��1 and "ifor 0 � i � n

�bLAn � LBn � < 0. (8)

The central bank may alternatively at date n: not deviate from the 0-commitment equilibrium and

plan never to deviate from it (option A1); not deviate from this equilibrium and consider deviating

from it at a later date (option A2); deviate from this equilibrium (option B). Inequalities (8)

imply that Ln is lower with option A1 than with option B whatever�n, ��1, "0, ..., "n

�. Option

A2, which implies that option B might be chosen at a later date, is therefore not time-consistent

(i.e. not compatible with rational expectations), contrary to option A1. As a consequence, the

only admissible values for the rational expectations of�zAn+k; x

An+k

�k�1 are

�zc;0n+k; x

c;0n+k

�k�1

and

therefore LAn = bLAn whatever ���1, n, "0, ..., "n�. Inequalities (8) then imply that S � 0 and, if itexists, M < 0.

E Determination of the sign of �bS�D

when � = 0 or x� = 0

Whatever���1, n, "0, ..., "n

�, bLAn does not depend on D, while

28

bLBn = En

�XD�1

k=0�k��zd;n+Dn+k

�2+ �

�xd;n+Dn+k � x�

�2��+En

�X+1

k=D�k��zc;n+Dn+k

�2+ �

�xc;n+Dn+k

�2��=

�2 + �

�En

�XD�1

k=0�k�zd;n+Dn+k

�2�+ �DEn

nLc;n+Dn+D

o.

Let bLB0n denote the value which bLBn would take if the punishment length were equal to D0 � D+1:

bLB0n =�2 + �

�En

�XD�1

k=0�k�zd;n+D

0

n+k

�2�+ �DEn

��2 + �

�

�zd;n+D

0

n+D0

�2+ �Lc;n+D

0

n+D0

�.

Whatever���1, n, "0, ..., "n

�and k 2 f0; :::; D � 1g,

�zd;n+Dn+k

�2is an increasing function of D if

� = 0 or x� = 0, so that

�2 + �

�En

�XD�1

k=0�k�zd;n+Dn+k

�2�� �2 + �

�En

�XD�1

k=0�k�zd;n+D

0

n+k

�2�.

Moreover, Lc;n+Dn+D is the minimal value of Ln+D so that whatever���1, n, "0, ..., "n

�,

�DEn

nLc;n+Dn+D

o� �DEn

��2 + �

�

�zd;n+D

0

n+D0

�2+ �Lc;n+D

0

n+D0

�.

These two inequalities imply in turn that bLBn � bLB0n whatever���1, n, "0, ..., "n

�, so that bLAn � bLBn

is a decreasing function of D whatever���1, n, "0, ..., "n

�. As a consequence, bS is a decreasing

function of D, i.e. �bS�D � 0.

F Coordination of the private agents on a punishment length when� 6= 0

Let us write zd;n+Dn+k and xd;n+Dn+k for k 2 f0; :::; D � 1g in the following form:

zd;n+Dn+k = c1;n+k + c2;n+k�D�k and xd;n+Dn+k = d1;n+k + d2;n+k�

D�k, (9)

where � � �� [1� � (1� �)]�2 + �

,

c1;n+k ���

�2 + � (1� �)x� +

�

�2 + � (1� ��)��n+k,

c2;n+k ����2 (1� !)2

� [�2 + � (1� �)]x� � ��! (1� !)

(1� ��!) [�2 + � (1� ��)]��n+k,

d1;n+k �(1� �)�

�2 + � (1� �)x� � �

�2 + � (1� ��)��n+k,

d2;n+k ��� (1� !)2

�2 + � (1� �)x� +

��! (1� !)(1� ��!) [�2 + � (1� ��)]��n+k.

29

This result holds when the value of D is known by all private agents from the start. Now let us

consider the alternative case where each private agent i has his or her own initial belief Di about

the value of D. Let � denote the set of all private agents and �t the set of the private agents who

set their prices optimally at date t 2 Z. The distribution of Di across individuals i 2 � is assumed

to be exogenous and independent of the distribution (across individuals i 2 � for any date t 2 Z)

of the variable which takes the value 1 if i 2 �t and the value 0 if i =2 �t. If n denotes the �rst

date of the �rst punishment interval, then equation (9) for k = 0 becomes

zn = c1;n +c2;n#�n

Xi2�n

�Di and xn = d1;n +d2;n#�n

Xi2�n

�Di ,

that is to say equivalently at the �rst order with the law of large numbers

zn = c1;n +c2;n#�

Xi2�

�Di and xn = d1;n +d2;n#�

Xi2�

�Di ,

where # represents the cardinal of set (i.e. the number of elements of ).

