Price Adjustment in a Model with Multiple-Price Policies
y
Luminita Stevensz
Columbia University
January 2012
Abstract
Understanding the patterns of prices in the micro data is a key step towards settling the debates on
the importance of nominal rigidities and the role of monetary policy. This paper presents the dynamic
price-setting problem of a rm that cannot observe market conditions for free. If it pays a xed cost,
the rm can obtain complete information on the state of the world and review its policy. In each period
between policy reviews, the
rm decides whether or not to pay the
and which price to charge from the current policy. The
xed cost and review its policy
rm can purchase additional information in
order to make each of these decisions, subject to a cost per unit of information. The paper shows that
the rm optimally chooses to only infrequently undertake policy reviews, and that between reviews it
implements a simple pricing policy that consists of a small set of prices. The model therefore matches
the empirical evidence of discrete multiple-price regimes documented in Stevens (2011). Although prices
change frequently, they are only partially related to market conditions at all times. Hence there is scope
for signi cant monetary non-neutrality.
This project has bene ted from the continued guidance and support of Mike Woodford. I am also especially greatful
to Ricardo Reis for valuable suggestions. I would like to thank Stefania Albanesi, Ryan Chahrour, Christian Hellwig, Filip
Matejka, Emi Nakamura, Jaromir Nosal, Ernesto Pasten, Bruce Preston, Dmitryi Sergeyev, Jon Steinsson, Heriberto Tapia, and
seminar participants at Columbia University, The Center for European Economic Research, Mannheim, and Toulouse School
of Economics for helpful comments. Part of this work was conducted while I was visiting the University of Mannheim. I thank
Klaus Adam for his hospitality.
y PRELIMINARY. New versions will be available at http://www.columbia.edu/~lld2108/research.html.
z Contact: lld2108@columbia.edu.
1
1
Introduction
How do rms set prices in response to constantly changing market conditions? In particular, how quickly
and accurately do they react to macroeconomic shocks? These questions have been central to monetary
economics ever since nominal price rigidities were introduced as a key link in the monetary transmission
mechanism. Di erent answers imply ascribing di erent roles to monetary policy, and yield a di erent account
of what drives business cycles. Not surprisingly, there is a large literature on both the evidence and the
theory of how prices are set.
Starting with the seminal paper of Bils and Klenow (2004), recent empirical work1 has focused on
characterizing pricing patterns at the product level as a way to discipline theories of price-setting. This
literature has documented broad-based high volatility of good-level prices, even under stable macroeconomic
conditions. The evidence of high volatility at the micro level poses a challenge to popular monetary models,
in which prices that remain unchanged for long periods of time are the key to obtaining signi cant real e ects
of monetary policy. Speci cally, while monetary models often assume stickiness of one year or longer, Bils
and Klenow (2004) show that in the US CPI, the prices of many products change at least every four months.
Nakamura and Steinsson (2008) further document that much of this volatility is driven by short-lived price
changes to and from a rigid price level. In response, recent theoretical work has sought to develop models of
price-setting that distinguish between regular and transitory prices.2
This paper proposes a di erent interpretation of product-level volatility, according to which rms choose
pricing policies in which both regular and transitory prices are chosen to be jointly optimal. My approach
is motivated by the stylized facts documented in Stevens (2011). That paper analyzes pricing patterns in
Dominick's grocery store data, and nds that price series are characterized by infrequent breaks that identify
\pricing regimes." Moreover, these regimes are de ned by a small set of prices relative to both the duration
of the regimes and the frequency of price changes within regimes. Approximately three quarters of products
contain regimes in which a small number of prices are revisited over the life of the regime. The left panel of
Figure 1 illustrates this pattern for the weekly price of frozen juice over a four-year period: prices change
1 The chapter by Klenow and Malin in the Handbook of Monetary Economics (2010) provides a comprehensive review of the
recent empirical literature on price setting at the micro level.
2 See, for example, Kehoe and Midrigan (2010) and Guimaraes and Sheedy (2011).
2
Figure 1. Sample price series from Dominick's.
very frequently and by large amounts, yet they alternate among a few distinct values over the life of each
regime. Approximately one quarter of products consist entirely of regimes in which prices either do not
change at all, or change very rarely (for example, around holidays), as in the right panel of Figure 1. While
the pattern of single sticky prices can be accommodated by existing sticky price theories, the pattern of
regimes with multiple rigid prices is inconsistent with existing theories of price-setting.
