When Should Policymakers Make

Announcements?�

Ricardo Reis

Columbia University

March 2010

PRELIMINARY AND INCOMPLETE

Abstract

If a policymaker learns that there will be a future change that a¤ects everyone, at

what time should he/she announce it? This paper presents a dynamic model with

agents that have a limited amount of attention to devote to news, and must divide it

between their individual circumstances and an aggregate regime variable. Announcing

a change in regime has the bene�t of allowing agents to start becoming better informed

and make better decisions when the change happens, but this comes at the cost of

paying less attention to current events and making worse decisions now. Moreover, the

earlier the announcement by the policymaker, the less precise it will be. This paper

characterizes the optimal timing of policy announcements, both from the perspective of

individual private agents and from social welfare. It show that: (i) announcements too

ahead in time are ignored, (ii) attention grows between the announcement and the actual

change and falls after it, (iii) unnecessary announcements lower welfare, (iv) it may be

best to keep quiet in spite of public interest, and (v) rumors should be squashed early.

Another contribution is the development of rational inattention theory in continuous

time.

�Contact: rreis@columbia.edu.

1

1 Introduction

Economists usually focus on the design of optimal policies, but policymakers are often as

worried with the optimality of a given policy as they are with its practical implementation.

A crucial step in implementing a policy is to communicate it e¤ectively and at the right

time, and there is a large literature outside of economics devoted to this problem (Kaid,

2004). In particular, it is sometimes argued that communicating a policy intention too early

may risk it being ignored by people, and may lead to its unpopularity as subsequent policy

changes lead to confusion (Cobb and Elder, 2001). Policymakers seem to respond to these

theoretical concerns, and in spite of commitments to transparency, often keep privileged

information and internal debates away from the public�s eyes until the right time arrives.

This paper provides an economic model to answer an important question in the com-

munication of public policies: what is the optimal timing for policy announcements? In

particular, consider the following scenario: a policymaker learns (or decides) today that at a

future date there will be an aggregate shock. The policymaker can choose to communicate

this to the public, but only imperfectly. Either because she only has an imperfect signal

of the future change, or because the channel of communication to the public is limited, all

the policymaker can provide is a public signal that something has changed. When should

the policymaker make this announcement?

If agents can immediately absorb the policy announcement at no cost, then it does

not matter whether the announcement is made today, at the date of the change, or at

anytime in between. In this paper instead, I assume that, following the announcement,

agents can choose to invest to learn more about the change, but at the cost of paying less

attention to current conditions. Each agent faces a private trade-o¤ between becoming

better informed about the future and responding to the change better when it arrives,

versus being better informed about conditions today and making better decisions today.

Because future bene�ts are discounted, there is an optimal cut-o¤ date at which agents

start paying some attention to learn more beyond the announcement.

On top of this private trade-o¤, the policymaker must also add two considerations in

determining what is best for social welfare. First, as the date of the change gets closer, the

policymaker�s signal becomes more precise, either because it learns more about the change or

2

because it makes progress in reaching a �nal decision. It is plausible to assume that learning

by the policymaker is less costly than learning by the public, perhaps because of economies

of scale in collecting information, perhaps because it has access to privileged information, or

perhaps because the object of the learning are the policymaker�s preferences or the political

balance of power. It may then be optimal to delay making imperfect announcements to

the public for some time, while the information on these announcements becomes better.

Second, if agents�choices are strategic complements, then they may put too much weight

on the policy announcements as they permit coordination of actions and expectations. In

this case, there may be an interval of time when agents �nd it privately advantageous to pay

attention to the announcement, but this is socially ine¢ cient. Delaying the announcement

gives the policymaker a tool to move towards a more e¢ cient outcome.

One important feature of our model is that agents endogenously choose what to pay

attention to in two ways. First, in the traditional Bayesian sense that the informativeness of

signals gets weighted by their relative precision. Second, agents endogenously choose these

relative precisions subject to an information capacity constraint. Agents are rationally

inattentive, following Sims (2003), and ignoring an announcement is understood here as

devoting no mental capacity to improving on this piece of information.

Related literature and summary of contributions. This paper makes contribu-

tions to four existing literatures in economics. First, there is a growing empirical literature

on the communication of monetary policy. In a recent survey, Blinder et al. (2008) summa-

rize it as follows: �Over the last two decades, communication has become an increasingly

important aspect of monetary policy. These real-world developments have spawned a huge

new scholarly literature on central bank communication� mostly empirical and almost all

of it written in the last decade.� This paper contributes to the theory of central bank

communication by focussing on the question of when should central banks announce their

actions. There are several applied policy questions that would �t into the model in this

paper, for instance: Having decided to tighten monetary policy, should this be announced

ahead of time or done immediately? Learning that there is a problem in �nancial markets,

when should the central bank communicate this to the public? And, having decided on

�exit strategies�or changes in policy regime, when should these be announced?

3

Following the important work of Morris and Shin (2002), a few papers have characterized

scenarios under which policy may want to be less than fully transparent because of strategic

complementarities inducing an over-weighting of public signals. Hellwig (2005), Roca

(2006), Morris et al. (2006), Svensson (2006) and Lorenzoni (2010) discuss this result

when agents learn from signals on exogenous variables, while Amador and Weill (2009)

show that this is also possible when agents learn from the distribution of prices. Angeletos

and Pavan (2007) characterize more generally when is releasing public information socially

harmful. In most of these papers the choice of releasing information is essentially static

as the policymaker must choose whether to release the information or not, once and for

all. In this paper, instead, the question is when to release the information, and the focus

is on the dynamic trade-o¤s involved. Moreover, in this paper, the precision of signals is

an endogenous choice variable, whereas in all of this literature private agents take signals

as given.

Eusepi and Preston (forthcoming) also provides a theory of optimal communication

by central banks with private agents that are less than fully rational. There are three

key di¤erences between this paper and theirs. First, their model of agent�s expectations

is least-squares learning, whereas in this paper agents are Bayesian learners with rational

inattention. Second, they focus on the stability and determinacy of equilibrium, whereas

this paper focus on optimal policy according to social welfare. Third, they ask what vari-

ables and coe¢ cients the central bank should communicate, whereas this paper again asks

when to communicate. Morris and Shin (2007) provide another theory of optimal com-

munication as a trade-o¤ between more precise information and common understanding.

This paper shares with theirs the important role played by strategic complementarities and

coordination. However, their analysis is static, while the question in this paper is inherently

dynamic, and the key choice studied in their paper is how to allocate policy signals across

the population, whereas in this paper, the key choice is on privately allocating attention.

This paper also provides a contribution to the literature on rational inattention.1 The

model builds on Sims (2003) and Mackowiak and Wiederholt (2009), although they typi-

1Recent surveys of this literature are Sims (forthcoming), Mankiw and Reis (forthcoming) and Veldkamp(2010).

4

cally focussed on interior solutions to the attention allocation problem, whereas the main

focus of this paper is on the corner solution where agent choose to completely ignore pub-

lic announcements. Van Nieuwerburgh and Veldkamp (forthcoming) also focus on cases

where rationally inattentive agents choose to completely ignore some information, but their

application is on one-period models of asset pricing, whereas in this paper, agents gradually

choose to become inattentive.

One contribution of this paper that may be important for future work in this area is

the development of the theory of rational inattention in continuous time. Moscarini (2004)

also has information arriving in continuous time, but assumes that agents observe it only at

discrete intervals. Instead, in the model in this paper both the arrival of information and

the allocation of attention by agents occurs continuously in time. I introduce a new result

to the economics literature on the mutual information of di¤usion processes, and show that

rational inattention problems can be analyzed in a simple way using deterministic control

problems.