We assume for simplicity that each private agent: i) observes the aggregate variables zn and xn

after prices are set at date n and before date n+1 (say, as o¢ cial statistics are publicly disclosed);

ii) believes that all other private agents are coordinated on a given value of D, whether this value

coincides or not with her initial belief24 . Since � 6= 0, the probability that (c2;n; d2;n) = (0; 0) is

equal to zero whatever the distribution of shock ". The observation of (zn; xn) therefore reveals

to each private agent j the value taken by 1#�

Pi2��Di , so that she deduces that all other private

agents are coordinated on

D� =

ln

�1#�

Pi2��Di

�ln�

,

which is well de�ned25 since � 2 ]0; 1[ when � 6= 0, and as an �expectations-taker� she will

therefore revise her initial belief Dj accordingly at the onset of period n + 1. This implies that

all private agents are coordinated on D� as soon as the second date of the punishment interval,

that is to say that the coordination problem is limited to the �rst date of the punishment interval.

Moreover, the aggregate variables behave as if all private agents were coordinated on D� from the

start: zn+k = zd;n+D�

n+k and xn+k = xd;n+D�

n+k for k 2 f0; :::; D� � 1g, while the increase in price24Assumption ii) requires in particular that each private agent does not observe the disaggregated variables, i.e.

the price and quantity of each di¤erentiated good (say, because of prohibitive data collection costs), since he or shecould otherwise learn from this observation the distribution of Di across individuals i 2 �. Without assumption ii),i.e. if each private agent were aware of the possibility of an initial general disagreement about the value of D, thecoordination problem would require a more complicated treatment which is beyond the scope of this paper.25We choose to disregard the problem raised by the fact that D� is generally not an integer, for the sake of

simplicity, on the ground that this problem is arti�cially due to the discrete nature of our model and would notarise in the continuous-time version of this model.

30

dispersion due the lack of coordination at date n adds the term �t�1 (1� �)2 Vn to the social loss

function Lt for t � n, where � is the proportion of prices not optimally reset at each date and

Vn = c22;nvari2�n��Di

�the empirical variance of the logarithm of the prices �optimally� reset

at date n, as can easily be shown from Woodford�s (2003a, chapter 6) analysis. This implies

that the results of the paper, obtained under the assumption that the value of D is known by all

private agents from the start, underestimate the social cost of deviation from the precommitment

equilibrium and hence overestimate D if in reality the private agents do not initially know the

value of D but instead behave as assumed in this appendix.

G Proof of proposition (4)

bLBn does not depend on n while bLAn = Lc;0n is a strictly increasing function of n, so that

bS = limn�!+1

�bLAn � bLBn � = �x�2

1� � ��2x�2

�2 [�2 + � (1� �)]2

"�2h�2 + � (1� �)2

i 1� �D1� �

+�2��2 + �

�� (1� !)4

���

�2 + �

�2D 1� (�2+�)2D�D�2D

1� (�2+�)2

��2

�2�D+2�2 (1� !)2"1�

��

�2 + �

�D##� �D

� (1� !)2h�2 + � (1� �!)2

ix�2

(1� �)�2 (1� �!) (1� �!2) .

Since bS is attained only asymptotically, cM does not exist. As a consequence, bS � 0 is a neces-

sary and su¢ cient condition for the precommitment equilibrium to be a reputational equilibrium.

Finally, bS � 0 is equivalent to �2

�2x�2bS � 0 which is inequality (4).

H Proof of proposition (5)

Expressing bLAn � bLBn as a function of ���1, n, "0, ..., "n�, we get:bLAn � bLBn = k1 (�n � �n)2 + k2 (�n)2 + k3

and bS = Sup

��1 2i�"1�� ;

"1��

h, n 2 N,

"i 2 [�"; "] for 0 � i � n

hk1 (�n � �n)

2+ k2 (�n)

2+ k3

i,

31

where k1 =!2 (1� !)

(! � �)2 (1� ��!)2,

k2 =

�1� �D�2D

�!

(1� ��2) (1� ��!)2�

��2 + �

��

[�2 + � (1� ��)]2XD�1

i=0�i�2iq2i

and k3 =��1� �D�1

�!V"

(1� �) (1� ��2) (1� ��!)2���2 + �

�V"

� (1� �2)XD�1

i=1�i�1� �2i

�q2i

with qi = 1��! (1� !)1� ��!

����

�2 + �

�D�ifor 0 � i � D � 1.