This paper explains both of these patterns using a theory of dynamic price setting in which prices
are determined by pricing policies that are sticky and simple, namely they are updated infrequently and
consist of a small set of prices. Both the stickiness and the coarseness of the pricing policy are a result of
costly information. I consider the problem of a monopolistically competitive rm that sets prices subject to
uncertainty in its demand and its production technology. Obtaining any information about the state of the
world is costly in two ways. First, both the rm's prices and its acquisition of information are determined
by a policy that can be reviewed subject to a xed cost. As in Reis (2006), payment of this cost enables
the rm to collect complete information about the state of the world at the time of the review. The xed
cost represents the managerial resources required in order to acquire and process the information, design the
policy, and communicate it within the rm, as documented by Zbaracki, Ritson, Levy, Dutta, and Bergen
(2004). Due to the xed cost, the rm does not update its policy in every period.
Second, in every period between policy reviews, the rm acquires additional information, based on which
it makes two decisions: whether or not to review its policy and, if the policy consists of a menu of prices,
which price to charge. The additional information acquired between policy reviews is subject to a cost per
3
unit of information, which captures the cost of monitoring market conditions. The measurement of the
amount of information acquired for each decision follows the rational inattention literature (Sims, 2003).
The rm chooses what aspects of the state of the world to monitor and how much information to acquire for
each decision. The signals that the rm chooses to acquire compress the state into a simpler representation,
given the rm's objective, the xed and variable costs of information, and the market conditions that the
rm expects to encounter under the current policy, until the next review. For each decision, the rm has
access to no other information except that received through the corresponding signal: the review signal and
the price signal act as the only interface between the rm and its environment at the time of each decision.
I rst show that the rm's optimal policy consists of three elements: (1) a hazard function that speci es
the probability of conducting a policy review conditional on the current state, (2) a set of prices, and (3) a
conditional distribution that speci es which price to charge conditional on the current state. The optimal
policy has the same form for all periods until the next review. Moreover, the optimal set of prices chosen
at each review di ers across reviews. Hence, every policy review starts a new regime, and every regime is
identi ed by a new distribution of prices.
I then characterize the properties of the solution. Prices vary stochastically with the current state,
and policy reviews are stochastically state-dependent, as in a generalized Ss model (in the terminology
of Caballero and Engel, 2007), and independent of the time elapsed since the last review. The random
relationship between each of the two decisions and the current state is a result of the rm's need to economize
on information. Obtaining more precise signals requires purchasing a larger quantity of information in each
period. Hence, the
rm faces a trade-o
between economizing on information expenditure and pricing
accuracy. The degree to which prices respond to concurrent market conditions depends on this trade-o .
Moreover, the hazard function for policy reviews and the conditional distribution of prices depend on
each other. If the cost of conducting a policy review is small, such that the rm can undertake frequent
reviews, then it need not design a complex pricing policy to be implemented between reviews. If the cost
of acquiring information for its pricing decision is small, then the rm can implement a pricing policy that
discriminates more nely among di erent future states, which in turn implies that a policy review can be
undertaken less frequently.
4
I derive conditions under which the
rm chooses to implement a single-price versus a multiple-price
policy: if the cost of information for the rm's pricing decision is higher than a certain threshold, the rm
charges a single price between reviews. Below this threshold, the rm implements a multiple-price policy.
This threshold is decreasing in the volatility of the shocks a ecting the rm's pro ts, and also depends on
the shape of the pro t function, in particular, on the elasticity of demand. Hence, the theory can generate
heterogeneity in the types of pricing policies pursued by di erent rms as a function of deep parameters that
plausibly vary across rms. For information costs below this threshold, I show conditions under which the
rm's pricing policy is a discrete set of prices. This allows me to generate simple pricing policies that consist
of a small set of prices. I present the optimality conditions that determine how the set of prices increases as
the cost per unit of information falls: given an asymmetric objective function (as is the case treated in this
paper), new prices are added one by one as the support of the price distribution spreads out; for symmetric
objective functions (and subject to shocks drawn from distributions that are also symmetric), a high and
a low price are added symmetrically. These conditions also yield an algorithm that can be used to solve
numerically for the optimal pricing policy. This novel algorithm makes it computationally feasible and fast
to nd the optimal policy.