Finally, while this paper approaches the question of when to release information from

the perspective of agents� limited attention, there are potentially other considerations at

play. The delay between policy announcements and their implementation may be due

to a modi�ed version of the war of attrition mechanism in the seminal paper of Alesina

and Drazen (1991). As in Dewatripont and Roland (1992) and Dewatripont and Roland

(1995), policy reforms may proceed gradually in order to gain support form di¤erent groups.

Moreover, the optimal transparency in communication may be related to credibility and

reputation as in Moscarini (2007) or Athey et al. (2005). These papers do not speak

directly to the optimal timing of announcements that is the focus of this paper, but it

would be useful to develop them in this direction in future research. The same applies to

the model of bounded rationality of Bolton and Faure-Grimaud (forthcoming), where agents

take time to deliberate their current and future decisions. While they do not address the

question of timing policy announcements, their model would provide a limited-rationality

alternative to the limited-attention story put forward here.

Outline. I start in section 2 with a simple discrete-time, two-period, white-noise model

to develop the basic intuition behind the key trade-o¤s in the model. Section 3 introduces

5

the general problem in continuous time. Section 4 solves for the equilibrium and for op-

timal policy. Section 5 considers several extensions to the basic framework, and section 6

concludes.

2 A two-period white noise version of the model

There are two periods, t = 0 and t = 1, a continuum of private agents indexed by i and

uniformly distributed over the unit interval, and a single policymaker. A limitation of this

analysis is that the choice of whether to make an announcement is only between these two

dates. Moreover, since shocks are white noise, the only relevant intertemporal trade-o¤ is

on whether to learn about the regime change ahead of time or not. The other side of the

coin to these two caveats is that the analysis is simple and can all be done analytically in

closed-form. Sections 3 and 4 will present a richer model without these limitations (or

these virtues).

2.1 Payo¤s and actions

Each agent chooses a pair of actions fai0; ai1g 2 <2 to maximize:

�1Xt=0

�tEit [ait � !uit � (1� �)at � �rt]2 ; (1)

where � 2 [0; 1] is the discount factor, and Eit(:) denotes the expectations operator condi-

tional on the information of agent i at date t. Each agent wants to track the movements

in three variables, at; rt; and uit, su¤ering a quadratic loss when his actions deviate from

target.2

The exogenous stochastic processes fuitg refer to idiosyncratic shocks to each agent.

They are pairwise independent across agents and have mean zero,Ruitdi = 0, each being

a random i.i.d. draw from a normal distribution with variance �2. They are meant to

capture all of the individual circumstances that each agent must pay attention to.

The other exogenous process rt denotes the policy regime. At date 0, r0 = 0 and all

2The action can be thought of as prices set by �rms, capital investments, or portfolio choices.

6

agents know about it and expect it will be the regime in place at date 1. However, the

policymaker learns at date 0 that the regime will change to r1 = �r 6= 0.3 She cannot

communicate this perfectly to agents, but can only provide them a signal n drawn from a

normal distribution centered at �r but with positive variance �2t . This policy regime may

correspond to something external to the policymaker that she has learned about (e.g., a

shock to money demand), or to a decision that she has made (e.g., a change in money

supply). The focus of this paper is not on the change per se, which I will take as given,

but rather on communicating it subject to only being able to make noisy announcements.

The �nal factor a¤ecting the choices of each agent is the aggregate behavior of all:

at =

Z 1

0aitdi (2)

The parameter � 2 [0; 1] determines the strength of strategic complementarities in this

economy. If � = 0, agents only care about the policy regime via what other agents are

doing, and this becomes an extreme beauty game. If � = 1, then the actions of others are

unimportant, and actions are strategically independent. In between, the smaller is � the

stronger are strategic complementarities. The coe¢ cient in front of the aggregate actions

and the policy regime add up to one so that the parameter ! > 0 governs the relative weight

of idiosyncratic versus aggregate conditions in agents�payo¤s.

With full information at = rt. Motivated by this result, I start by guessing that with

incomplete information a0 = 0 and a1 = �n + (1 � )n, where is a coe¢ cient to be

determined. After solving for the equilibrium, I will verify that this guess is correct and

solve for .

Maximizing (1) with respect to ait gives the standard certainty-equivalence result that

each agent chooses its action equal to the expectation of the variables it wants to track.

Replacing this optimal action back into the expression in (1) gives the payo¤ function:

�!2Ei0(a2it)� �!2Ei1�a2i1�� � [(1� �) + �]2 Ei1(r � �r)2; (3)

3From the perspective of the agent, before an announcement, r1 = 0 with probability one, so the regimechange is a zero-measure event. It therefore correponds to a policy regime change in the sense of the Lucascritique, unanticipated by all, as clari�ed by Sims (1982).

7

2.2 Announcements and information

Learning that there will be a regime change at date 1, the policymaker may choose to

announce this immediately by releasing the public signal r. An announcement is equivalent

to giving agents a normal prior for �r centered at the signal n but with positive variance �2.

The variance captures both uncertainty that the policymaker may have on the new policy

regime, as well as her inability to communicate perfectly with the public.

The policymaker can instead choose to keep its information to herself, releasing it instead

at date 1. At this time, the signal gives agents a normal prior for �r with variance equal to

�2=�. The parameter � � 1 captures an improvement in the policymaker�s information or

in her ability to transmit it.4

Private agents receive signals on individual and aggregate circumstances and choose

how to allocate attention to them. Entering date 1 with a prior variance of �2 for the ui1,

agent�s i posterior variance is �2e�yi1 , with yi1 � 0. The variable yi1 is the (logarithm of

the) reduction in variance from devoting attention to learning more about the individual

shock.5 It is monotonically related to the precision of the signal: if the signal has zero

precision then yi1 = 0; if it has in�nite precision then yi1 ! 1. Likewise, if at date 1,

agents start with a prior variance for r1 of v2i1; their posterior variance is v2i1e

�xi1 , where

xi1 � 0.

At date 0, the problem of inferring ai0 is the same so the posterior variance is �2e�yi0 .

There is nothing to learn about n0, since it equals zero and this is known, nor about

idiosyncratic conditions next period since they are i.i.d. However, if there was been an

announcement, the agent can devote attention to learning more about n1 and this way

improving on the prior for next period. The posterior variance with respect to the aggregate

change is then v2i1 = v2i0e

�xi0 , and the agents may choose xi0 > 0 because of the bene�t of

having lower prior variance tomorrow,

4 In principle, the policymaker could also not release any information at date 1. This is a peculiar casethough, where there has been a change, but no one notices it in the economy, and continues behaving asif n1 = 0 for sure. I exclude it for this reason. Including it though would not make much of a di¤erencethough, as long as I assumed that �n is su¢ ciently di¤erent from zero. In that case, the bias of agents inforecasting n1 would cause such large losses, that policymakers would never choose this option.

5For readers more used to seeing this expression in terms of relative variances of signal and noise, if �iuis the variance of the invididual signal on ui1, then yi1 is de�ned by the expression: yi1 = (�2 + �iu)=�iu.

8

The allocation of attention between aggregate regime and idiosyncratic conditions fxit; yitg,

or equivalently the precision of the two signals at each of the two dates, is not taken as

given. Instead, agents choose them subject to the following upper bound on the reduction

in variance that they are able to achieve:

xit + yit � 2k: (4)

This constraint is motivated by the literature on rational inattention, which shows that the

left-hand side is the reduction in entropy about the two shocks that can be achieved with

their respective signals.. In that literature, k is the amount of information (or �nats�) that

the agent can observe. More generally, this can be seen as a constraint on the precision of

signals that prevents the agent from ever learning everything, and that involves a trade-o¤

between learning about the aggregate regime or the individual circumstances.

I make the following assumption on !, the relative weight on the volatility of the idio-

syncratic fundamentals:

Assumption 1: ! > e�k�=�.

This is su¢ cient to ensure that agents will always want to devote at least some attention

to their individual circumstances. This is the plausible case, for unless the policy regime

involves near nuclear meltdown, even searching for the next meal is a problem that most

people cannot completely ignore.

