Managing Expectations�

Robert G. King, Yang K. Lu and Ernesto S. Pasten

Boston University

May 2007; revised December 2007

Abstract

The idea that monetary policy is principally about "managing expectations"

has taken hold in central banks around the world. Discussions of expectations

management by central bankers, academics and by �nancial market partici-

pants frequently also include the idea that central bank credibility is imperfect.

We adapt a familiar macroeconomic model so as to discuss key concepts in

the area of expectations management. Our work also exempli�es a model con-

struction approach to analyzing the dynamics of announcements, actions and

credibility which we think makes feasible a wide range of future investigations

concerning the management of expectations.

Keywords: managing expectations, imperfect credibility, monetary policy

JEL codes: E3, E5, E6

�Prepared for the Universitat Berne-Swiss National Bank Conference honoring Ernst Bal-tensperger, June 8, 2007. We thank Huberto Ennis for many conversations about this topic, whichled to a stronger paper. We also thank Matthew Canzoneri, Harris Dellas, Andreas Fischer, BartLipman, Antonio Miralles, Leo Martinez, Pierre Sarte and Alex Wolman for useful comments and/ordiscussions. This research has bene�tted from presentations at Boston University, the Federal Re-serve Bank of Richmond, and the SNB/JMCB conference.

1

1 Introduction

It is increasingly standard for central bankers, �nancial market participants, and

academic researchers to describe management of expectations as central to mone-

tary policy. The idea of "managing expectations" arises in many speci�c contexts

including the rationale for particular policy actions, the optimal choice of the mon-

etary instrument, the form of central bank behavior during leadership transitions,

the desirable central bank response to frequent or unusual shocks, and the nature

of monetary policy necessary for disin�ation. Analysts who stress management of

expectations frequently also suggest that credibility for low and stable in�ation is

imperfect, making expectations management subtle.1

It is therefore desirable to have macroeconomic models in which active manage-

ment of expectations can be systematically examined. In this paper, we describe a

strategy for doing constructing such models. Adapting a simple and familiar model,

we use our strategy to exposit the relationship between policy announcements, beliefs,

and policy actions that we think is at the heart of the management of expectations.

The model construction approach re�ects our view is that there are �ve features crit-

ical to the interplay between expectations management and credibility2. We think

that private agents do not know the underlying nature of the central bank and, in

particular, whether it will take the actions necessary to produce low and stable in-

�ation in the longer run: we de�ne long-term credibility of the central bank as the

private sector�s likelihood that the central bank is of such a strong type. We think

that private agents also are uncertain about whether near-term central bank actions

1A recent �nancial market discussion sets the tone: �If monetary policy is managing expecta-tions, as the current doctrine has it. . . �Central Banking News, April 2006. A former central bankerand academic, Alan Blinder uses the terminolgy in a more speci�c context: �Forward-looking infor-mation helps condition (�manage�) expectations�Making Monetary Policy and Talking about it, apresentation at UC-GSB in March 2007.Phillip Hildebrand, vice chairman, SNB board has recently written that "Monetary policy could

therefore be de�ned as the process of managing expectations about the future path of the monetarypolicy instrument and the probable e¤ects of that path on the economy.�He also stresses the issuesthat concern us in this paper: �Monetary policy works primarily through expectations. Transparencyand credibility render monetary policy more e¤ective. However, they are no substitutes for action.If a credible central bank uses words with the explicit aim of substituting them for action, it willrisk losing credibility.�

2We distinguish between policy that manages expectations and policy that a¤ects the economyvia the coordination of expectations. In other work in progress, Lu and Pasten (2007a) explore how�scal policy can improve economic performance by coordinating expectations. Policy may also a¤ectthe incentives that agents have to accumulate information, as in Lu and Pasten (2007b).

2

will be those consistent with the central bank being of a strong type: we de�ne the

short-term credibility of the central bank as the private sector�s likelihood that the

central bank will take such actions. We think that short-term credibility is generally

higher than low-term credibility because of mimicking, the possibility that the central

bank will take the same near-term actions as a low-in�ation central bank even it is of

a weak type, i.e., even if low long-term in�ation will not be the outcome under its op-

timal behavior.3 We think that the announced in�ation plans of a strong central bank

will in�uence in�ation expectations and the central bank will use its plans to manage

in�ation expectations at a point in time and the evolution of its long-term credibility.

Finally, we think that in�ation plan announcements are subject to imitation by weak

central banks, so that such plans are not perfectly credible.

To display how these core ideas about expectations management can be introduced

into macroeconomic models and the consequences of doing so, we adapt a framework

familiar from the work of Kydland and Prescott [1977] and Barro and Gordon [1983]

on central bank behavior under commitment and discretion. This choice re�ects the

fact that the starting point for our research is work by Barro [1986], Backus and

Dri¢ ll [1985a,b], and Cukierman and Leviatan [1991], which studied variants of the

KPMG model in which policymaker type was assumed not observable. This prior re-

search considered two types of monetary authority � a strong type that was capable

of commitment and a weak type that was not �and focused on the interplay between

central bank behavior and private sector learning about policymaker type. In turn,

that analysis drew heavily on prior game-theoretic work by Kreps and Wilson [1992],

which had developed a basic model of reputation acquisition under imperfect informa-

tion about player type. In common with this prior work, we aim to produce models

that have a unique Markovian equilibrium, so that unambiguous predictions can be

made. However, to apply the Kreps and Wilson approach, prior research adopted

behavioral speci�cations which precluded analysis of important aspects of the man-

agement of expectations in monetary policy. For Barro [1986] and Backus and Dri¢ ll

[1985a,b], the strong monetary authority was essentially an automaton, interested

in securing zero in�ation, while their analysis focused understanding the interplay

3The idea that mimicking is important for the dynamics of credibility is stressed by Phelan [2006].That framework is poorly suited to our research focus in this paper, because the strong (low tax)policymaker is an automaton. However, with its focus on unobserved shifts in policymaker type �contrasting with our observed regime replacement resulting in an unknown new type policymaker �there are components of that model which we �nd very interesting directions for future work.

3

between weak authority mimicking and private sector beliefs. For Cukierman and

Leviatan [1991], the weak authority was an automaton, choosing in�ation unrelated

to the actions of the committed monetary authority and the beliefs of private agents.

Further, the Kreps and Wilson approach has important drawbacks as a framework for

the analysis of the interplay between expectation management and macroeconomic

activity, since it involves randomized strategies by the weak policymaker and does

not make interesting predictions about the evolution of a credibility state variable

when decisionmaker horizons are driven to an in�nite horizon limit..

We therefore adopt an alternative modeling strategy, in which the weak mone-

tary authority is subject to privately observable random discount factor shocks, i.e.,

shifts in the urgency attached to present as opposed to future conditions. Techni-

cally, this exogenous randomness serves as an alternative "convexi�cation" device

for model economies so that randomized strategies are not a feature of our analysis.

However, more basically, we think that it captures the idea that time-varying, ex-

ogenous factors in�uence the likelihood that a weak decisionmaker will behave in a

discretionary manner. Using this approach, we are able to exemplify core channels

of expectation management by a strong central bank in simple static models and

quantify expectations management e¤ects in more elaborate dynamic models.

Within a static model, we begin by showing that the strong central bank will

not choose zero in�ation in a setting of imperfect credibility, an idea stressed by

Cukierman and Leviatan [1991]. Instead, the strong central bank will choose to

announce and carry out a positive rate of in�ation, with the extent of this in�ation

bias depending on inversely on the extent of its short-term credibility. Further, in

our framework, the strong central bank will take into account the e¤ect that its

plan will have on private sector beliefs about what a weak central bank would do,

both in terms of the rate of in�ation that it would select if it opts to behave in

a discretionary manner and the likelihood that it would instead mimic the strong

central bank�s in�ation action.4 We also use this static model to discuss the role of

in�ation plan announcements: like Cukierman and Leviatan [1991], we view these

announcements as an essential element of central bank transparency which allows

for a strong central bank to bring about its desired management of expectations as

part of a signalling game with an imperfectly informed private sector. Although

4In the static model, we use a random �xed cost of deviation to stand-in for the random discountfactor in the full model.

4

signalling games with costless imitation � such as that by a weak central bank in

our framework �have not been much studied, we describe the natural equilibrium

selection approach and explain why it leads to the same outcome that we derive

using a more direct and familiar rational expectations approach. We conclude that

management of expectations is generally not possible without some signals and that

in�ation reports are the necessary signal in our setting.