I leave for future work the analysis of the degree of non-neutrality implied by the model in a general
equilibrium framework.3 Nonetheless, it is already clear that non-neutrality in this model will necessarily
be higher than that implied by a model with the same frequency of price adjustment, but in which all price
changes are based on full information regarding market conditions at the time of adjustment.
Relative to the existing literature, this is the rst model to generate pricing regimes that consist of a small
set of prices and that are infrequently updated. Moreover, consistent with the data, prices within regimes
change frequently and by large amounts. The most commonly used models of pricing cannot generate these
patterns.
Full information exible price models, in which prices are continuously re-optimized, do not generate
regimes except to the extent that there are regimes in the underlying shocks, and do not generate mass
points in the distribution of prices observed over time, except to the extent that the underlying shocks are
3 Extending the model to general equilibrium requires keeping track of the entire distribution of prices, and hence increases
the computational complexity of the numerical algorithm. The derivation of the algorithm in general equilibrium is in progress.
5
themselves drawn from distributions with mass points. By disregarding the substantial rigidity in price
levels apparent in Figure 1, and documented more broadly in Stevens (2011), these models may overstate
the degree of exibility in the pricing data.
Sticky price models, such as time-dependent models (Taylor, 1980 or Calvo, 1983) or state-dependent
models (Sheshinski and Weiss, 1977, Golosov and Lucas, 2007), generate single-price regimes. As in the case
of exible price models, there is no reason for past prices to be revisited once the rm re-optimizes its policy,
hence these models cannot explain the discreteness of prices observed in the data. Moreover, sticky price
models that abstract from transitory price changes within regimes may overstate the degree of rigidity in
the pricing data. As others have documented, a signi cant portion of rms' revenues is derived from sales
at the non-modal prices, which suggests that rms should have a strong incentive to tie transitory prices to
concurrent market conditions, at least partially. Klenow and Willis (2007) further document that transitory
prices have macro content that does not wash out with aggregation.
It is important to note that in the model proposed here there are no physical costs of price adjustment; in
fact, prices can change all the time in this model. Rigidity arises because they are always drawn from a xed
set of prices over the life of the regime, and are based on noisy information about market conditions. There
are also no a priori constraints on the rm's ability to change "regular" versus "temporary" prices, thus
distinguishing this model from those proposed by Kehoe and Midrigan (2010) and Guimaraes and Sheedy
(2011).
The model brings together di erent features of the growing literature on imperfect information in price
setting. In particular, the introduction of both xed and variable costs of information combines two competing approaches to modeling information acquisition. However, the model departs from both literatures
by generating simple pricing policies that consist of a small set of prices.
First, as in the inattentiveness model of Reis (2006), the rm faces a xed cost of conducting a policy
review and the strategy that is used to decide when to conduct the next review is itself part of the policy
that is chosen at the time of a review. In the model of Reis (2006), the policy speci es the path of prices to
be charged until the next review, and the date of the next review. Between reviews, the rm cannot obtain
any information about market conditions, other than information regarding the passage of time, which is
6
available for free. In contrast, I allow the rm to acquire information between reviews, but all information,
including knowledge about the number of periods since the last review, is subject to the same cost per
unit of information. The resulting timing of reviews and the price charged in each period are stochastically
state-dependent rather than time-dependent. In my model, a perfectly precise review signal would generate
the triggers in an Ss model of policy reviews, as in the model of Burstein (2006). Conversely, if the rm
acquired no information through its review signal, the timing of policy reviews would be completely random,
as in the model of Mankiw and Reis (2002).