2.3 The individual problem of allocating attention

Using the notation for posterior variances, and replacing the information constraint (4) in

(3), the payo¤ functions from the perspective of dates 0 and 1 are, respectively:

W (xi0; v2i0) = �!2�2e�2kexi0 + �V (xi1; v2i1) (5)

V (xi1; v2i1) = �!2�2e�2kexi1 � [(1� �) + �]2 v2i1e�xi1 : (6)

The attention allocation problem at date 1 is to choose xi1 to maximize V (xi1; v2i1)

9

subject to the constraint that xi1 � 0.6 It is easy to verify that this problem is concave,

so the �rst-order condition gives the optimal allocation of attention:

x�i1 = max

�k + ln

�vi1�

�+ ln

�(1� �) + �

!

�; 0

�: (7)

Three factors determine the allocation of attention. First, the higher is capacity, then

the more attention agents can devote to all activities. Second, the higher is the relative

prior volatility of aggregate vis-a-vis individual conditions, the more important it is to learn

about aggregate conditions. Third, agents will devote more attention to learning about the

policy regime if there is a higher relative weight of aggregate vis-a-vis individual volatility.

This weight is partly determined by exogenous coe¢ cients (! and �), and partly by the

endogenous response of others�actions to the policy regime captured by .

The problem at date 0 is to choose xi0 to maximize W (xi0; v2i0) knowing that the state

evolves according to

v2i1 = v2i0e

�xi0 : (8)

Now, if x�i1 = 0 it is easy to see that x�i0 = 0. If it does not pay to learn about the policy

regime at the date when it happens, it is also not worthwhile learning about it the period

before. With x�i1 > 0, the value function in the second period is:

V (x�i1; v2i1) = �2vi1�e�k! [(1� �) + �] (9)

Intuitively, higher capacity raises welfare, whereas more volatile conditions or higher losses

from this volatility lower welfare. Maximizing (5) subject to equations (8) and (9) with

respect to xi0 gives the optimal choice of attention at date 0 as the solution to the following

equation:

x�i0 = maxfln� + x�i1; 0g: (10)

The �xed-point solution to this equation can be visualized graphically. The left-hand

side of the equation is a 45o line from the origin. The right hand-side is downward-sloping

because more attention devoted to the policy regime at date 0 (x�i0), lowers the prior variance

6Recall that assumption 1 ensures that the upper bound xi1 � 2k will never bind in equilibrium.

10

at date 1 (s2i1), which lowers the attention at date 1 (x�i1). The two intersect at most once.

Beyond the same three factors that in�uence attention at date 1, there is an extra factor

a¤ecting attention at date 0: the discount factor �. The more agents discount the future,

the less costly is their imperfect knowledge of the future regime vis-a-vis the cost of being

imperfectly informed about individual circumstances right now. Therefore, the lower is �,

the more attention they devote to learning about conditions at date 0, and less attention

is spent on learning about the change in regime the next period. If � is small enough,

agents will choose to not invest any attention at learning anything else beyond the public

announcement.

2.4 The competitive equilibrium

All agents are identical, so I focus on symmetric equilibrium, and drop the i subscripts from

now onwards. Agents�optimal attention choices are given by equations (7) and (10) if an

announcement has been made. I refer to ignoring an announcement at date t as choosing

x�t = 0, since in this case the agents devote no attention to learning more beyond the public

signal.

All that remains to solve for is the coe¢ cient that characterizes a1. Using Bayes

formula, the weight that each person agent puts on its signal on r1 is equal to (1� e�xi1).

Intuitively, the more attention is devoted to the policy regime, the better is agent�s signal

about it, so the more weight it gets, more people know about it. Aggregating to obtain

the aggregate action from equation (2) then gives the �xed point equation for :

= [(1� �) + �] (1� e�x�1) (11)

There are two noteworthy features of this equation. First, note that < 1 since the

agent cannot devote in�nite capacity to learning about the aggregate regime. Second, note

that because agents behave competitively, they take as given when they allocate their

attention. However, their choice of x�1 a¤ects the equilibrium value from equation (11),

and in turn this a¤ects the choice of x�1 in equation (7). More attention to the policy regime

implies a higher ; that is a stronger response of aggregate actions to the regime, which in

11

turn raises the importance of aggregate volatility in the individual�s payo¤ function as long

as � < 1. This payo¤ externality therefore makes paying attention to the policy regime

more costly socially than it is privately.

Equations (7), (10) and (11) jointly solve for the competitive equilibrium for aggregates

and attention fx�0; x�1; g, as a function of the policy announcements. I describe the di¤erent

cases next.

2.5 When are announcements ignored?

It is convenient to introduce a new variable at this stage:

Assumption 2: Let z = �ek=!� and call it the relative importance of the policy regime.

Note that z increases with capacity and with the prior variance for the new regime, while it

falls with the relative welfare weight and variability of idiosyncratic shocks. Therefore, it

measures how important it is for agents to track the policy regime vis-a-vis their individual

circumstances

If the policymaker makes an announcement at date 1, then it gives agents a prior

v21 = �2e��. Combining the three equations describing the competitive equilibrium, this

will be ignored (x�1 = 1) if:

�z � e� (12)

Intuitively, if agents have too little capacity or if the uncertainty regarding the announce-

ment is too small relative to the expected losses from individual uncertainty, then the policy

regime is relatively unimportant (z is small). In this case, agents prefer to ignore the an-

nouncement and will be entirely focussed on their individual circumstances.7 In turn,

because no one pays attention to the announcement, aggregate actions are independent of

the policy regime ( = 0) even though this comes at a cost to everyone.

If instead the policymaker makes an announcement at date 0, the prior at that date

7Combining this expression with assumption 1, note that a su¢ cient condition for this not to be anequilibrium is thast �e2k > 1, so agents have enough attention capacity.

12

becomes: v20 = �2. This will be ignored at date 0 if:

�z � 1=�: (13)

Note that, relative to the condition in (12), this condition will be looser if �e� < 1, and

tighter otherwise. There are two forces at play. On the one hand, if agents discount the

future losses from not knowing the policy regime, they are more inclined to ignore earlier

announcements. On the other hand, if earlier announcements are more imperfect, then

agents will choose to devote more attention to obtaining signals that clarify the message.

Combining the previous two results, if 1=� > �z > 1, agents would ignore an announce-

ment made at date 0, and would only pay attention to it at date 1. The public information

would lay unused by all at date 0. Clearly, as long as the policymaker can improve her

announcement in between dates (� > 0), this would be ine¢ cient. This takes us to the key

question of this paper.

2.6 When is it socially optimal to make announcements?

I take an ex ante perspective on welfare, with the policymaker having its prior on �n and

wishing to maximize the expected utility of the ex ante identical agents subject to their in-

formation limitations. The policymaker�s announcement problem is then to choose whether

to make the announcement at date 0 or at date 1, in order to maximize W (:) in equation

(2), subject to the competitive equilibrium choices of attention {x�0; x�1}, and the commu-

nication constraint that announcing at date 0 implies v20 = �2 while announcing at date 1

implies that x�0 = 1 and v21 = �

2e��.

It is useful though to start by solving a di¤erent problem: the Pareto optimum. Fol-

lowing Angeletos and Pavan (2007), the Pareto optimum solves the problem:

maxx0;x1

W (x0; �2) = �!2�2e�2kex0 + �V (x1; v21) subject to: (14)

V (x1; v21) = �!2�2e�2kex1 � [(1� �) + �]2 v21e�x1 (15)

v21 = �2e�x0 if x0 > 1, v21 = �2e�� otherwise (16)

=�(ex1 � 1)

1 + �(ex1 � 1) (17)

13

The last two constraints show the two main di¤erences relative to the private agent�s prob-

lem. First, if no attention e¤ort or resources are expanded at date 0, still the prior variance

will shrink as long as � > 0. Therefore, even if an individual agent might have wished

to pay attention at date 0, it would be better for social welfare in some cases to delay the

announcement and reap the free arrival of information. Second, as long as � < 1, there

is a payo¤ externality in this economy. Agents do not internalize the fact that when they

devote more attention to the policy regime, aggregate actions depend more on the regime,

and this in turn raises the incentive to pay attention to it. This provides another reason

for why sometimes agents may wish to pay attention, when this is socially undesirable.