We then extend the analysis to a setting in which the central bank has a very long

horizon. Central bank tenure is, however, random. Each period, replacement occurs

with a �xed probability: agents know whether there has been a replacement, if one

occurs, but do not know the type of a new central banker. Accordingly, our focus

is on how a strong central bank builds long-term credibility, e¢ ciently investing in

reputational capital. We �nd that a strong central bank delivers a path of in�ation

that declines toward zero through time, with expected in�ation also declining toward

the same level. Private agents recognize that there is a chance of a surprise in�ation by

a weak policy-maker. But, as time passes, it becomes less likely that a weak central

bank is in place (long-term credibility increases) and the likelihood of mimicking

stabilizes at a positive level. When a replacement event occur, the new central banker

again invests in credibility, either to ultimately secure zero in�ation or to burn it in

a surprise in�ation.

Our dynamic analysis yields results that are quite di¤erent from the earlier lit-

erature in its predictions about the dynamics of in�ation after a new policymaker is

introduced. In that work, the tenure of the central bank is a �xed length. Early in

the tenure, incentives for mimicking are highest and these are eroded as time passes,

with weak central banks being more likely to engage in discretionary behavior as time

passes, as stressed by Backus and Dri¢ ll [1985] and Barro [1986], but the likelihood

of a weak central bank being present lower. When the strong policymaker can choose

his in�ation rate, as in Cukierman and Leviatan [1991], the implication is that actual

and expected in�ation rises with the time that an apparently strong central bank

has been in o¢ ce. This contrasts with our model, but occurs for an understandable

reason: the increase of time-in-o¢ ce reduces the time left in o¢ ce in a model with a

�xed tenure.

A virtue of our framework is that it can be readily extended, which we illustrate

with two important example applications. First, it is sometimes hypothesized that

central banks misunderstood the linkage between in�ation and real activity during

5

the 1960s and 1970s, due to an incorrect expectations mechanism, resulting in mis-

management of economic activity. We illustrate how central bank behavior would

di¤er if incorrectly thought that long-run credibility was relevant for short-term in-

�ation expectations rather than short-term credibility. Second, temporary positive

oil price shocks have sometimes been suggested as a reason for lasting changes in the

in�ation rate, an idea investigated by Ball [1995] within a model that also features

learning about policymaker type. The idea is that such a shock can induce a weak

central bank to behave in a discretionary manner, revealing its type and leading to

a period of high in�ation until a new central bank occurs, as stressed by Ball. How-

ever, in his analysis, the strong type always chooses zero in�ation, as in Barro [1986]

and Backus and Dri¢ ll [1985a,b]. Our framework allows us to investigate how the

strong policymaker�s incentives for accommodation to energy shocks depend on ex-

tent of his credibility, in turn, how this response a¤ects the nature of private sector

learning about policymaker type. At high levels of credibility, we �nd that there is

little accommodation. However, when credibility is low, then the strong policymaker

faces a large shift in expected in�ation: private agents know that a weak central bank

could well be in place and that it may choose to in�ate if it places a lot of weight

on current circumstances. Therefore, to avoid a large fall in output, the optimizing

strong central bank will also choose a higher in�ation rate, partially accommodating

the shift in expected in�ation.

The organization of the discussion below is as follows. In section 2, we provide

an overview of key ideas in our work. In section 3, we discuss a simple static model

in which there are two types of central banks, with the present type unknown to the

public. In section 4, we provide an extension of that static model with endogenous

mimicking, with a random �xed cost of deviation standing in for the random discount

factor of the full model. In section 5, we brie�y consider the links between our rational

expectations approach and signalling games, explaining why our equilibrium is the

natural one from the perspective of modern game theory and indicating why a public

in�ation plan announcement report is a central part of a game-theoretic interpretation

of our rational expectations equilibrium. In section 6, we detail the dynamic model,

using it to explore time series behavior of credibility, in�ation, and real activity.

In section 7, we discuss extensions to the research involving incorrect expectations

functions and optimal response to shocks to the macroeconomy. In section 8, we

conclude and provide a discussion of desirable future lines of inquiry.

6

2 Commitment, discretion, plans and credibility

The original analysis of Kydland and Prescott [1997] focused on the debate over

"rules versus discretion" and on the "time inconsistency of optimal plans" within

economies in which private agents form expectations rationally. They stressed that

the use of optimal control theory � with its implicit assumption that the policymaker

could commit to an entire future path of policy � led to time inconsistency of optimal

plans if the policymaker actually operated in a discretionary setting, making decisions

on a period-by-period basis. Since that time, it has become standard to contrast

macroeconomic equilibrium under the alternative assumptions of commitment and

discretion.

Our analysis also focuses on the idea that policymakers can be of a strong type,

capable of commitment to following a plan, or a weak type, capable of engaging in

discretionary behavior. However, there are some particular features that are impor-

tant to detail. First, we view strong policymakers as formulating an in�ation plan

p at the start of each period within our discrete time model. This plan is assumed

to be observable to private agents, but we consider alterations of this assumption

below. Second, we view strong decisionmakers as always able to execute the plan,producing in�ation � = p at the end of the period. A weak decisionmaker is assumed

to state the same plan as a strong decisionmaker would �also a provisional restric-

tion that we consider further below �but to be able to depart from the plan if it is

subsequently in his self-interest. Third, we view the type of the policymaker, strong

or weak, as unobserved by the private sector. However, the in�ation action � reveals

useful information about policymaker type.

Each of the periods of our discrete time model is broken into four subperiods, as

in displayed in panel A of Table 1. Just prior to the start of the period, a policymaker

transition occurs: the existing policymaker is continued with probability � and there

is a replacement of the policymaker with probability 1 � �. The transition outcome

is a publicly observable. If replacement occurs, then there is strong policymaker with

probability and a weak one with probability 1� , but the type of the new policy-maker is not observed. At the start of the period, an in�ation plan p is formulated.

In the middle of the period, the private sector forms in�ation expectations e. In the

end of the period, the in�ation action � is chosen. Panel B of Table 1 also shows

the two types of central banks. The strong central bank always chooses an in�ation

7

action equal to the planned level: � = p. The weak bank may choose to deviate,

selecting � = d, or it may choose to mimic, selecting � = p.

2.1 Expectations and credibility

Our strong central bank is concerned with managing expectations: it therefore must

have a model of how expectations will respond to its in�ation action p and depend

on long-term credibility �. We call this the expectations function,

e(p; �): (1)

Ultimately, we require that this expectations function is a rational one: it must be

consistent with the information, technical, and behavioral structure of the economy.

It is frequently the case that discussions of the management of expectations involve

considerations of credibility. In our model, as stressed above, there are actually two

notions of credibility which are important; these notions are displayed in Table 2..

First, there is the likelihood that the decisionmaker at date t will actually choose

� = p: we denote this likelihood as and call it short-term credibility.

A central bank also must have a model of how its actions a¤ect its short-term

credibility: we call this the short-term credibility function,

(p; �);

and also impose a rational expectations requirement on this function below. Second,

there is the start-of-the-period likelihood that the central bank is of the strong type,

which we denote as � and call long-term credibility. This probability is a key state

variable of models of the variety that we study in our research on the management

of expectations.

We think that the distinction between short-term and long-term credibility is

an important one conceptually and empirically. In the framework that we develop

below, short-term credibility is relevant for date t expected in�ation and the Phillips

curve slope, while � would be relevant for longer-term expected in�ation and the term

structure of interest rates in extensions of the framework.

Bayes�law implies that long-term credibility evolves according to

8

�t+1 =

8><>:�(pt; �t) = �t= (pt; �t) if �t = pt and r = 0

0 if �t 6= pt and r = 0

if r = 1

9>=>; (2)

The notation �(p; �) is meant to suggest that long-term credibility is a type of capital

for the economy 5 To minimize notational clutter, we sometimes write this equation

as �0 = �(p; �) when considering the non-replacement dynamics. The form of (2)

emphasizes that a regime replacement event wipes out existing long-term credibility

and resets the value to .

2.2 Strong versus weak central banks

Strong and weak central banks will not di¤er in the momentary objective that they

have, but will in two other ways. First, a strong central bank is assumed to choose its

in�ation action before expectations are formed and a weak central bank to choose its

in�ation action after expectations are formed. Thus, behavioral di¤erences will arise

even in a static model. Second, strong and weak central banks are assumed to di¤er

in terms of how they discount the future. The strong bank has a �xed discount factor

of b, which is known to all participants in the economy. The weak central bank has

a �uctuating discount factor of �, which is independently and identically distributed

according to a continuous probability distribution function F (�) with support (0; b).