Now consider the upper bound bS � bS of function bLAn � bLBn when "n is arti�cially allowed to be

higher than " or lower than �":

bS � Sup

��1 2i�"1�� ;

"1��

h, n 2 N,

"n 2 R, "i 2 [�"; "]for 0 � i � n� 1 if n � 1

hk1 (�n � �n)

2+ k2 (�n)

2+ k3

i.

The value of "n 2 R maximizing bLAn � bLBn is "n = ���n�1 (so that �n = 0) because bLAn � bLBndepends on "n only via its term k2 (�n)

2 and because

k2 ����

�1� �D�2D

�! (1� !)

(�2 + �) (1� ��2) (1� ��!)2< 0

since qi � qD�1 =�2 + � (1� ��)(�2 + �) (1� ��!) > 0 for i 2 f0; :::; D � 1g .

Since k1 > 0, bS is attained for ��1 �! "1�� and (if n � 1) ("0; :::; "n�1) = ("; :::; ") or equivalently

��1 �! �"1�� and (if n � 1) ("0; :::; "n�1) = (�"; :::;�"), so that we eventually obtain:

bS = Supn2N

"k1(! � �)2 (1� !n)2 "2

(1� �)2 (1� !)2+ k3

#= k1

(! � �)2 "2

(1� �)2 (1� !)2+ k3

which is attained only asymptotically (for n �! +1). In the general case 0 � � < 1, we thus havebS � bS and cM < bS (if cM exists) since bS is attained only asymptotically, so that bS � 0 =) (bS � 0and cM < 0 if cM exists), i.e. bS � 0 is a su¢ cient condition for the precommitment equilibrium to

be a reputational equilibrium. In the speci�c case 0 � � � 12 , the value lim

n�!+1

����n�1

�= � �"

1��

of "n 2 R maximizing bLAn � bLBn belongs to [�"; "], so that bS = bS and therefore (bS � 0 and cM < 0

if cM exists) =) bS � 0, i.e. bS � 0 is a necessary condition for the precommitment equilibrium to

be a reputational equilibrium.

32

We have thus shown that bS � 0 is a necessary and su¢ cient condition when 0 � � � 12 and

a su¢ cient condition when 12 < � < 1 for the precommitment equilibrium to be a reputational

equilibrium. Finally, bS � 0 is easily shown to be equivalent to inequality (5).I Determination of function F�s variations and limits when � = 0

Equation ��!2 ����+ �2 + �

�! + � = 0 implies

@!

@�=�!2 (1� !)1� �!2 < 0,

@!

@�=

�2�!2� (1� �!2) < 0 and

@!

@�=(1� !) (1� �!)� (1� �!2) > 0,

from which we easily get

@F

@�> 0,

@F

@�=�2�1� �D�1

�� (1� !) [(1� �) + 1 + �!]

(1� �) (�2 + �)2 (1� �!2)> 0

and@F

@�= �

�2�1� �D�1

��2 (1� !)

���!2 (1� !) + 2�! + �2! + �2

�(1� �) (�2 + �)2 �! (1� �!2)

< 0.

Moreover, the limits

lim��!0+

! =�

�2 + �, lim��!1�

! =

��2 + 2�

��q(�2 + 2�)

2 � 4�2

2�, lim��!0+

! = 1 and lim��!+1

! = 1

imply respectively

lim��!0+

F (�; �; �;D) = 0, lim��!1�

F (�; �; �;D) =�2 (D � 1)�2 + �

,

lim��!0+

F (�; �; �;D) = 0 and lim��!+1

F (�; �; �;D) = 0.

Finally, the equivalences

! s��!+1

�

�2and ! s

��!0+

�

�2

imply respectively

lim��!+1

F (�; �; �;D) =�2�1� �D�1

�1� � and lim

��!0+F (�; �; �;D) =

�2�1� �D�1

�1� �

with l�Hôpital�s rule.

33

J Calibrations used in the literature

Table 4 presents a few calibrations of our standard New Keynesian model used in the literature,

for quarterly data with the in�ation rate measured as an annualized percentage. Most of them

are calibrations of the canonical New Keynesian model as they set to zero. We retain only the

calibrations with a model-consistent value of �, i.e. such that � = 4�� where � is the elasticity of

substitution between di¤erentiated goods26 . Most studies choose for � Rotemberg and Woodford�s

(1997) estimated value, roughly equal to 0; 10, and all studies use the value � = 8 taken from

Rotemberg and Woodford (1997) to derive � from �, except Aoki and Nikolov (2004) who implicitly

use the value � = 6.

Table 4: calibrations used in the literature.