Second, as in the rational inattention literature, the acquisition of information between reviews is subject
to a cost per unit of information, using entropy as a measure of information. Allowing the rm to occasionally
review its policy, subject to a cost, enables me to generate regime changes, distinguishing this setup from
other rational inattention papers, such as those of Sims (2003, 2006), Mackowiak and Wiederholt (2009), or
Matejka (2011). In those models, the rm speci es the optimal policy once and for all at some initial date,
and then receives signals in accordance with that policy. In contrast, I model both the decision to change
the price and the decision to change the overall policy, and hence to move to a new regime. The fact that
the rm can occasionally review its policy means that it can implement simple policies between reviews.
Moreover, other rational inattention models assume that the cost per unit of information only applies
to current market conditions, whereas the full history of past signals and knowledge about the number of
periods since the policy was rst chosen are both available for free. In contrast, I assume that all information,
including memory and knowledge of the number of periods elapsed since the last review, is subject to the same
cost per unit of information. This assumption identi es the information friction directly with the limited
attention of the decision-maker processing the information from a particular signal. This assumption is
critical in generating regimes that are identi ed by a single distribution of prices: without it, the rm would
charge prices from a di erent pricing policy in every period; moreover, the optimal policy would not generate
a discrete distribution of prices, except in the special case of i.i.d. variations in market conditions, as assumed
in Matejka (2011).
This treatment of time and memory is the same as in Woodford (2009), who also models policy reviews
that are subject to a xed cost and whose timing is determined by a stochastically state-dependent hazard
7
function. The present model di ers from Woodford (2009) along two dimensions. Firstly, I relax that
model's assumption that between policy reviews the rm charges a single price. Introducing the price signal
generates price volatility between policy reviews, consistent with the empirical evidence of multiple-price
regimes documented in Stevens (2011). Secondly, I allow the rm to redesign its signals at each review,
whereas in Woodford (2009), the rm's information acquisition policy is chosen once and for all at some
initial date.
Although in my in nite horizon model, the optimal prices of individual rms eventually drift in nitely
far apart, the occasional policy reviews enable rms to choose from a small set of prices in each regime (under
certain assumptions). Hence within regimes, the model maintains the pattern of discrete prices obtained
in a static context by of Matejka (2011). That paper, instead, obtains a discrete set of prices only under
the additional assumption that the range over which the rm's prices can vary remains forever bounded.
However, such an assumption is inconsistent with evidence that both individual and relative prices exhibit
unit roots.
Section 2 presents the setup and introduces the information costs, starting from the full-information
frictionless benchmark. Section 3 presents the acquisition of information between reviews and de nes the
rm's problem. Section 4 derives and discusses the optimal policy. Section 5 maps a standard monopolistic
competition model with Dixit-Stiglitz preferences into the problem introduced in section 2. Section 6 presents
numerical results. Section 7 concludes.
2
The Setup
A monopolistic rm producing a non-durable good must choose the price to charge for its output in
every period, subject to a demand function and a production technology that vary stochastically. The rm's
per-period pro t in units of marginal utility, (p
x), is a function of the rm's actual log-price, p, and its
target log-price, x. The pro t function is a smooth real-valued function with a unique global maximum at
p = x.
All the information about rm-speci c and aggregate market conditions that the rm needs in order to
choose its optimal price is summarized in the target price, xt . This target is a linear combination of the
8
exogenous disturbances in the economy, both transitory and permanent. It evolves over time according to:
xt
x
et
= x
et +
= x
et
1
where the permanent and transitory innovations, et and
(1)
t
+ et
t,
(2)
are drawn independently from known distribu-
tions ge and g , with continuous, but bounded support. After both et and
t
have been realized, the period
t price is set and orders are ful lled.
Section 5 maps a standard monopolistic competition model with Dixit-Stiglitz preferences into this
speci cation.
2.1
The Setup Under Full Information
In the frictionless benchmark, the rm chooses a pricing policy that speci es what price to charge in
each period and state of the world, to maximize its discounted pro t stream,
E0
1
X
t
(pt
xt )
(3)
t=0
where
2 (0; 1) is the discount factor. In the absence of information costs, the rm perfectly observes the
realization of xt in each period. If there are no other frictions, such as physical costs of price adjustment,
the rm's optimal policy is to charge
pt = xt , 8t
2.2
(4)
The Setup Under Costly Information
This paper departs from the frictionless benchmark by assuming that although complete information
about the state of the economy is available in principle, the
rm must expend resources to receive any
information. The rm chooses how much information about market conditions to acquire. Acquiring a larger
quantity of information in turn translates into more precise information about market conditions. The
measurement of the quantity of information is based on the literature on rational inattention (Sims, 2003).