A good starting point is when these two di¤erences are not present. This happens when

� ! 1, so there is no free information for the policymaker, and � = 1, so that does not

a¤ect the payo¤s because there are no strategic complementarities. In this case, the Pareto

optimal allocation of attention is:

x0 = max

�2

3(ln� + ln z) ; 1

�; (18)

x1 = max fx0 � ln�; 1g ; (19)

which coincides with the competitive equilibrium fx�0; x�1g in the previous section, if there

is an announcement at date 0.

Since, in this case x1 = x�1, then the solution to the original announcement problem is

to announce at date 0 if x0 > 0 and announce at date 1 if x0 = 0. Therefore, making

an early announcement is optimal if � is not too small so that �z > 1. That is, early

announcement are desirable if agents are su¢ ciently patient and the aggregate regime is

su¢ ciently important.

Next consider the case where � > 1 but keep � = 1. The decision at date 1 is

still the same as before. Agents will privately choose to devote attention to the regime

change (x�1 > 0) if z > 1, and because there is no payo¤ externality, this is the Pareto

optimum amount of attention (x�1 = x1). However, at date 0, the problem perceived by

the policymaker is di¤erent from either the one we just considered or the private problem,

since the policymaker takes into account the free �ow of information that will come from

14

keeping quiet.

The solution is derived in the appendix and it is as follows: let �z be the only solution in

[1=�;1) to the equation 2��z=h3 (��z)2=3 � 1

i= e�=2, noting that �z is increasing in � and

decreasing in �. Then, if z > �z it is optimal to make an announcement at date 0 and the

allocation of attention will be x0 = x�0 = (2=3)(ln� + ln z). If z � �z is it best to not make

an announcement at date 0, but leave it for date 1. Note that this result generalizes the

condition in the previous paragraph: when � = 0, then �z = 1=� so the condition reduces to

�z > 1.

Intuitively, there are two main di¤erences relative to before. First, the condition for

early announcements to be optimal is more stringent. With a free �ow of information,

it may be optimal to wait and learn more. This is stronger if the amount of in�ation

that comes from free is large (� large). Second, now as we cross the threshold making

early announcements worthwhile, attention jumps from 0 to (2=3)(ln� + ln �z) > 0. That

is, there is a range of values for the parameters where private agents are curious about a

regime change and would like to devote attention to learning about it, but the policymaker

thinks it is best to keep the change secret.

Finally, I turn to the general case where � < 1. This case involves messier algebra, and

the steps to solve it are in the appendix. Starting with the Pareto optimum, a �rst result is

that now x�1 > x1: private agents devote more attention to an announcement than what is

is socially optimal. Intuitively, they do not privately take into account the extra cost from

exerting more attention that comes from the excess sensitivity of aggregate actions to the

policy regime. As for attention at date 1, the appendix proves the following result:

x�0 = maxfln� + lnx�1; 1g; (20)

x0 = max

�ln� + ln x1 + ln

��x1 + 1� ��x1 � 1 + �

�; 1

�: (21)

As before, private agents want to set the ratio of attention at date 0 and date 1 equal to

the discount factor. Insofar as x�1 > x1, this would suggest that at the social optimum

attention is also lower at date 0. However, the term in the parenthesis (which is larger than

1 if � 2 (0; 1)) pushes the socially optimum level of attention up relative to the competitive

15

equilibrium, so in general whether x0 is larger or smaller than x�0 depends on the parameter

values. The reason for this second e¤ect is that substituting attention at date 0 for attention

at date 1 minimizes the ine¢ cient coordination e¤ect of attention because at date 0, there

is no change in regime so there is no loss from coordination in terms of aggregate dynamics.

Turning to the optimal announcement date, making an announcement at date 0 is now

only optimal if z > ~z, where the threshold ~z is again increasing in � and declining in �.

Importantly, ~z > �z, so that there are now fewer circumstances when it is worth making an

early announcement. The presence of strategic complementarities provides an alternative

reason to keep mum: as announcements lead agents to start trying to forecast what others

are forecasting about it, they became ine¢ ciently too curious and are best left in the dark.

3 A general model of rational inattention and announce-

ments in continuous time

The model in the previous section is insu¢ cient in some respects to address the general

question on when to optimally time policy announcements. First, the decision was only

whether to make an announcement ahead of time or not; ideally, one would want a model

where announcements can be made farther or less ahead in time. Second, the assumption

that all stochastic processes are white noise is not realistic for most applications. This

section therefore presents a more general model that addresses these two concerns. One side

product of the setup is the introduction of some results on �ltering and mutual information

of stochastic processes that allow for the development of rational inattention theory in

continuous time.

3.1 The setup: environment and payo¤s

Time is continuous, indexed by t 2 <, so that the choice of the optimal timing of announce-

ments comes from a continuous set. All random processes are de�ned in a probability

space (;F ;P), where is the space of outcomes, F is the �-�eld of events (or the mea-

surable subsets of ), and P : F ! [0; 1] is a probability measure. A stochastic process

(or random variable) Xt : ! <, that has a subscript t; is understood to be adapted

16

to the sub-�ltration Ft with an associated measure Pt and its expectation is de�ned as

Et(Xt) =RXt(!)dPt. Each agent observes a di¤erent set of events (or signals) leading to

individual-speci�c Fi and therefore likewise individual speci�c Fit, Pit, and Eit(Xt).

There are two exogenous stochastic processes (or fundamentals). The aggregate regime,

rt, is equal to zero for t < T , and afterwards to a positive constant rt = �r for t � T . Date

T > 0 is the date of the regime change, and the policymaker learns about it at date 0.

However, she is either is not sure of it, or she cannot communicate it perfectly, for all he

can communicate is one noisy signal r to everyone that is an unbiased draw from a normal

distribution:

r � N(�r; �2t ): (22)

The variance of the announcement �2t is equal to �2 at date 0, but after that it declines

exponentially at an exogenous rate � � 0:

d(�2t )=dt

�2t= ��: (23)

The assumption that the policymaker can make more precise announcements as we get

closer to the date of the change may be due to the policymaker learning herself better

about the regime change. The policymaker�s choice is the date � 2 [0; T ] when to release

the n public signal with precision ��2� .

The second set of fundamental are the individual circumstances, uit. These are individual-

speci�c stochastic processes, independent from rt, as well as pairwise independent. I assume

thatRuitdi = 0, although this does not have a signi�cant e¤ect the on results that follow.

The stochastic process has an integral equation for t > 0:

uit = ui0 � �Z t

0uisds+ �Bt; (24)

with an initial condition ui0 that has mean zero and is normally distributed, independent of

Bt. The scalars � and �, capture the persistence and volatility of the process, respectively,

and Bt is a standard Brownian motion. Crucially for what follows, � > 0, so that the

17

process is stationary.8

There is a continuum of private agents, indexed by i and distributed in the unit interval,

i 2 [0; 1]. Their objective function is:

Ui = Ei0�Z 1

0e��t [ait � (1� �)at � �rt � !uit]2 dt

�: (25)

Agents discount the future at rate � > 0, and choose an action ait 2 <, to track movements in

the exogenous policy regime and the individual circumstances, as well as on the endogenous

aggregate action of everyone in the economy: at =R 10 aitdi. The parameter � 2 [0; 1]

measures strategic interdependence in the economy, where the higher is � the higher is the

weight given to the exogenous aggregate regime vis-a-vis the endogenous aggregate actions.