No. Study � � � � pV" x�

1a Woodford (2003a, chapter 7) 0; 99 0; 10 0; 05 NE 0; 0 NE 0; 21b Woodford (2003a, chapter 7) 0; 99 0; 10 0; 05 NE 0; 5 NE 0; 2

2a�Woodford (2003a, chapter 7)Schaumburg and Tambalotti (2003)

0; 99 0; 10 0; 05 0; 00 � NS 0; 0

2b Adam and Billi (2004a, 2004b) 0; 99 0; 10 0; 05 0; 00 0; 0 0; 006 0; 02c Woodford (1999) 0; 99 0; 10 0; 05 0; 00 0; 0 0; 010 0; 03 Aoki and Nikolov (2004) 0; 99 0; 12 0; 08 0; 35 0; 0 0; 015 0; 04 Adam and Billi (2004a, 2004b) 0; 99 0; 23 0; 11 0; 36 0; 0 0; 007 0; 05 Schaumburg and Tambalotti (2003) 0; 99 0; 10 0; 05 0; 70 0; 0 NS 0; 06 Woodford (2003a, chapter 7) 0; 99 0; 10 0; 05 0; 80 0; 0 NS 0; 0

The notations NE, NS and � stand respectively for �non-existent, �non-speci�ed� and � 2

f0; 0; 0; 5; 0; 8; 1; 0g�.

26We therefore do not consider the calibrations used by Evans and Honkapohja (2002), McCallum and Nelson(2000), Vestin (2000) and Walsh (2003a, chapter 11; 2003b) in particular.

34

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February 2004. 108. E. Jondeau and M. Rockinger, “Optimal Portfolio Allocation Under Higher Moments,”

February 2004. 109. C. Bordes et L. Clerc, « Stabilité des prix et stratégie de politique monétaire unique », mars

2004. 110. N. Belorgey, R. Lecat et T. Maury, « Déterminants de la productivité par employé : une

évaluation empirique en données de panel », avril 2004. 111. T. Maury and B. Pluyaud, “The Breaks in per Capita Productivity Trends in a Number of

Industrial Countries,” April 2004.

112. G. Cette, J. Mairesse and Y. Kocoglu, “ICT Diffusion and Potential Output Growth,” April

2004. 113. L. Baudry, H. Le Bihan, P. Sevestre and S. Tarrieu, “Price Rigidity. Evidence from the

French CPI Micro-Data,” September 2004. 114. C. Bruneau, O. De Bandt and A. Flageollet, “Inflation and the Markup in the Euro Area,”

September 2004. 115. J.-S. Mésonnier and J.-P. Renne, “A Time-Varying “Natural” Rate of Interest for the Euro

Area,” September 2004. 116. G. Cette, J. Lopez and P.-S. Noual, “Investment in Information and Communication

Technologies: an Empirical Analysis,” October 2004. 117. J.-S. Mésonnier et J.-P. Renne, « Règle de Taylor et politique monétaire dans la zone euro »,

octobre 2004. 118. J.-G. Sahuc, “Partial Indexation, Trend Inflation, and the Hybrid Phillips Curve,” December

2004. 119. C. Loupias et B. Wigniolle, « Régime de retraite et chute de la natalité : évolution des mœurs

ou arbitrage micro-économique ? », décembre 2004. 120. C. Loupias and R. Ricart, “Price Setting in France: new Evidence from Survey Data,”

December 2004. 121. S. Avouyi-Dovi and J. Matheron, “Interactions between Business Cycles, Stock Markets

Cycles and Interest Rates: the Stylised Facts,” January 2005. 122. L. Bilke, “Break in the Mean and Persistence of Inflation: a Sectoral Analysis of French CPI,”

January 2005. 123. S. Avouyi-Dovi and J. Matheron, “Technology Shocks and Monetary Policy in an Estimated

Sticky Price Model of the US Economy,” April 2005. 124. M. Dupaigne, P. Fève and J. Matheron, “Technology Shock and Employement: Do We Really

Need DSGE Models with a Fall in Hours? ,” June 2005. 125. P. Fève and J. Matheron, “Can the Kydland-Prescott Model Pass the Cogley-Nason Test? ,”

June 2005. 126. S. Avouyi-Dovi and J. Matheron, “Technology Shocks and Monetary Policy in an Estimated

Sticky Price Model of the Euro Area,” June 2005. 127. O. Loisel, “Central Bank Reputation in a Forward-Looking Model,” June 2005. Pour tous commentaires ou demandes sur les Notes d'Études et de Recherche, contacter la bibliothèque du Centre de recherche à l'adresse suivante : For any comment or enquiries on the Notes d'Études et de Recherche, contact the library of the Centre de recherche at the following address :

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