9
In an important departure from most models of rational inattention, I assume that both the rm's prices
and its acquisition of information are determined by a policy that can be occasionally reviewed. Reviewing
this policy entails payment of a xed cost, as further discussed below.
The rm's ability to review its policy implies that in every period it must make two decisions: (1) whether
or not to undertake a policy review, and (2) what price to charge from the current policy. Each of these
decisions is based on information acquired in the form of signals. The information acquired in order to make
the review decision is subject to a unit cost
r
. Letting Itr denote the quantity of information acquired for
this decision in period t, the information expenditure associated with the review decision in period t is
r r
It .
Similarly, the information acquired in order to decide which price to charge is subject to a cost per unit of
information,
p
, and the expenditure associated with the pricing decision in period t is
p p
It .
Modeling the
acquisition of information using a unit cost of information is equivalent to assuming that the decision maker
processing the information acquired for each decision has a capacity limit on the quantity of information
that he or she can process.
The
xed cost of conducting a policy review, denoted by
, is also a type of information cost. It
represents the managerial resources associated with acquisition of the information necessary to design a new
policy and with the decision-making and communication of the new policy. As documented by Zbaracki et
al (2004), rms spend a signi cant amount of resources acquiring information and deciding what type of
policy to implement. Payment of this cost allows the rm to acquire extensive information about the state
of the world, on the basis of which it designs its new policy. For simplicity, I assume that it enables the rm
to receive complete information about the state of the world at the time of the review. The assumption that
the cost is xed can be rationalized via economies of scale in the review technology. This assumption follows
Reis (2006).4
It is important to note that except for the information received at the time of the policy review, and
re ected in the policy adopted at that time, the rm has no additional information that can guide it in
making its two decisions for free.
4 The assumption of a xed cost of policy reviews is also similar to that of Burstein (2006), except for the fact that in that
model, the rm has full information at all times for free, and the xed cost represents only the resources required to design and
communicate the new policy.
10
The rm's objective under costly information is to maximize
E0
1
X
t
[ (pt
xt )
r
t
r r
It
p p
It ]
(5)
t=0
where
is the xed cost of a policy review,
r
t
reviews its policy in period t and 0 otherwise,
is an indicator function that takes the value 1 if the rm
r
is the unit cost of the information acquired in order to
make the review decision, Itr is the quantity of information that informs the review decision in period t,
p
is the unit cost of the information acquired in order to make the pricing decision, and Itp is the quantity of
information that informs the pricing decision in period t.
The two unit costs,
r
and
p
, are not necessarily equal. For instance, it may be the case that two
individuals with di erent costs of acquiring information make the two decisions within the rm. However,
for each decision, the same unit cost applies to all types of information that may be relevant to that decision.
The types of information potentially relevant to each decision include information about the current state,
the history of signals previously received, and the number of periods that have elapsed since the last review.
The equal-cost assumption identi es the information friction with the limited attention of the decision-maker
processing the information from a particular signal. Moreover, this information is assumed to require a small
fraction of the decision-maker's overall capacity to process information. Hence, regardless of the degrees of
complexity of di erent types of information, the e ort per unit of information required of the decision-maker
receiving a particular signal is taken as xed. As a result, the rm's problem involves a single signalling
mechanism for each of the two decisions. This equal-cost assumption follows Woodford (2009), and its
consequences are discussed in more detail in section 3.5
2.3
The Sequence of Events
The sequence of events that occur in each period t is as follows:
5 An alternative approach, implemented by Sims (2003), and other rational inattention papers, is to assume that the entire
history of past signals is available to the decision-maker for free. One consequence of that assumption is that it makes dynamic
problems stationary. However, that assumption is no longer necessary once I allow the rm to occasionally review its policy, as
in the current paper. Yet another approach, left for future work, would be to assume that some types of information are easier
to process than others. This approach would increase the complexity of the model, since it would require modeling a di erent
information acquisition strategy for each type of information.