Each period, the weak central bank receives a draw of this discount factor, which

is private information. There are two reasons for this assumption about random

discount factors, one substantive and one technical.

We think that a weak monetary authority is one that does not have the inter-

nal organizational capital to announce and carry through an in�ation plan within

a period. When actual decision-makers that have such di¢ culties, they appear to

make trade-o¤s between the present and the future that are short-sighted and vary

sharply through time. Our assumption of a random discount factor accords with this

5In perhaps more familiar terms, letting � = s be the event in which the decision-maker is of thestrong and � = w be the event in which the type is weak, then Bayes�rule is that

pr(� = sj� = p) =pr(� = s)

pr(� = s) � pr(� = pj� = s) + pr(� = w) � pr(� = pj� = w)

with the strong decision-maker always takes the action p, so that pr(� = pj� = s) = 1.

9

observation, in that E(�) < b and � is stochastic. Our assumed distribution of the

discount factor is shown in Figure 1.6 There is a substantial likelihood that the weak

central bank greatly discounts the future, in that there is a 50% chance that � < :125.

On the technical side, the continuously distributed random discount factor permits

us to generate mimicking, short-term credibility, and expectations functions that

depend smoothly on the in�ation plan and the state of the economy. More speci�cally,

the continuous distribution of the discount factor means that a strong central bank

views expected in�ation as a smooth function of its in�ation plan, e(p; �), and takes

this into account in viewing how in�ation expectations depend on its plan. When it

optimizes, it chooses an in�ation plan that is optimal given the state of the economy,

p(�).

2.3 Central bank policymaking

As discussed above, there are two types of central banks in our economy, strong and

weak: each makes in�ation decisions so as to maximize the welfare objective w(�; e),

which depends positively on in�ation and negatively on expected in�ation.

2.3.1 Strong central bank policymaking

The strong monetary authority thus maximizes welfare taking into account the in�u-

ence that its in�ation plan has on the likelihood and nature of discretionary policy.

The Bellman equation for its decision problem is:

W (�t) = maxptfw(pt; et) + b�W (�t+1) + b(1� �) W ( ) (3)

+b(1� �)(1� )V ( )g6The discount factor is distributed according to a beta distribution, with support (0,b). There

are two parameters of the beta distribution, which can be adjusted to give it a wide variety of shapes.If these parameters are �1 and �2 then the beta density is

f(z; �1; �2) =�(�1 + �2)

�(�1)�(�2)z�1�1(1� z)�2�1

where � is the gamma function. An additional convenient property of the beta distribution is thatthere are simple analytical formulae for the moments: the mean is �1=(�1 + �2).In our analysis, the distribution above governs the standardized random variable z = �=b. Hence,

when we choose �1 = 1 and �2 = 5 so that the mean discount factor is b/6 or .16 since b = :96.

10

where the maximization takes place subject to

et = e(pt; �t)

t = (pt; �t)

�t+1 = �(pt; �t) = �t= t

which represent the rational formation of private sector in�ation expectations, the

short-term credibility associated with mimicking by the weak central bank, and the

evolution of long-term credibility.

In addition to the momentary objective, the right hand side of the Bellman equa-

tion includes the discounted values of (i) future welfare if the strong central bank is

not replaced, W (�t+1); (ii) future welfare if replacement by a future strong central

bank occurs, W ( ); and (iii) future welfare if replacement by a weak central bank

occurs, V ( ). This formulation re�ects the assumption that the policymaker cares

about the state of the economy under other future policymakers, of both his own and

other types.

The solution to this maximization includes a time-invariant decision rule p�(�)

for in�ation that takes into account the management of expectations and the evo-

lution of long-term credibility. Notice that the sole state variable of our model is

long-term credibility: this re�ects our restricted focus on "minimal state variable

equilibria" from a rational expectations modeling perspective, using the terminology

of McCallum [1983], and "Markov perfect equilibria" from a game theory perspec-

tive, using the terminology of Maskin and Tirole [2001]. Operationally, we construct

equilibria by backward induction, imposing the state variable restriction at each step.

When we discuss results of the in�nite horizon problem below, it is thus actually an

approximate dynamic program with a very long, �nite horizon.

2.3.2 Weak central bank policymaking

The weak central banker has two levels of his decision problem. First, he decides on

an optimal rate of in�ation to set if he behaves in a discretionary manner, d. Second,

he decides whether it is desirable to behave in a discretionary manner (� = 1) or to

mimic (� = 0).

The Bellman equation takes the form

11

V (�; �) = max�t;�t

f(1� �t)Mt + �Dtg (4)

Mt = [w(pt; et) + ��V (�t+1) + �(1� �) W ( ) + �(1� �)(1� )V ( )] (5)

Dt = [w(�t; et) + ��V (0) + �(1� �) W ( ) + �(1� �)(1� )V ( )]g (6)

with �t+1 = �t= (pt; �t). The �rst bracketed expression, Mt; captures the welfare

if the weak central banks mimics its strong counterpart by choosing �t = pt. The

second bracketed term,Dt, captures the welfare if it departs from the plan and chooses

�t 6= pt. Notice that by acting in this discretionary manner, the weak central bank

has zero credibility in the future unless a regime-replacement event occurs.

The right hand side of the Bellman equation contains expressions V (�), re�ect-

ing our interest in notational simplicity. We de�ne V (�) as EV (�; �), i.e., as the

expectation over the possible realizations of the discount factor.

There are several points to be made about the decision rules resulting from this

optimization problem. First, the optimal in�ation action by the weak central bank

simply involves maximizing momentary utility, w(�t; et): let�s call that optimal action

dt. Second, there is a critical discount factor, b�, such that the values (5) and (6) areequated, given by b� = [w(dt; et)� w(pt; et)]

�[V ( �t t)� V (0)]

(7)

The numerator of this expression is the one-period gain from behaving in a discre-

tionary manner, sometimes called the temptation, so that we de�ne Tt = w(dt; et)�w(pt; et). The denominator is the expected cost of being identi�ed as a decision-

maker of weak type, sometimes called the punishment, so that we de�ned Pt =

�[V ( �t t)�V (0)]. For values � < b�, then, the temptation exceeds the punishment and

it is optimal to have an in�ation rate higher than planned. For values � > b�, it isoptimal to carry out the in�ation plan.

More precisely, taking as given e and p, the weak monetary authority�s optimiza-

tion results in a cut-o¤ rule of the form

b�t = c(pt; et; t; �t); (8)

where we have included all of the determinants of this cuto¤ rule suggested by (7).

12

In terms of decision, then,

�t = dt and �t = 1 if �t � b�t�t = pt and �t = 0 if �t > b�t (9)

That is, the weak central bank decides on those situations in which to deviate from the

in�ation plan. In deciding on when to deviate and how to deviate from an announced

in�ation plan, weak central bank treats in�ation expectations as predetermined and,

more speci�cally, as a function of the state of the economy, et = e(�t):

2.4 Rational expectations equilibrium

We now indicate rational expectations restrictions and other consistency conditions in

our economy that apply for each date t in our economy. First, short-term credibility

is given by

(p; �) = �+ (1� �)m(p; �), (10)

where m is the probability that a weak central bank will mimic a strong one by

choosing � = p. That is: specifying a short-term credibility function (p; �) is

equivalent to specifying a mimicking function m(p; �):

Second, imposing one layer of rational expectations, expected in�ation is a credibility-

weighted average of the two possible outcomes for in�ation, the weak central bank�s

plan and the discretionary action by the weak central bank,

e(p; �) = (p; �) p+ (1� (p; �)) d(e(p; �)): (11)

Third, imposing another layer of rational expectations, the mimicking function

must be consistent with the actual likelihood of deviations, i.e.,

m(p; �) = 1� F (T

P)

where the temptation T and punishment P were discussed above.

These three conditions serve to determine three "unknown middle sub-period

equilibrium functions" e(p; �); (p; �) and m(p; �) as a rational expectations �xed

point. Given these functions, the optimization of the strong policymaker determines

p(�), which in turn implies equilibrium functions e(�), (�), and m(�) as well as the

13

transition rule, �t+1 = �t= (�t) in line with (2).

3 A basic model of expectations management

To exposit the core ideas about expectations management, we specialize to a simple

framework familiar from the early work of Kydland and Prescott [1977] and Barro

and Gordon [1983a] as well as textbook presentations like that of Walsh [2003]. In

this section, we introduce key aspects of this familiar model and then we study how

it works under the assumption that there is no mimicking, so that = �.