The parameter ! > 0 measures how important are individual circumstances relative to the

aggregate movements.

The policymaker�s goal is to maximize social welfare. Because agents are all ex ante

identical, I focus on an utilitarian measure of welfare that weighs all agents identically:

EP0�Z 1

0Uidi

�: (26)

Note that the expectation is with respect to the information the policymaker has at date

0, crucially including his imperfect knowledge that a change will take place at date T .

3.2 Information

To make her choices at date t, an agent needs a forecast of the vector of fundamentals

ft = (at; rt; uit), knowing that they follow the stochastic processes just speci�ed. As is

typical in the imperfect information literature, the agents can only receive error-ridden

signals about the fundamentals, in the from of a vector of signals zt. That is, they want to

forecast f t = (fs; 0 � s � t) knowing their marginal distribution Pf , by using zt = (zs; 0 �

s � t). Unlike the majority of the literature that assumes a particular joint distribution

between the fundamentals and signals Pfz, Sims (2003) proposed letting agents choose Pfz.

8One can show that uit is normally distributed with mean equal to ui0e��t and variance equal to (1 �e�2�t)�2=2�. So, as t goes to in�nity, uit tends to a normal variable with mean zero and variance �2=2�.

18

Sims (2003) describes this as an attention-allocation problem in the sense that the agent is

endogenously choosing the channel by which he receives news about the outside world. The

only constraint in choosing the stochastic process for zt is that this is not too informative:

Zlog

�dPfz

d (Pf Pz)

�dPfz � kt: (27)

The left-hand side is Shannon (1948) measure of the information that zt gives about

f t. Is is called mutual information, and it equals the reduction in entropy in going from

the marginal distribution for f t to instead the conditional distribution on zt.9 If the

signal provides any information on the fundamental, that is if the two are not independent

random variables, then mutual information is always positive. Mutual information is

uniquely de�ned, and Shannon (1948) showed that it follows from a few axioms on the

information content of messages. It has become the universal measure of information in

the �eld of �information theory�and it is closely related to the Kullback-Leibler measure

used in statistics, as well as to Kolmogorov�s measure of complexity in complexity theory.10

The right-hand side is the amount of information agents are allowed to get, where k > 0

is the constant rate of information transmission per instant of time. I will take the base of

the logarithm in mutual information to be the natural number, so that k can be interpreted

as the number of �nats�, but all the results are robust to any other base for the logarithm.

While the general theory of rational inattention allows agents to choose Pfz from any

set of distributions, I will follow Mackowiak and Wiederholt (2009) and restrict this choice

by imposing two assumptions. First, I assume that agents must choose two indepen-

dent signals, one for the aggregate regime, and another for the individual circumstances.11

Mackowiak and Wiederholt (2009) argue for assuming that paying attention to aggregate

and idiosyncratic conditions are separate activities. Second, I assume that whatever noise

agents receive about uit is idiosyncratic. Intuitively, this says that while agents are all ex

9Recall that the entropy of a random process f t is �Rlog(dPf )dPf .

10Mutual information is equal to the Kullback-Leibler distance between the joint distribution Pfz and theproduct of the marginals Pf and Pz. The expected average Kolmogorv complexity of n i.i.d. draws of arandom variable from a distribution convergers to the entropy of that distribution as n goes to in�nity.11The key assumption is that the vector of signals can be clip into two independent sub-vectors, one having

non-zero mutual information only with rt, and the other with uit. Saying that these two sub-vector haveone element is without loss of generality.

19

ante identical, and set up channels with similar properties to receive information, the noise

that contaminates these channels is individual-speci�c.

In principle, the problem of picking Pfz to maximize (25) subject to (27) and the two

assumptions in the previous paragraph is quite hard. The choice is an in�nite-dimensional

distribution, and the capacity constraint does not ensure that the distribution will even be

continuous. Moreover, even after we know Pfz, the agent�s problem of picking ait will require

forming expectations of the f t, but for a general Pfz this non-linear �ltering problem does

not have analytical solutions. In spite of these formidable problems, the joint assumptions

that the objective function in (25) is quadratic and that the fundamental follow normally-

distributed stochastic processes (22) and (24) will lead to a surprisingly simple solution.

This is due to three fundamental results, which I will state next, applied to the special case

of the problem in this paper.

The �rst result is well-known in the theory of rational inattention, and was extensively

used by Sims (2003) and Luo (2008):

Proposition 1 (Shannon) The optimal signals (zrit; zuit) on (rt; uit) are unbiased but conta-

minated with additive Gaussian noise:

dzrit = rtdt+p�irtdB

0it; (28)

dzuit = uitdt+p�iutdB

00it: (29)

where B0it and B00it are standard Wiener processes, independent across any pair of i, inde-

pendent of each other, and independent of Bt. The noise in the two channels at every

instant, �irt and �iut, are choice variables of each private agent.

The more general result of Shannon (1948) is that if Pf is normal, and the objective

function is quadratic, then Pfz is a joint normal distribution as well. The proof of this

result has not appeared in the economics literature, even though it is frequently cited, so

the appendix provides it for this case. Perhaps the most remarkable feature of this result

is that the attention allocation problem of choosing the in�nite-dimension distribution Pfz

reduces to picking only the pair of scalar variances, �irt and �iut. These are then measures

of the (in)attention that agents pay to the two sources of uncertainty.

20

The second result is an application of the �ler due to Kalman and Bucy (1961), well-

known to most economists working in �ltering in continuous time.

Proposition 2 (Kalman-Bucy) Letting rit = Eit(�r) and �irt = Eit(�r�Eit(�r))2, respectively

the forecast mean and the forecast variance, then their evolution is described by the following

dynamic equations:

drit =�irt�irt

(dzrit � ritdt) ; (30)

d�irt=dt

�irt= �

��irt�irt

�: (31)

De�ning uit and �iut similarly:

duit = ��uitdt+�iut�iut

(dsuit � uitdt) (32)

d�iut=dt

�iut=

�2

�iut� 2� � �iut

�iut: (33)

A check on this result may provide some intuition that makes the equations easier to

digest. If the announcement was made at date 0, and the noise of the signal on the aggregate

regime, �irt, was constant, then equation (31) becomes a simple Ricatti equation that has

the closed-form solution:1

�rt=

1

�0t+t

�r: (34)

In this case, note that the problem is that of forecasting a scalar �r using observations

srit = rt +p�rB

0it, the sum of two normals where rt � N(�rt; �20t2) and

p�rB

0it � N(0; �r),

so that the standard signal extraction formula gives:

�rt =�r�

20

�r + �20t: (35)

Equations (34) and (35) are of course identical since �0t = �20.

Analyzing the solutions for the variances, note that the least attention is devoted to

the aggregate regime, the higher is the noise in its signals �irt and so, the slower does the

forecasting variance �irt fall. The same applies to �iut and �iut. Moreover, the higher is

the variance of the new shocks hitting individual circumstances �2, the higher will be the

21

forecast variance. Finally, the more predictable is the process, because it reverts to its mean

quickly, � is high, the faster again will the forecast variance fall with time.

Looking at the asymptotic limits of this �ltering provides some usual hints for when we

go to solve the attention problem, Assuming that the allocation of attention converges to

some �nite level: limt!1 �irt = �r and limt!1 �iut = �u, then:

limt!1

�rt = 0; (36)

limt!1

�ut = ���2 +q�2�4 + �u: (37)

Intuitively, the agents need to learn about the regime, but this is a constant. If they always

devoted non-zero attention to it, they would eventually learn about the regime for sure.

Only if at some point, they stop devoting attention, so �irt =1, will �irt be positive (and

constant) from then onwards. Individual circumstances, instead are hit by shocks every

period. therefore, there is always some uncertainty on what the current uit is.