11
1. The value of et is realized.
2. The rm receives the review signal, based on which it decides whether or not to undertake a review,
in accordance with its current policy:
(a) if it decides to undertake a review, it acquires complete information about the current state of
the world, and chooses a new policy that consists of a strategy for its review decision, to be
implemented starting in period t + 1, and a strategy for its pricing decision, to be used starting
in period t;
(b) otherwise, the existing policy is maintained.
3. The value of
t
is realized.
4. The rm receives the price signal, based on which it decides what price to charge in the current period,
in accordance with its current policy.
5. Period-t demand is met and pro ts are realized.
The assumption that in each period the rm makes its review decision before that period's transitory
shocks are realized, and hence that this decision cannot depend on these shocks, is a simpli cation that
reduces the state space relevant for this decision, while only having small quantitative implications. If,
instead, both permanent and transitory shocks were realized at the beginning of the period, the review
decision would depend on both types of shocks. However, the extent to which the transitory shock would
impact the review decision would be small: only particularly large transitory shocks would justify triggering a
review, despite the transient character of the shock. The timing assumption abstracts from this complication
by eliminating the possibility of such an e ect.
3
The Firm's Problem
This section reformulates the rm's objective (5) in terms of the choices that the rm makes each time
it undertakes a review, and de nes the rm's complete optimization problem.
12
First, I de ne the signals that inform the rm's two decisions, and the quantities of information required
by each signal, starting from general de nitions for each signalling mechanism. A crucial determinant of the
optimal signals is the way in which we measure the quantity of information required by each signal, Itr and
Itp . Following the rational inattention literature, I use a measure derived from information theory (Shannon,
1948), which quanti es the reduction in the agent's uncertainty about a random variable.
I then simplify the de nition of each signalling mechanism, using some preliminary results that exploit
the information theoretic framework. In particular, the optimal signalling mechanism for each of the two
decisions is shown to generate signals that specify the action that the rm should take. In the case of the
review decision, the review signal directly indicates whether or not the rm should undertake a review in
the current period, and in the case of the pricing decision, the price signal directly tells the rm what price
to charge in the current period. These results ensure that the rm implements the most e cient signal
structure. They also allow me to rede ne the rm's objective (5) in terms of a more tractable set of rm
choices. Finally, I employ a normalization that makes the rm's problem stationary.
3.1
The Review Signal
Let !
e t denote the state of the world at the time of the receipt of the review signal in period t. It includes
the realization of the permanent shock in the current period, et , as well as the full history of shocks and the
full history of signals and the decisions made by the rm in response to these signals, through the end of
1. Suppose that the rm conducts a policy review in an arbitrary state !
e t in period t. As noted
period t
in the previous section, the review policy is implemented starting in period t + 1, since there can only be
one review per period.
De nition 1 The review signal, implemented following a policy review in an arbitrary state !
e t in period t,
is de ned by a signalling mechanism that speci es
1. Rt , the set of possible review signals;
2.
t+
(rje
! t+ )
, the sequence of conditional probabilities of receiving the review signal r for all
all r 2 Rt , and all !
e t+ that follow the policy review in state !
e t , period t, until the next review;
13
> 0,
3.
t
(r), the overall frequency with which each review signal is received, until the next review, for all
r 2 Rt ;
4.
t
(r) : Rt ! [0; 1], the decision rule for conducting a policy review, which speci es the probability of
conducting a policy review when the review signal r is received, for all r 2 Rt .
At the time of each policy review, part of the rm's problem is to choose these four objects, which make
up the rm's optimal review policy. Before the receipt of the review signal in each period, the rm has no
additional information, except for that information received at the time of the policy review, and re ected
in the current policy. Hence, the overall frequency of the review signals,
signal,
t
t
(r), and the decision rule for each
(r), can depend only on !
e t , the state at the time of the policy review in period t. Therefore, these
two objects are only indexed by t, the period in which the last review took place. In particular, because
awareness of the passage of time is not treated as free information, the signals received in all periods
in
which the policy may apply must be treated as part of a single information structure. Conversely, if the rm
had independent knowledge of
before receiving the review signal in each period, either for free, as in Reis
(2006), or through a separate signalling mechanism on the passage of time, the rm would design separate
signalling mechanisms for each period, with period-speci c anticipated frequencies of review signals,
and period-speci c decision rules,
t+
t+
(r),
(r).