We assume that the central bank �strong or weak �maximizes a social welfare

function of the form (12)

u(�; x; e) = u� + !1[x� x�]� 12f�2 + !0[� � e]2 + !2[x� x�]2g (12)

That is, the central bank likes increases in output towards the e¢ cient level x� .It

dislikes departures of in�ation (�) and real activity (x) from �rst-best values of � = 0

and x = x� , as well as in�ation surprises The weights !i � 0 express the importanceof these latter three e¤ects, relative to in�ation e¤ects. The parameter u� captures

the level of welfare at the �rst best, but plays no substantive role in the analysis

below.

The private sector is simply described by

x = �(� � e); (13)

which links variations in real activity (x) to surprise in�ation ��e as in the analyses ofLucas [1973] and Fischer [1977]. This simple speci�cation is used in many discussions

of commitment and discretion in monetary policy, such as Persson and Tabellini

[1990].

Following the literature, we will treat in�ation as the decision variable for the cen-

tral bank. It is therefore convenient to work with a reduced form objective obtained

by combining (12) and (13),

w(�; e) = u� + !1[�(� � e)� x�] (14)

�12f�2 + !0[� � e]2 + !2[�(� � e)� x�]2g:

14

as we did in section 2 above. To maximize this reduced form objective, it is frequently

convenient to use the following �rst order condition,

�� + (1��)[�!1 � (!0 + !2�2)(� � e) + �!2x

�] = 0 (15)

where� is the partial derivative of expected in�ation e with respect to actual in�ation

�, � = @e=@�, when a particular expectations function has been imposed.

3.1 Standard analysis of commitment and discretion

We begin with a quick discussion of standard models of policy under commitment

(a strong central bank) and discretion (a weak central bank), by way of review and

to highlight aspects of expectations management in each case. Our models are very

simple, without any real or nominal shocks, so as to focus attention on the interplay

of central bank plans and actions with outcomes for real activity.

3.1.1 Commitment

A strong central bank can perfectly manage expectations, since it is a large player

and private agents care only about forecasting the central bank�s action. In terms of

the framework above, then, � = 1 and the �rst order condition dictates that � = 0.

In our context, we say that the strong central bank formulates an in�ation plan of

p = 0 at the start of the period, that expectations are thus perfectly managed so that

e = 0, and that actual in�ation is � = 0 since the central bank executes the plan.7

3.1.2 Discretion

A weak central bank takes its action after the private sector, so that it cannot manage

expectations at all (� = 0). Using (15), we �nd that the central bank�s decision rule

7

With zero actual and expected in�ation, the real activity measure xt is also zero and the welfaremeasure under commitment is

w(0; 0) = u� � !1x� �1

2f!2[x�]2g:

so that there is the level of utility under commitment is less than u� due to real distortions presentin the economy that the monetary authority cannot eliminate.

15

is

d = d(e) ='

1 + 'e+

1

1 + 'B (16)

with ' = !0 + !2�2 and B = �!1 + �!2B. Thus, the discretionary in�ation policy

it partially accommodates expectations of in�ation and partly tries to stimulate the

economy because output is ine¢ ciently low, as stressed by Kydland and Prescott

[1977] and Barro and Gordon [1983a].

Imposing rational expectations (e = d), the equilibrium discretionary in�ation

rate is given by

d = !1�+ !2�x� = B (17)

where the compound parameter B represents the extent of the celebrated "in�ation

bias under discretion".

Under pure discretion, there is no sense in which the in�ation plan is relevant to

either the private sector or the monetary authority: it could be anything.8

3.2 Uncertainty about commitment or discretion

We are interested in environments in which private agents do not know the type

of the central bank �whether it is strong or weak �when they form expectations.

If a strong central bank is in place at the start of the period, then it will choose

an in�ation plan p knowing that it will subsequently carry through on this plan as

its in�ation action. In line with our discussion in section 2, we analyze this model

by working backwards from the actions of a weak central bank, should one exist,

choosing in�ation after expectations have been formed. Since there is no mimicking,

long-term and short-term credibility are identical: we use t rather than �t in this

section because that is the better interpretation in fully dynamic extensions of our

analysis.

8Moreover, with in�ation accurately forecasted in equilibrium, there are no e¤ects on real eco-nomic activity (x = 0) and the level of welfare is

w(B;B) = u� � !1x� �1

2fB2 + !2[x�]2g:

As stressed in the literature, welfare in equilibrium is consequently lower under discretion than undercommitment due to the cost of fully anticipated in�ation.

16

3.2.1 End: The weak central bank

At the end of the period, the weak central bank takes its action after expectations

are formed, so that � = 0 as in the analysis of pure discretion above. Using the

�rst order condition, we can determine that the weak central bank�s decision rule is

unchanged from (16) above. That is, the weak central bank�s action continues to be

given by (16), d = '1+'

e+ 11+'

B.

3.2.2 Middle: Expectations and discretion

In the middle of the period, private agents will form expected in�ation e knowing

that there is some chance that there will be a strong central bank in place ( ) and

some chance that there will be a weak central bank in place (1 � ), choosing the

action d. Hence, expected in�ation will be e = p+(1� )d in line with (11). Thus,even though the weak central bank�s decision rule is the same, the equilibrium with

imperfect credibility is di¤erent because there is a di¤erent speci�cation of beliefs in

(11). That is, the equilibrium in�ation action of the weak central bank arises as a

�xed point between beliefs and actions, just as with the situation of pure discretion

studied above, but with a di¤erent structure for beliefs. In the current context, the

degree of short-term credibility in�uences the nature of this �xed point, since it it

a¤ects the relative weight placed on the strong central bank�s plan and the weak

central bank�s action in the formation of expected in�ation. It is possible to simply

solve for the equilibrium weak central bank action since the equations that de�ne the

�xed point are both linear, with the result being,

d(p; ) ='

1 + ' p+

1

1 + ' B = �p+ (1� �)B (18)

withB de�ned above as the in�ation bias if there is a weak central bank with certainty

(that is, if = 0).

There is thus an "middle subperiod equilibrium function" d(p; ) describing the

weak central bank�s response to the in�ation plan under the requirement that expec-

tations are rational: the intensity of this reaction, which we call �, is between 0 and 1

so long as ' > 0, since � = ' 1+'

. Further, � increases with the degree of short-term

credibility . Correspondingly, with > 0 and thus � < 1, monetary discretion gives

rise to a smaller degree of in�ation bias �at a given p �because expected in�ation is

17

held down by the chance that there is a strong central bank in place.

With one con�guration of model parameters, it is not necessary to solve the ra-

tional expectations �xed point. If ' = !0 + !2�2 = 0 because the objective function

weights on quadratic terms involving expectations are zero, then the weak monetary

authority simply always chooses d = B. In that case, assumed by Barro [1986],

Backus and Dri¢ ll [1985a,b], and Cukierman and Leviatan [1991], the in�ation plan

p has no e¤ect on the behavior of the weak central bank.

3.2.3 Start: The strong central bank

We next consider the in�ation plan formulated by the strong central bank, which

optimizes taking as given the equilibrium expectations rule

e(p; ) = p+ (1� )d = �p+ (1��)B

where �( ) = [ + (1 � )�( )] and thus 1 � � = (1 � )(1 � �). Notice that

imperfect short-term credibility ( < 1) implies that the central bank can no longer

perfectly manage expectations ( < 1 implies � < 1): a part of expectations is

invariant to its plan because of the possibility that the in�ation action will be taken

by a weak central bank. However, in its expectations management, the strong central

bank takes into account the direct e¤ect of its in�ation plan on expectations ( p) and

also the indirect e¤ects that operate through the plan�s in�uence on a private sector

beliefs about the weak central bank�s in�ation action, as captured by (1� )�p.

Using (15), we determine that the optimal in�ation plan takes the form

p( ) = [1� �( )

1 + '(1��( ))2 ] B: (19)

There are a number of interesting aspects of the strong central bank�s in�ation plan

in a setting of imperfect short-term credibility. First, there continues to be some

in�ation bias: a strong central bank does not fully o¤set the in�ation bias from

expectations about the weak central bank�s behavior, because it would face losses

from doing so due to unexpected in�ation. Second, the coe¢ cient attached to the

bias term B coe¢ cient has the property that it is 1 when = 0 (no short-term

credibility) and that it is 0 when = 1 (full short-term credibility). That is, a strong

central bank with low short-term credibility will behave very much like a weak central

18

bank in terms of its in�ation plan.