The last step remaining is to measure how di¤erent choices of �rt and �ut a¤ect mutual

information in order to impose the constraint in (27). The third result answers this and is,

as far as I am aware, new to the economics literature:

Proposition 3 (Duncan) Mutual information is determined by the signal-to-noise ratio of

each fundamental-signal pair:

Zlog

�dPfs

d (Pf Ps)

�dPfs =

1

2

Z t

0

�vrs�rs

+vus�rs

�ds (38)

This result was �rst proven by Duncan (1970). Most of the literature in economics so

far has used only discrete-time setup so no result for the mutual information of continuous-

time processes has been stated, with only two exceptions. Sims (2003) brie�y discusses

the formula for mutual information in continuous time, and shows how to express it in

terms of the spectral density of the processes. Moscarini (2004) sets up a continuous-time

problem, but where agents can only obtain noisy signals at discrete times. Moreover, he

notes (Moscarini (2004), pages 2015-15) that the classic textbook of Lipster and Shiryaev

(2001) requires that the fundamentals have �nite quadratic variation almost surely, an

22

assumption that fails for di¤usions but is common in engineering applications of information

theory. Duncan (1970) result is not as well-known in the literature, but he shows that with

normal fundamentals and normal signals, as long as the expected quadratic variation is

�nite, which is satis�ed by most di¤usions, then mutual information exists and can be

easily computed from the mean-squared error of the prediction.

3.3 The competitive equilibrium in actions

Working with quadratic objective functions has another bene�t beyond allowing for easy

computation of mutual information. They imply that a separation principle applies,

whereby we can solve for the problem of choosing actions faitg taking as given the choices of

attention f�irt; �iutg, and then solve for the choice of attention taking as given the optimal

actions.

I focus on linear symmetric competitive equilibrium. Linear because, given that the

problem is linear in the fundamentals and signals, I look for equilibria where the path of

aggregate variables depend only on the aggregate fundamentals and signals. Symmetric

because, since all agents are ex ante identical and only receive individual signals, I focus

on the case where agents may take di¤erent actions but ex ante make the same decisions

allocating attention. Competitive because agents take the evolution of at as given.

Starting with the actions problem, private agents choose ait each period to maximize

(25). The �rst-order condition is:

a�it = (1� �)Eit(at) + �Eit(rt) + !Eit(uit): (39)

Before date T , rt = 0 and I assume that all agents knew about it. Since this is the only

aggregate variable, it is reasonable to conjecture that at = 0 for t < T as well. Plugging

this into equation (39) and integrating over the unit mass of agents, veri�es the conjecture.

For t � T the new regime is �r and agents have received a public announcement r. Since

these are the only two aggregate variables, I conjecture that:

at = t�r + (1� t)r; (40)

23

where t is a deterministic process that agents take as given.

With this guess, all that remains is to verify it, and to solve for t. Using the guess in

equation (39) gives:

a�it = [(1� �) t + �]rit + (1� �)(1� t)r + !uit; (41)

recalling the de�nitions of rit and uit in proposition 2. Integrating over the unit mass of

agents and letting rt =Rritdi:

at = [(1� �) t + �]rt + (1� �)(1� t)r (42)

Finally, combining equations (28) and (30) shows that drt=dt = �rt�rt(�r � rt) for t � � with

r� = r. From this it follows that:

rt = �tr + (1� �t)�r, where (43)

�t � e�R t0 (�rs=�rs)ds (44)

Combining equations (41), (42) and (43) veri�es the guess and gives the solution for t:

t =��t

1� �t + ��t:(45)

Combining these results gives the competitive equilibrium for actions, taking as given

the choices of information.

Proposition 4 Taking as given the allocation of attention (�rt and �ut) and the variances

they lead to (�rt and �ut), the competitive equilibrium for the actions is in equation (40),

with t de�ned in equation (45) and �t de�ned in equation (44). The individual payo¤

functions at the optimal actions are identical for all agents:

W =

Z T

0e��t!2vutdt+

Z 1

Te��t

�!2vut + [(1� �) t + �]�rt

�dt (46)

24

3.4 The competitive equilibrium in attention

The individual attention problem is to choose the allocation of attention f�irtg1t=0 and

f�iutg1t=� to maximize (46) subject to the evolution of the variances in equations (31), (33)

and the information capacity constraint in equation (38). To start, note that the laws of

motion for the variances do not depend on the individual signals, so that if agents all make

the same attention decisions, then the variances will be identical across agents. But in turn,

from the objective function, if agents face the same evolution for the variances, they all

choose the same allocation of attention. Therefore, I drop the i subscripts, and focus on

the symmetric equilibrium.

By date T , the latest, agents have become aware that there has been a regime change.

At that time, their best estimate of �r is riT , and the variance of this estimate is �rT , so these

are initial conditions for the problem from then onwards. The problem of an individual

agent is to picking non-negative values for f�rt; �utg1t=T to maximize the second term in

(46) subject to the evolution of the variances in (31) and (33), and the information capacity

constraint in (38). Collecting all of these ingredients in one place, and de�ning the indirect

utility function of the agent as V (vrT ; vuT ; T ):

V (vrT ; vuT ; T ) = max

��Z 1

Te��(t�T )

�!2vut + [(1� �) t + �]�rt

��(47)

vrt�rt

+vut�ut

� 2k (48)

d�rt�rt

= ���rt�rt

�dt (49)

d�ut�ut

=�2

�ut� 2� � �ut

�ut(50)

�rt; �ut � 0 (51)

Turning to the date of the announcement � , the problem facing the agents is:

W (�2t ; v�u; �) = max

f�rt;�utgTt=�

��Z T

�e��(t��)!2vutdt� e��(T��)V (vrT ; vuT ; T )

�(52)

subject to the same constraints as in the problem for V (:) but with the initial conditions

�r� = �2t and �u� = v�u. Because at date � , the agent only has the signal n about the

25

new regime, this is his best forecast of the new regime �n, and it comes with the exogenous

variance �2t . As for the idiosyncratic shocks, agents have been devoting all their attention

to them for an in�nite amount of time before, so they have converged to the steady-state

value, which combining (33) and (48) is v�u = �2=2(� + k):

Finally, the problem of the agent at date 0 < t < � is straightforward: he devotes all his

attention to the idiosyncratic circumstance, so the variance of the signals is �ut = 2k=v�u,

and the variance stays at the steady state v�u. The payo¤ is:

U� = �!2v�u�1� e���

�

�+ e���W (�2t ; v

�u; �) (53)

Proposition 5 The competitive equilibrium in attention taking aggregate actions t as

given, is the allocation of attention f�rtg1t=0 and f�utg1t=� and the resulting variances fvutg1t=0and fvrtg1t=� that solve the Bellman equations in (47)-(51), (52) and (53).

One important feature of the competitive equilibrium is that, in spite of all the uncer-

tainty and the multiple signals and shocks hitting the economy, the competitive equilibrium

boils down to solving a sequence of deterministic control problems, which are part of the

standard toolkit of most economists. This is the main result of this section, which is more

general than the study of the optimal timing of announcements: as long as we are willing

to keep the assumption of quadratic objective functions that is pervasive in the literature

on imperfect information, then modelling information endogenously through the theory of

rational inattention leads to tractable problems.

4 The optimal timing of announcements

After the regime change, each agent allocates attention to solve the following Bellman

equation, which captures the optimization problem in equations (47)-(51):

�V � @V@t

= max0�xt�2k

8>>><>>>:�!2vut � [(1� �) t + �]�rt

��@V@vrt

�xt�rt

+�@V@vut

���2

�ut� 2� � 2k + xt

��ut

9>>>=>>>; : (54)

26

I introduced the notation xt � vrt=�rt to denote the attention devoted to the aggregate

regime. When xt = 0 this corresponds to an in�nite variance of the signal, that is no

attention devoted to learning about the regime change at all. When xt = 2k, then the

signal on the aggregate regime is as precise as possible while there is no new information

about the individual circumstances. In between, the larger xt the more attention is devoted

to the aggregate regime and the more precise are the signals on it.

One feature of the competitive equilibrium is that people take t, the aggregate dynam-

ics, as given. As long as attention changes over time, then so will t and this implies that

the relative weights in the payo¤ function changes. As a result, the Bellman equation in

(54) is a partial di¤erential equation, that can generally only be solved numerically. Before

attacking that problem, therefore, I focus on the case where � = 1. Now, t drops out

of the individual problem, since there are no longer any strategic complementarities. The

Bellman equation becomes an ordinary di¤erential equation that can be studied analyti-

cally. Moreover, the two components of the competitive equilibrium in propositions 4 and

5 become independent of each other. The problem then becomes much easier, and so I

start with it before turning to the general case at the end of the section.

4.1 Optimal behavior after the regime change

Without strategic complementarities, the Bellman equation in (54) becomes:

�V = max0�xt�2k

8<: �!2vut � �rt ��@V@vrt

�xt�rt

+�@V@vut

���2

�ut� 2� � 2k + xt

��ut

9=; : (55)

The appendix derives the optimality conditions and presents the phase diagram associated

with this problem. The main result are collected in the next proposition:

Proposition 6 After the regime change at date T :

a) If the uncertainty on the aggregate regime is low:

�rT <�!�2

2(� + k) [�+ 2(� + k)]; (56)

then xt = 0 for all t � T , that is agents never pay any attention to the regime change.

27

b) Otherwise, xt approaches 0 asymptotically, and as it does:

�ut ! �2

2(� + k)(57)

�rt ! �!�2

2(� + k) [�+ 2(� + k)]: (58)

The �rst case in the proposition refers to a situation in which agents have a very precise

understanding of the new regime. In comparison, they still face uncertainty on their

current individual circumstances, which are changing every period. Ideally, they would

like to trade some of their knowledge of the aggregate regime for a more precise knowledge

of the individual circumstances. Not being able to do this, the next best thing is to never

devote any attention to the aggregate regime, and never learn more than what they started

with.

The initial precision on the aggregate regime is not an exogenous variable though, but

is endogenously determined by agent�s attention choices between the announcement date

and the regime-change date. It would never be optimal for the agent to reach the situation

described in part a) of the proposition, since he could just have devoted less attention to

the aggregate regime in the past. Therefore, as long as the initial announcement is not too

precise, the situation in part a) of the proposition would never be observed; since the most

precise announcement possible is done at date T inducing a variance �2e��T , the following

assumption is su¢ cient to rule this case out:

Assumption 3: �2e��T > �!�2= [2(� + k) [�+ 2(� + k)]].

Focussing instead on case b) in the proposition, �gure 1 shows the typical dynamics

of attention about the aggregate regime after date T . Initially, uncertainty about the

regime is high, so people devote all of their attention to learning more about the change.

After some time, they have learned enough to start devoting some attention to individual

circumstances, at which point xt jumps down. From then on, the system has saddle-path

dynamics, with attention to the aggregate regime slowly converging to zero and uncertainty

converging to its steady-state.

28

Noticeably, �r does not go to zero, so agents never learn for sure what the new policy

regime is. After they have a good enough estimate, it is just not worthwhile improving on

it instead of devoting all their attention to learning about the new idiosyncratic shocks that

arrive every instant.12

4.2 Optimal behavior between the announcement date and the regime

change

Given an announcement at date � , sometime between when the policymaker learned about

the regime change (date 0) and the date when the change took place (date T ), the private

agents problem has the following Bellman equations:

�W � @W@t

= max0�xt�2k

8<: �!2vut ��@W@vrt

�xt�rt

+�@W@vut

���2

�ut� 2� � 2k + xt

��ut

9=; : (59)

@W

@vrt

����T

=@V

@vrt

����T

and@W

@vut

����T

=@V

@vut

����T

(60)

There in one main di¤erence relative to the problem after the regime changes. First,

note that only the uncertainty on the individual circumstances a¤ects the payo¤ today. The

private agents may wish to devote attention to the aggregate regime only because of the

gains that would bring in the future. Therefore, if at the date of the regime change, more

precise information about the regime was worthless (@V=@vrtjT = 0), then this information

would also be worthless at all other dates (@W=@vrtjT = 0) and the optimal allocation of

attention would be xt = 0 at all dates. Otherwise though, the agent may wish to devote

attention to the regime change at a cost today because this will be valuable later.

The appendix solves this dynamic optimization and proves the following result:

Proposition 7 Between the announcement date � and the regime change at date T , if T��

is large enough, there are three stages in the evolution of attention

Stage 1) Initially, from � to ��, the optimal xt = 0, so agents devote no attention to the

12The reader might wonder why did I then assume that information about the old regime was perfect atdate 0. This assumption made the presentation easier, but has no in�uence on the results. As long as theuncertainty about the old regime had reached its steady-state, agents would never expand any attention onit ever again, so this would not a¤ect any of the attention allocation decisions studied in this paper.

29

announcement.

Stage 2) Between �� and T �, xt increases continuously and monotonically from 0 to 2k.

Stage 3) Between T � and T , xt = 2k and the agents are devoting their full attention to the

regime change.

As � becomes closer to T , we observe only stages 2 and 3, or only stage 3.

Figure 1 combines the results from the previous two propositions to show the life-cycle

of attention to aggregate regime changes. If the announcement comes very early, people

ignore it. The bene�ts of learning about the future are being discounted at rate �, while the

costs in terms of reacting worse to current individual circumstances are felt immediately.

As the regime change approaches, discounting is not so heavy and people start investing

some capacity in learning about the change. The amount of attention increases with time

as the change gets nearer and peaks at the maximum just before the change. After the

change, and when the agents have a precise enough estimate of the new regime, they start

allocating attention slowly back to individual circumstances.

To give the policymaker the largest set of possibilities to choose from when picking the

optimal announcement date, I add the following assumption

Assumption 4: T is large enough that an announcement at date 0 with precision �2 would

lead to observing all three stages in proposition 7.

4.3 Optimal announcement date

In the tradition of Ramsey, the policymaker takes the competitive equilibrium de�ned in

propositions 4 and 5 and described in propositions 6 and 7 as given, and chooses the

date of the announcement to maximize his unweighted utilitarian expectation of welfare.

Mathematically the policy problem is to choose � 2 [0; T ] to maximize:

U� = �!2v�u�1� e���

�

�+ e���W (�2t ; v

�u; t) (61)

30

The �rst-order conditions from this problem is:

�!2v�ue��� + e����@W

@t+@W

@�2�

@�2�@�

�= 0 (62)

The �rst term is the loss in terms of having an extra instant of time when attention to the

problem is zero and the variance of the individual circumstance is at its steady-state. The

second-term is the gain that comes from two sources: the allocation of private attention

to the regime change and the improvement in the precision of the signal. Both sources

are straightforward to calculate, since an important result in optimal control theory is that

@U=@t is equal to the negative of the maximized Hamiltonian.

Consider then two cases. First, when �! 0, so that there is no improvement in public

precision, then the optimal announcement date is equal to the date at which agents would

start paying attention to an announcement. That is, � is equal to the �� at which stage

2 in proposition 7 starts. Intuitively, because � > 0, the policymaker will never want to

release an announcement that is ignored; by waiting an extra instant, he could make a

better announcement. If � ! 0 though, as soon as agents privately would choose to pay

some attention to the aggregate regime, then the policymaker wants to allow them to do

so.