3.2.4 Implications

We now discuss some aspects of the expectations management equilibrium that we

have constructed.

Equilibrium behavior by the weak central bank Given the in�ation plan just

determined, the weak central bank�s behavior is

d( ) = �( )p( ) + (1� �( ))B = p( ) +�( )(1� �( ))

1 + '(1��( ))2B (20)

Thus, in equilibrium, the weak authority chooses a higher rate of in�ation than the

in�ation plan, although the e¤ect is minor when short-term credibility is low (because

the coe¢ cient on B is close to zero in that setting). But it also chooses a lower rate of

in�ation than under full information, as seen in (18) unless the nonnegative response

coe¢ cient � is zero. This occurs only when expected in�ation enters linearly in the

reduced form objective, as assumed by Barro [1986], Backus and Dri¢ ll [1985] and

Cukierman and Leviatan [1991], and requiring !0 = !2 = 0 in our model.

In�ation plan, expectation, and action Our �ndings about the nature of the

in�ation plan, the in�ation expectation, and the in�ation action are summarized

in Figure 2.9. The top panel (A) shows the reaction coe¢ cient � that governs

how a weak central bank�s equilibrium in�ation action depends on the in�ation plan

and the reaction coe¢ cient � that governs how expected in�ation depends on the

in�ation plan. As can be veri�ed directly from de�nitions above, these reaction

9This �gure is drawn under the assumption that � = 3, so that a one percent unexpected in�ationis associated with a 3% increase in real output. Additionally, the �gure is constructed under theassumption that !1 = :03, that !2 = :3, and that x� = :1 (so that it is desirable to stimulate realactivity to 10% above its normal level). Taking these coe¢ cients together, the model produces anin�ation bias of B=.18. The percentage counterpart of 18% is used in the �gure.These values are suggested by U.S. annual macroeconomic analysis and data from prior to 1970:

if there is an annual Phillips curve linking in�ation and unemployment with a slope of negative one,as was widely believed at the time, and if there is an Okun�s law linkage between unemploymentto real output with a coe¢ cient of three, then � = 3. The idea that output was perhaps 10% toolow in a nonin�ationary environment is captured by x� = :1: The welfare weights are selected todisplay a high discretionary in�ation rate (18%) and also to generate a period in which discretionaryin�ation remains high for some time during the transition (see Figure 6 below).

19

coe¢ cients are increasing in short-term credibility. The second panel (B).shows how

the three measures of the in�ation rate depend on short-term credibility : (i) the

solid line is the strong central bank�s in�ation choice p, which declines from the

in�ation bias level (B=.18) to zero as � rises; (ii) the weak central bank�s in�ation

choice d, which is higher than p but also declining with credibility; and (iii) expected

in�ation, which is a credibility-weighted average of the plan and the discretionary

action (e = p+ (1� )d)

Real activity Given the expectations function, e = p + (1 � )d, it is now easy

to determine the behavior of real activity as x = �[� � e] = �[� � p � (1 � )d].

This simple expression implies that there is a recession when there is a strong central

bank in place,

x = ��(1� )(d� p)

since we just saw that d > p. The above analysis, though, also shows that this

recession will be small when short-term credibility is very high (since = 1) and

when it is low (since d � p is small because the coe¢ cient on B in (20) is close to

zero). By contrast, when there is a weak central bank in place, there is a boom

x = � (d� p):

This is a small e¤ect when short-term credibility is small ( close to 0) but it is

increasing as short-term credibility increases, both because the "slope" � increases

and because the size of (d � p) increases. Hence, short-term credibility bene�ts the

weak government when he produces an in�ation surprise.10

3.2.5 Summing up the benchmark analysis with imperfect credibility

The strong central bank actively manages beliefs given the expectations function

that it faces. However, with imperfect credibility, it cannot exert the same degree

of control on expectations as with perfect credibility: private agent think that there

is some chance that in�ation will be chosen in a discretionary manner. This simple

10The third panel (C) of Figure 2 provides more detail on these real activity outcomes: themagnitude of the real contraction brought about by a strong government is largest when there is anintermediate level of short-term credibility, since this is when the e¤ective slope (�(1� )) and thesize of the in�ation di¤erence (d�p) are both large in magnitude. With high short-term credibility,alternatively, there is a large gap between d and p, for example, but there is a small e¤ective slope.

20

model delivers a number of interesting implications about managing expectations in

an environment of imperfect credibility, as follows:

Management of expectations: A strong central bank in a setting of imperfect

credibility will manage the beliefs of agents in part by changing what is rational to

believe that a weak central bank will do;

Imperfect credibility and in�ation bias: Even a strong and forward-looking central

bank will not eliminate the in�ation bias arising from discretionary policymaking if

it is in a setting of imperfect credibility;

Failure of in�ation plans: In�ation plans will turn out to fail, when central banks

turn out to be discretionary in their actions, even though these plans were consistent

with optimal management of expectations by a strong central bank;

Real e¤ects of in�ation policies: Under imperfect credibility, there will be a real

boom when a weak central bank generates a surprised in�ation and a real recession

when a strong central bank maintains its in�ation plan.

Cukierman [1986] and Cukierman and Leviatan [1991] stress the latter three im-

plications. The �rst implication does not appear in their analysis because the weak

central bank�s in�ation action, which we call d, does not depend on expected in�ation

because of their assumption about the momentary objective.

4 Managing mimicking

We now consider an extension of our static model to allow for management of en-

dogenously determined short-term credibility. That is, there is now an expectations

function of the form

e(p; �) = (p; �)p+ [1� (p; �)]d(p; �) (21)

Expected in�ation continues to be a weighted average of the in�ation rates of strong

(p) and weak (d) central banks, but the weight now depends on both the level of long-

term credibility and on the in�ation plan. Generally, this setup means that part of

the e¤ect of an in�ation plan is its e¤ect on mimicking incentives. Speci�cally, a

lower in�ation plan can consequently have a heterodox e¤ect on expected in�ation,

raising it because the likelihood of discretion increases.

21

To generate endogenous mimicking in a static setting, we suppose that a weak

monetary authority faces a utility cost of � if it deviates, with this �xed cost being

drawn from a continuous distribution F with support [0;�]. Accordingly, an authority

will deviate if the temptation exceeds the cost, i.e., T (p; e) = w(d; e) � w(p; e) > �

and otherwise it will mimic, so that the mimicking function is

m = 1� F (T (p; e))

which is decreasing in temptation.

4.1 An analytical example

To generate a very simple example, we assume that the reduced form objective func-

tion w is quasi-linear in expected in�ation (!0 = !2 = 0), so that weak government�s

decision rule is just d = !1� = B. Given the discretionary in�ation rate, it then

follows that the welfare under discretion is w(B; e) = u� � 12B2 + !1�(B � e) and

that under the in�ation plan is w(p; e) = u�� 12p2+!1�(p� e), so that the gain from

deviating is

w(B; e)� w(p; e) = �12(B2 � p2) + !1�(B � p) =

1

2(B � p)2.

Hence, the mimicking depends just on the in�ation plan, m = 1�F [12(B� p)2], with

some interesting properties. First, if planned in�ation is zero, then mimicking will

be positive so long as there are su¢ ciently high mimicking costs, � > 12B2. Second,

mimicking will be zero when p = B. Third, if the distribution of costs takes the form

F (T ) = (T=�)1=2 then mimicking depends linearly on the gap between discretionary

and planned in�ation,

m = 1� 1p2�(B � p);

which is increasing for p between 0 and B. By increasing the planned in�ation rate,

the monetary authority increases short-term credibility (p; �) = � + (1 � �)m(p)

because it increases mimicking, i.e., it increases the likelihood that the in�ation plan

will be carried out even if a weak government is in place.

The speci�c objective function thus makes it possible to represent expected and

unexpected in�ation as quadratic functions of the in�ation plan. Unexpected in�ation

22

is

p� e(p) = (1� (p))(p�B) = (1� �)(1�m(p))(p�B)

= �(1� �)1p2�(p�B)2.

so that it always be negative if the strong central bank selects a lower planned rate

of in�ation than its weak counterpart would choose. The ex ante likelihood that the

central bank is of a weak type (1� �) dictates the scale of unexpected in�ation.

Optimal in�ation for the strong central bank is found by maximizing u(p; x) =

u� � 12(p)2 + !1(x � x�) subject to x = �(p � e(p)), with the expectations function

given above. The plan is

p =�!1(1� �)p

�=2 + �!1(1� �)B

Hence, with imperfect information about its type (� < 1), a strong central bank will

choose an in�ation plan that does not fully eliminate the in�ation bias, even when it

can manage expectations, echoing the results of section 3.

4.2 Managing mimicking can be tricky

We now evaluate mimicking when, as in section 3, there is an e¤ect of the in�ation

plan p on the action of the weak authority d if it deviates. We also assume that

there is a mimicking cost distribution of the form displayed in Figure 1.11

The mimicking function m(p; �) is the solution to a �xed point problem at each

candidate in�ation plan p, which we must solve numerically rather than analytically

when we depart from a reduced form objective that is linear in expected in�ation.

11The �xed cost is distributed according to a beta distribution, with support [0,1]. There are twoparameters of the beta distribution, which can be adjusted to give it a wide variety of shapes. Ifthese parameters are �1 and �2 then the beta density is

f(z; �1; �2) =�(�1 + �2)

�(�1)�(�2)z�1�1(1� z)�2�1

where � is the gamma function. An additional convenient property of the beta distribution is thatthere are simple analytical formulae for the moments: the mean is �1=(�1 + �2).In our analysis, the distribution above governs the standardized random variable z = �=�. When

we choose �1 = 1; �2 = 5 and � = :05 so that the mean is is �=6 or just under .01. The � = :05value exceeds 12B

2 = 12 (:18)

2 = :0162, so that there is indeed mimicking when there is zero in�ation.

23

At any given mimicking rate m and long-term credibility level �, there is an implied

level of short-term credibility, = � + (1 � �)m. In turn, the analysis of section 3

shows us that there is then a particular e; d pair consistent with that level of and

a level of the in�ation plan, p. Further, at the triplet (m; �; p), there is an implied

temptation, T = w(d; e)�w(p; e): a point on the the equilibrium mimicking functionm(p; �) arises when the 1 � F (T ) = m. The lower panel of Figure 3 illustrates the

determination of this �xed point at two di¤erent levels of p (p = 2% and p = 2:5%)

and two di¤erent levels of long-term credibility (� = 0:2 and � = 0:5).

There is a particularly strong e¤ect of the in�ation plan on mimicking when

credibility is low. The upper panel of Figure 3 shows an illustrative case, where � is

respectively 0:2; 0:5 and 0:95. When the in�ation plan is around 2:5%, a variation of

only 0:5% in in�ation creates a di¤erence of 8% in mimicking probability as a result

of the steep slope associated with that low �. By contrast, if � is at 0:5, the same

0:5% variation in the in�ation plan only creates a 1% di¤erence in m. The relatively

big impact of the in�ation plan on the mimicking probability when � is low makes

the choice of the plan p particularly important and subtle for an authority with low

credibility, perhaps a new authority, since it is not very trusted by the private sector.

In turn, the e¤ect of the plan on short-term credibility also depends on p in a

similar manner as mimicking (panel A in Figure 4), since = � + (1� �)m (p; �).

Therefore, a higher planned in�ation rate is more credible because it is more likely

to be mimicked. Moreover, when long-term credibility � is low, the impact of p on

is particularly signi�cant. Having this optimal response of to p in hand, we now

turn to its consequence for the optimal in�ation plan.

We have previously stressed that the strong monetary authority evaluates the con-

sequences of its in�ation plan on in�ation expectations. Since in�ation expectations

take the form (21), the e¤ect of a marginally higher in�ation plan is

@e(p; �)

@p= + [1� ]

@d(p; �)

@p+ [p� d]

@ (p; �)

@p

The �rst term in this expression is simply the direct e¤ect of the plan on expectations:

it produce a higher rate of expected in�ation. The second one looks familiar from

the discussion in section 3 above: it is the indirect e¤ect via the weak government�s

response to the plan. However, due to the endogenous , the weak government�

response is more complicated. Recall that the response function (18) speci�es that d

24

is the credibility-weighted average of p and B. In section 3, the weight is exogenous

so that an increase in in�ation plan also increases the in�ation action of the weak

government. But, now, the weights are a¤ected by the in�ation plan through its

e¤ects on : an increase in p raises , and in turn lowers the weight on B in d. If

this e¤ect is strong enough, as in Figure 4, d falls rather than rises for some values

of � and p.

The third term is new to the current setup: it also concerns the e¤ect of a mar-

ginally higher in�ation plan on the likelihood that the plan will be carried out, i.e., on

its credibility. We have seen above that a higher rate of in�ation raises the mimicking

rate thus increases short-term credibility. Since the discretionary in�ation rate d is

always higher than the in�ation plan p it follows that this third channel involves a

heterodox e¤ect: a higher planned in�ation rate lowers expected in�ation because it

raises the likelihood that the plan will be carried out. A strong monetary author-

ity with low long-term credibility (� = :2) that had an in�ation plan of about X

percent in Figure 4 would be bedeviled by rising rapidly output losses if it sought

further reductions in in�ation. Panel A shows that expected in�ation becomes �rst

unresponsive to declines in p and then actually increases; Panel D shows that the

combined e¤ect of lower p and higher e lead to substantial output losses.

Relative to the monetary authority described in section 3, who faced exogenous

short-term credibility, the strong decision-maker will choose a higher planned rate

of in�ation: he knows that low rates of in�ation will decrease the current credibility

of the in�ation plan and move expected in�ation against him, so that there will be

higher output losses when he sticks to his plan. Hence, the endogenous reaction of

to p sets a barrier for the strong government in selecting a low in�ation plan. We note

that the optimal p (indicated by ���) always lies on the right of the nonmonotonicityregime of e.12

12Eliminate former Figure 5? Since we can determine the optimal p and the associated levels ofm; e; d; and at each level of �, it is also possible to depict the way in which these variables dependon the state, long-term credibility, as in Figure 5. Note that increasing long-term credibility reduceseach of the three in�ation variables p; e and d, while reducing the output loss that a strong centralbank faces (this is xc in the bottom panel). Notice also that the extent of mimicking is initially highat low credibility, then decreasing at higher levels of �.

25

5 Reports, rules and in�ation commitments

We have so far depicted a strong monetary authority that knows that private agents

will respond to his announced plan recognizing that there are two types of monetary

authorities, one type that will execute the in�ation plan for sure and another that

will (or may) deviate from the plan. We now reconsider aspects of this structure,

with speci�c emphasis on the market interpretation of central bank signals and the

role of transparency in central bank management of expectations.

5.1 Signalling equilibria and in�ation reports

Our approach to this point has simply been to restrict our attention to a rational

expectations equilibrium of a Markovian form, rather than to consider an explicit

game between the central bank and the private sector. By contrast, we could have

begun by describing a signalling game, with the central bank being the sender of the

message p and a large number of private agents being the receivers of the message.

Since the central bank is of unknown type, the relevant equilibrium concepts would

be the sequential equilibrium of Kreps and Wilson [1982] or the perfect Bayesian

equilibrium of Fudenberg and Tirole [1991]. While there are subtle di¤erences in

these approaches, we will treat them as equivalent for our purposes.

The key di¢ culty with using a game-theoretic approach is that any signal �any

value of p �can be an equilibrium so long as it is supported by su¢ ciently pessimistic

out-of-equilibrium beliefs. It is clear that the only equilibria of the game in our

economy is a pooled one, in which both types of central banks send the same message:

if they send di¤erent ones, so that they are distinguished, then the weak central bank

can always improve its welfare by sending the same message as the strong bank.

However, rather than the value p� which we determined above, suppose that an

alternative equilibrium p = a were considered. Then, if private agents interpret

other messages as indicating that government type is weak, then p = a is indeed an

equilibrium. This bewildering set of equilibria has led many to avoid modeling settings

with private information and limited commitment, which seem essential features from

the standpoint of analyzing the management of expectations and the evolution of

credibility.

>From the standpoint of game theory, the announcement game present in this

model is unusual in that the strong authority generates a publicly observable in�ation

26

plan as a by-product of his action and that imitation is optimal for the weak authority.