The second case is when � is above this limiting case. The appendix proves the following

result:

Proposition 8 If � > 0, then the optimal time of an announcement is strictly after the

time when agents would start paying attention to it: � = ��. If �+ �=2 < 2k, then:

� = �+�

2; (63)

and at that point, attention starts at a strictly positive value.

The solution shows that, as expected, the faster the policymaker learns about the regime

change, then the later she should make an announcement. Moreover, the more agents

discount the future, the later they will wish to start paying attention to the regime change,

so again the later the optimal announcement should be. Remarkably these are the only two

31

determinants of the optimal announcement date.

4.4 Optimal announcements with strategic complementarities

To be written...

5 Policy analysis

The theory of attention allocation and policy announcements developed in the previous

three sections and represented in �gure 1 leads to several principles for the communication

of policies.

Policy principle 1: Do not confuse people. If there is no change in the aggregate regime,

do not make announcements.

In terms of the theory, this corresponds to the case where in fact �n = 0, so the new

regime is identical to the old one. However, if the policymaker makes an announcement,

all that the agents observe is a noisy signal n and must then spend precious attention to

gradually �gure out that in fact the regime has not changed. This wasted attention is

costly because it leads to worse decisions regarding everyday idiosyncratic shocks.

Policy principle 2: If can �x it, do it, don�t talk about it. If the policymaker can react to

o¤set the regime change, it is best to do it than to allow the change and make announce-

ments.

Imagine now that nt = ot + pt where ot is exogenous but pt is controlled by the policy-

maker. In response to a change in ot, it is best to set pt to o¤set it, than to announce the

change, since the latter requires spending scarce attention.

Policy principle 3: Be relevant but keep mum in spite of public interest. There is an

optimal time to make an announcement.

This summarizes the main result from the previous section. Announcements that are

made too early are ignored and socially costly. Announcements that are made too late do

32

not give people enough time to get ready for the coming changes. The optimal timing of

the announcement balances these two forces, and occurs after the time when people would

start paying attention to it for two reasons. First, because the policymaker can make more

precise announcements as we get closer to the date of the change. Second, because people

put too much weight on announcements as they try to �gure out what others know or think

they know.

Policy principle 4: Rumors should be squashed early. If an imprecise rumor starts that

a regime change is coming, the policymaker should respond by making an announcement as

soon as people start paying attention to the rumor.

Imagine now that at date 0, a rumor starts that there will be a regime change at date T .

Rumors can be treated just like announcements, as signals with a given variance, and their

imprecision is captured by that variance being higher than �2. Without any announcement,

this will lead agents to start devoting attention to the aggregate regime earlier than what

they would have in response to an announcement at date 0. The policymaker loses control

over this initial date at which attention will be devoted to the change. The optimal response

is to make an announcement as soon as this is about to happen, in order to give agent a

more precise signal than the rumor.

6 Conclusion

When should a policymaker make an announcement about an impending change that will

a¤ect everyone? This paper provided an answer to this question based on the notion that

people have a limited amount of attention that they choose to allocate (or not) to policy an-

nouncements. I focussed on two trade-o¤s involved in optimally timing announcements, one

private and the other public. The private trade-o¤ is between obtaining more information

early on about the future regime change to make better choices when it arrives versus in

exchange paying less attention to individual circumstances today and making worse choices

now. The public trade-o¤ is between satisfying agents�demand for information versus wait-

ing to obtain a more precise signal of the impending change and �ghting the over-reliance

of agents on public signals.

33

The aim was to make three contributions. First, to suggest a new question to the small

literature on policy announcements, focusing on the optimal timing of announcements.

Second, to propose a limited-attention theory to the topic of optimal communication of

policy changes leading to several applied policy principles. Third, to develop the theory of

rational inattention in continuous time in linear-quadratic-gaussian environments showing

that it leads to deterministic optimal control problems that �t into the standard economics

toolbox. In none of these three areas, this paper will surely be the last word.

34

Appendix

Still to be written...

35

References

Alesina, Alberto and Allan Drazen (1991), �Why are stabilizations delayed?� American

Economic Review, 81, 1170�88.

Amador, Manuel and Pierre-Olivier Weill (2009), �Learning from prices: Public communi-

cation and welfare.�Stanford University manuscript.

Angeletos, George-Marios and Alessandro Pavan (2007), �E¢ cient use of information and

social value of information.�Econometrica, 75, 1103�1142.

Athey, Susan, Andrew Atkeson, and Patrick J. Kehoe (2005), �The optimal degree of dis-

cretion in monetary policy.�Econometrica, 73, 1431�1475.

Blinder, Alan S., Michael Ehrmann, Marcel Fratzscher, Jakob De Haan, and David-Jan

Jansen (2008), �Central bank communication and monetary policy: A survey of theory

and evidence.�Journal of Economic Literature, 46, 910�45.

Bolton, Patrick and Antoine Faure-Grimaud (forthcoming), �Thinking ahead: The decision

problem.�Review of Economic Studies.

Dewatripont, M and G Roland (1992), �Economic reform and dynamic political con-

straints.�Review of Economic Studies, 59, 703�30.

Dewatripont, Mathias and Gerard Roland (1995), �The design of reform packages under

uncertainty.�American Economic Review, 85, 1207�23.

Duncan, Tyrone E. (1970), �On the calculation of mutual information.�SIAM Journal of

Applied Mathematics, 19, 215�220.

Eusepi, Stefano and Bruce Preston (forthcoming), �Central bank communication and ex-

pectations stabilization.�American Economic Journal: Macroeconomics.

Hellwig, Christian (2005), �Heterogeneous information and the welfare e¤ects of public

information disclosures.�UCLA manuscript.

Kalman, R. E. and R. S. Bucy (1961), �New results in linear �ltering and prediction theory.�

Transactions of the ASME, 83, 95�107.

36

Lipster, Robert S. and Albert N. Shiryaev (2001), Statistics of Random Processes. Springer-

Verlag New York.

Lorenzoni, Guido (2010), �Optimal monetary policy with uncertain fundamentals and dis-

persed information.�Review of Economic Studies, 77, 305�338.

Luo, Yulei (2008), �Consumption dynamics under information processing constraints.�Re-

view of Economic Dynamics, 11, 366�385.

Mackowiak, Bartosz and Mirko Wiederholt (2009), �Optimal sticky prices under rational

inattention.�American Economic Review, 99, 769�803.

Morris, Stephen and Hyun Song Shin (2002), �Social value of public information.�American

Economic Review, 92, 1521�1534.

Morris, Stephen and Hyun Song Shin (2007), �Optimal communication.� Journal of the

European Economic Association, 5, 594�602.

Morris, Stephen, Hyun Song Shin, and Hui Tong (2006), �Social value of public information:

Morris and shin (2002) is actually pro-transparency, not con: Reply.�American Economic

Review, 96, 453�455.

Moscarini, Giuseppe (2004), �Limited information capacity as a source of inertia.�Journal

of Economic Dynamics and Control, 28, 2003�2035.

Moscarini, Giuseppe (2007), �Competence implies credibility.�American Economic Review,

97, 37�63.

Roca, Mauro (2006), �Transparency and monetary policy with imperfect common knowl-

edge.�Columbia University manuscript.

Shannon, Claude E. (1948), �A mathematical theory of communication.�The Bell System

Technical Journal, 27, 379�423, 623�656.

Sims, Christopher A. (2003), �Implications of rational inattention.� Journal of Monetary

Economics, 50, 665�690.

37

Svensson, Lars E. O. (2006), �Social value of public information: Comment: Morris and shin

(2002) is actually pro-transparency, not con.�American Economic Review, 96, 448�452.

Van Nieuwerburgh, Stijn and Laura Veldkamp (forthcoming), �Information acquisition and

under-diversi�cation.�Review of Economic Studies.

38

Figure 1. The life‐cycle allocation of attention to the aggregate regime change

x*

τ* τ^0 T t