It turns out that many of the re�nements that have been developed to reduce the set

of equilibria do not "work" in this context. Notably, the intuitive criterion of Cho

and Kreps [1987] produces the counter-intuitive conclusion that the only signalling

equilibrium involves full in�ation bias, � = p = B.13 There is no equilibrium at all

under the "perfect sequential equilibrium" approach of Grossman and Perry [1986],

which introduces the idea of using a rational, Bayesian approach to discipline on out-

of-equilibrium beliefs. However, an approach due to Mailath, Okuna-Fujiwara, and

Postelwhaite [1993], which imposes strongly coherent out-of-equilibrium beliefs of a

Bayesian form, can be used to show that the unique equilibrium is exactly the p� that

we have derived above. There are two essential features of that approach. First, any

out-of-equilibrium message must be one that is sent in some alternative equilibrium

by some set of agents. Second, the incentives that various types of agents have to

send an alternative message m are evaluated by comparison of bene�ts in a candidate

and alternative equilibrium and these induce coherency restrictions on beliefs, which

specify an out-of-equilibrium probability distribution of sender type. Third, if these

coherent beliefs are inconsistent with the support of the candidate equilibrium, it is

"defeated" by the alternative and, hence, discarded.

To illustrate equilibrium selection along these lines, we use a special case of the

model of section 3, which is the quasilinear utility function so that the weak central

bank selects d = B. We structure our discussion around the two panels of Figure

5. The horizontal axis in this �gure is the in�ation plan (p) and the vertical axis

is expected in�ation (e). Indi¤erence curves are drawn in with dashed lines: these

correspond to values of p and e such that welfare is held constant at two di¤erent

levels, with higher values of welfare corresponding to a lower value of e for a given

p.14

13Cukierman and Leviatan [1991, appendix] study the same signalling game. They use of reason-ing along Cho-Kreps line but depart from Cho-Kreps by assuming that message-sending incentivesare evaluated in an alternative pooled equilbrium, which anticiipates aspects of the approach ofMailath et al [1993]. For analysis of economies with more than two types, as with our random dis-count factor or random cost models, Bayesian reasoning along the lines of Mailath et al. is a moreappropriate approach. However, as we stress in King, Lu, and Pasten [2007], a direct application ofthe Cho-Kreps approach leads to selection of the worst equilibrium, rather than the optimal one.14The quasilinear reduced form objective takes the form w(�; e) = � 1

2�2+ !1�(� � e), so that

the discretionary action is � = !1� = B and the objective may also be written as w(�; e) = � 12�

2+B(�� e). An indi¤ernce curve at welfare w is then e = �� 1

Bw�12B�

2, so that it is increasing andstrictly concave on the interval (0,B) as shown in the �gure.

27

The locus of potential candidate equilibria is the solid line: with no distinction

between short-term and long-term credibility, it is e = p+ (1� )d = �p+ (1� �)dwith d = B resulting from quasi-liner utility. In section 3, this line was the expectation

function faced by the strong central bank, but we now treat it as a locus of pooled

equilibria. We thus allow for a separate speci�cation of the expectations function,

as we discuss in detail below. The equilibrium that we have so far considered is

marked with a "*" in the �gure: it involves an indi¤erence curve just tangent to the

expectations function.

There is a one-to-one correspondence between expected in�ation and belief about

type. That is, if expected in�ation is e, the in�ation plan is p, and discretionary

in�ation is d, then the implied belief that a strong central bank is in place is (d �e)=(d� p).

To illustrate the multiplicity of signalling equilibria, suppose that there is an

expectations function of the form e = max(�h+ (1� �)B; �p+ (1� �)B), illustratedby the dotted line in panel A of Figure 5. Then, an optimizing strong central bank

would choose p = h instead of p = p� because a lower in�ation plan would not reduce

in�ation expectations. Such an expectations function would implicitly assign a value

of short-term credibility

=B � h

B � p� < �

to in�ation plans p < h.15 That is, for such in�ation plans, it must be the case that

the plan p < h is interpreted as signalling that a strong central bank is in place with a

lower probability than the actual likelihood that one is present (�). Yet, since such an

o¤-equilibrium signal will never be sent, it is the case that these o¤-equilibrium beliefs

cannot be invalidated. That is: it is possible for signalling to lead to an "expectation

trap" in this one period model, when there is latitude to specify out-of-equilibrium

beliefs.16

Re�nements of signalling equilibria all necessarily involve the interpretation of out-

of-equilibrium messages that might be sent and the incentives that various parties

would have to send such messages. We �nd the approach of Mailath et al [1993]

compelling, as it places a strong coherency on out-of-equilibrium beliefs. In our

context, if an out-of-equilibrium message is sent, then its implication for belief about

15This implicit value is derived by equating e = p+ (1� )B and e = �h+ (1� �)B.16Chari et al on expectations traps

28

type cannot be inconsistent with the incentives that types have to send the message.

This approach eliminates candidate equilibrium h, as follows. Working with panel

A of Figure 5, we consider h as the candidate equilibrium and consider p� as the

alternative equilibrium. Then, we ask whether the beliefs that support h are coherent

with sender incentives in an actual alternative equilibrium p�. We �nd that both

strong and weak central banks would send p� because each has higher welfare in that

alternative equilibrium relative to the h equilibrium.17 Thus, the coherent out-of-

equilibrium belief would be = � rather than = B�hB�p� < � as with the adverse

expectations function above. Hence, we say that the supporting belief for h is not

strongly coherent at p� and we rule out the equilibrium h. In the terminology

of Mailath et. al.[1993], the candidate equilibrium h is defeated by the alternative

equilibrium p�.

Next, consider the alternative equilibrium in panel B, which is a low in�ation

plan outcome (p = l < p�). This must be supported by beliefs consistent with an

expectations function that makes it undesirable for the strong government to pick

p�, as illustrated by the dotted line. However, since this expectation lies above solid

line at p�, it again involves the belief that a message of p� is more likely to be sent

by a weak bank. Considering sender incentives once again, we �nd that only the

strong central bank would send this message (its welfare is higher at p�, while the

weak central bank is worse o¤ with higher expected in�ation). So the candidate

equilibrium l is ruled out.

A potential di¢ culty with using "belief based re�nements" along the lines of

Grossman and Perry [1986] and Mailath, Okuno-Fujiwara and Postlewaite [1993]

is that there can be no equilibrium. We have described so far why there cannot

be an equilibrium h > p� or l < p� but this does not guarantee that p� itself is

an equilibrium. If it were the candidate equilibrium and it were supported by the

expectations function,

e = p� + " for p < p�

e = p+ (1� �)B for p � p�

then there would be no con�ict with the requirements of Mailath, Okuno-Fujiwara

17The weak central bank bene�ts because e� < eh and the strong central bank bene�ts becausethe indi¤erence curve through (p�; e�) lies above that throught (h; eh).

29

and Postlewaite [1993]. Neither central bank type would send a message of h > p� or

l < p�, so that no restriction is placed on beliefs for such messages.

Overall, we have found our equilibrium p� is the sole surviving signalling equilib-

rium with coherent out-of-equilibrium beliefs.18 In our analysis of section 3 above, we

assumed that p� would be the choice of a strong central bank under a rational expec-

tations perspective and that a weak central bank would just mimic this announced

plan. We now have an equivalent �nding in the context of a signalling game, un-

der a perspective that essentially imposes rational expectations on out-of-equilibrium

beliefs.

5.2 In�ation rules

The game-theoretic perspective also suggests that the announcement plays a central

role, even though it is not informative about central bank type. Within that frame-

work, a strong central bank has no in�uence on private sector expectations unless it

sends a message and it has no incentive to choose low in�ation unless it has an in-

�uence on in�ation expectations. Thus, if it can disclose its planned in�ation action,

then a strong central bank should do so. Lack of transparency, in this sense, would

lead to the fully discretionary outcome � = B if the strong central bank did not make

the announcement. Intrinsic strength is not a substitute for communication within

this model.

The view that announcements are central contrasts with an alternative perspec-

tive which has some credence among monetary policy analysts working from rational

expectations perspective, which we summarize as follows. First, the central bank is

in an environment with imperfect credibility, but the participants in the model all

understand exactly that a disciplined central bank would issue the in�ation report

p(�); and that a weak central bank would issue the same report, so that the report

is not informative. Second, as a result, a disciplined central bank that uses the in�a-

tion rule p(�), implemented with an appropriate interest rate or monetary aggregate

instrument policy, would also secure the same performance. In the jargon of current

policy discussions, the alternative perspective would rationalize a system of implicit

in�ation targets and downplay the importance of announcements of explicit in�ation

targets.

18We discuss the application of alternative re�nements to a simple microeconomic model withmany of the same properties of our current framework in King, Lu and Pasten [2007].

30

We think it is important to carefully investigate whether there is a logical coher-

ence to the alternative perspective, with some modi�cation of the above signalling

game in which it does not make sense. Consequently, we are presently working to

detail the game-theoretic approach to our model and to contrast the rational expecta-

tions perspective. However, at present, we stress that a basic game-theoretic analysis

of our model both supports our focus on p(�) as the equilibrium in�ation plan and

indicates that transparency about in�ation plans is critical to their success.

6 Evolution of long-term credibility

We have provided basic examples of two aspects of expectations management. In

section 3, we studied how a strong monetary authority with a short-horizon would

optimally choose in�ation when there was exogenous imperfect credibility, with its

in�ation plan in�uencing expected in�ation both directly and via its e¤ect on weak

central bank�s discretionary in�ation action.. In section 4, we show this strong au-

thority would manage credibility with endogenous mimicking by a weak central bank

with random costs of behaving in a discretionary manner. We think that these ex-

amples �each of which contains a closed form solution under certain parameters �

shed light on important aspects of managing expectations.

However, the simple models do not describe the dynamic evolution of credibility

that seems essential to understanding many issues in macroeconomics or the related

choice of an in�ation plan by a strong central bank that is managing its stock of

long-term credibility. It is to this topic that we now turn.

6.1 Mechanical dynamics and general principles

Speci�cation of a mimicking function m(�) allows us to write short-term credibility �

the probability that the action will be that of a strong central bank �as

(�t) = �t + (1� �t)m (�t) :

This expression makes clear that short-term credibility will be higher than long-term

credibility because of mimicking behavior. As stressed in the discussion in section

3 above, it is short-term credibility that plays a role in the determination of output

and in�ation in the economy at a given point in time.

31

Further, as discussed in section 2 above, since the observed in�ation rate is the

only information in the economy about the nature of central bank type, then Bayesian

learning implies that

�t+1 =

(�t t= �t

�t+(1��t)m(�t)if �t = pt

0 if �t 6= pt

)

if there is no replacement. That is, Bayes�rule governs the linkage between in�ation

plans �which in�uence the rate of mimicking �and the evolution of the state variable

of our model, which is long-term credibility

In terms of general principles, there is thus an interesting interplay between short-

term credibility and long-term credibility �. If short-term credibility is high (close

to one) then long-term credibility evolves very slowly (if = 1, then Bayesian learning

speci�es that �t+1 = �t). This is because there is little information in the observation

that in�ation is at the planned rate p. On the other hand, if short-term credibility

is low, then there is a good deal of information in the fact that in�ation is at the

planned rate. Further, short-term credibility can be high either because long-term

credibility is high or because the likelihood of mimicking (m) is high. In terms of

the latter case, = 1 if m = 1 irrespective of the level of �. So, if there is substantial

mimicking, then long-term credibility will evolve very slowly.

[RK stopped].

7 Summary and conclusions

We provide a basic analysis of expectations management in an environment of im-

perfect credibility, extending a standard macroeconomic framework for this purpose.

In our model, a monetary authority pursuing low in�ation but concerned about real

activity manages expectations taking into account imperfect credibility and, in par-

ticular, gauging the e¤ect of his policy action on private sector beliefs about the

likelihood and intensity of discretionary policy actions that would be taken by an

alternative type of policymaker.

Our analysis highlights the nature of optimal expectations management by a cen-

tral bank in a very simple model and displays its implications for the evolution of

credibility over time. It leaves open a number of important topics, which never-the-

32

less seem feasible to explore in extensions of our approach.

First, our analysis concerns a model in which all public and private decision-

makers have one period horizons, even though the economy operates for many peri-

ods. Modern central banks consider in�ation policy from the standpoint of intertem-

poral objectives that incorporate the consequences of current actions for future eco-

nomic performance. It is therefore important to extend the analysis to situations in

which central banks have present discounted value objectives: this is the topic of our

companion research, which explores expectations management in�uences transitional

credibility dynamics and the response of the economy to shocks.

Second, our analysis introduces imperfect information about central bank type

only in a very limited manner. It delivers the conclusion that expectations manage-

ment can be undertaken via explicit binding plans, announcements such as in�ation

reports, or simply via a well understood regular operating method such as an implicit

in�ation target. It seems important to enrich the extent of private agent uncertainty

about central bank type in future work, with an eye to understanding when each of

these institutional arrangements is desirable.19

Third, the macroeconomic model that we employ is deliberately "old school" in

its in�ation dynamics and does not spell out detailed microeconomic foundations.

It seems important to explore the nature of in�ation management in more realistic

macroeconomic models with better micro foundations.

Fourth, issues of expectations management frequently are suggested to be impor-

tant in the choice of the monetary instrument in general and the setting of the chosen

instrument at a point in time. By assuming that the central bank directly controls

in�ation, we have abstained from consideration of these issues, but it seems important

to do so systematically.

Fifth, imperfect credibility and associated topics of expectations management

have been suggested to be important for understanding speci�c historical periods in

the monetary histories of the U.S., the U.K., and other countries. It is of interest to

begin a more detailed exploration of these connections.

19Stein (1989) shows that unobserved decision-maker type may lead to imprecise policy announce-ments, using the "cheap talk" approach of game theory.

33

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[13] Kreps, David A. andWilson, and RobertWilson, "Sequential Equilibria", Econo-

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35

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36

Table 1: Basic Model Ingredients

Panel A: subperiod structure

prior start middle end

transition plan (p) expectations (e) action (�)

Panel B: types and actionsType: �

strong weak

Plan p p (mimicking)

Action: �

Deviation � d

Table 2: Long-term credibility, short-term credibility and mimicking

Concept de�nition

Long-term credibility �t =prob(� t = strong)

Short-term credibility t =prob(�t = ptjpt)Mimicking probability mt= prob(�t = pt)j (pt; � t = weak)

Note to Table 2: All probabilities are minimally conditioned on infor-

mation up to the start of period t. Long-term credibility is the start-of-the

period probability that the decision-maker is of the strong type. Short-

term credibility is the probability that the in�ation plan will actually be

carried out, conditional on the plan�s level. The mimicking probability

is the likelihood, conditional on the decision-maker actually being of the

weak type, of in�ation being at the planned level.

37

Notation Table

Symbol De�nition

� in�ation

p in�ation plan

d in�ation deviation (discretionary in�ation)

e expected in�ation

x real activity

x� e¢ cient real activity

u welfare (u(�; x))

w reduced form welfare (w(�; e))

!0 welfare weight on unexpected in�ation

!1; !2 welfare weight on output terms

� slope of Lucas-Fischer supply function

� partial derivative of e wrt �

B in�ation bias (B = !1�+ !2�x�)

short-term credibility

� reaction coe¢ cient ( !2�2 1+!2�2

)

� long-term credibility

m mimicking probability

W value function for strong central bank

V value function for weak central bank

T temptation

� random discount factor for weak central bank

b discount factor for strong central bank

� random �xed cost of deviating

� largest �xed cost

F distribution of �xed costs or discount factors

38

0 0.01 0.02 0.03 0.04 0.050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ξ

cdf o

f fix

cos

t

Figure 1: Distribution of �xed costs

39

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1A. Reaction to inf lation plan

discretionary inf lation (θ)expectations (∆)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2B. Inf lation measures

plan (p)dev iation (d)expectation (e)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

0

0.1

0.2

0.3C. Real activ ity

Short-term credibility (ψ)

commitmentdiscretion

Figure 2: In�ation and real activity with imperfect credibility

40

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1A

m

p

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

m

1-F

B

ρ=.2ρ=.5ρ=.95

ρ=.5,p=2.5%

ρ=.5,p=2%

ρ=.2,p=2.5%

ρ=.2,p=2%

Figure 3: Short-horizon monetary authority. Panel A shows the e¤ect of plan (p)on mimicking (m) at three di¤erent levels of credibility. Panel B shows the e¤ect ofchanges in the in�ation plan on the �xed point for mimicking at two di¤erent levelsof credibility.

41

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

p

e

A

0 0.05 0.1 0.15 0.20.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p

ψ

C

0 0.05 0.1 0.15 0.20.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

pd

B

0 0.05 0.1 0.15 0.2-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

p

xc, x

d

D

ρ=.2ρ=.5ρ=.95

Figure 4: Short-horizon monetary authority: Panel A e¤ect of plan on expectedin�ation (e), Panel B on discretionary in�ation (d), Panel C on short-term credibility( ) and Panel D real activity (x).

42

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

inflation plan (p)

expe

cted

infla

tion

(e)

expectations (pooled)indifference curve (optimum)indifference curve (worse)adverse expectations

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

inflation plan (p)

expe

cted

infla

tion

(e)

expectations (pooled)indifference curve (optimum)indifference curve (worse)adverse expectations

Figure 5