The information transmitted by this signalling mechanism is the average amount by which receipt of
the signal reduces the decision-maker's uncertainty about the state. As proposed by Shannon, uncertainty
of a random variable is measured by entropy. For convenience, both in the de nition of the review signal
and in the later de nition of the price signal, I employ an equivalent de nition, namely the average amount
by which uncertainty about the optimal signal would be reduced by observing the state.6 This leads to the
following de nition:
De nition 2 For any state !
e in which a review signal is received, the amount of information that is required
6 These two de nitions are equivalent since the information about a state variable contained in the signal is equal to the
information about the signal contained in the state variable. Exploiting this symmetry simpli es the exposition.
14
in order to implement a signalling mechanism de ned by the conditional probabilities
Ir ( ; )
X
(rje
! ) [log (rje
!)
(rje
! ) is de ned as
log (r)]
(6)
r2R
This quantity is the relative entropy between the conditional distribution
bution
( je
! ) and the default distri-
(r). More precise information about !
e implies a bigger di erence between the distribution of the
signal conditional on !
e , (rje
! ), and the frequency with which the rm anticipates receiving the signal r prior
to the realization of the state, (r). For expository purposes, R is a countable set, though the de nition can
be extended to allow for continuous signal distributions. It will be established below that the optimal set of
review signals is not only countable, but nite.
The quantity of information expected, at the time of the review in an arbitrary state !
e t and period t, to
be required by the implementation of the signalling mechanism de ned in De nition 1, in each period t + ,
> 0, over all states !
e t+ that follow !
e t with positive probability, is given by
r
It+
Et I r
t+
(rje
! t+ ) ;
t
(r)
(7)
where Et [ ] is always de ned as expectations conditional on !
e t and on a policy review having taken place in
state !
e t , period t.
I simplify the problem using a preliminary result from Woodford (2008), who proves that the optimal
review signal is binary and directly speci es whether or not a review should be undertaken. Hence, without
loss of generality, the review signal received in any period and state of the world is drawn from the set f0; 1g,
and the decision rule is simply
t
(r) = r, such that the rm conducts a policy review upon receiving the
signal r = 1, and continues with its existing policy upon receiving the signal r = 0.
The quantity of information conveyed by this signalling mechanism in a particular state, relative to the
probability of a policy review anticipated before receiving the review signal, is expressed as
Ir
;
log
log
+ (1
15
) log (1
)
log 1
(8)
This quantity is the relative entropy between two binary random variables for which the probabilities of
observing the signal r = 1 are
and
respectively.
As a result, I can express (7) in terms of a sequence of hazard functions, in the terminology of Caballero
and Engel (2007), f
t+
(e
! t+ )g , and an anticipated frequency of policy reviews,
r
It+
= Et I r
t+
(e
! t+ ) ;
t:
(9)
t
The signalling mechanism for the review decision is accordingly rede ned in the following lemma:
Lemma 1 The optimal review signal, implemented following a policy review in an arbitrary state !
e t in
period t, is always drawn from the set f0; 1g, and is described by a signalling mechanism that speci es
1. f
t+
(e
! t+ )g , the sequence of conditional probabilities of receiving the review signal rt+ = 1 (con-
ducting a policy review) for all
until the next review;
2.
t,
> 0 and all !
e t+ that follow the policy review in state !
e t , period t,
the anticipated frequency with which the review signal rt+ = 1 is received, over all states and
periods, until the next review.
r
required by this signalling mechanism in each period t + ,
The quantity of information It+
> 0, is
given by equation (9).
This result is not only intuitive, but it also formally de nes the cheapest signalling mechanism that the
rm can employ in order to make its review decision. Any other signal structure would require a quantity
of information weakly greater than (9). Reformulating the signalling mechanism in this way also leads to
a simpli cation in solving for the rm's review decision: rather than choosing the four objects de ned in
De nition 1, the rm chooses the sequence of hazard functions, f
of policy reviews,
t.
16
t+
(e
! t+ )g , and the anticipated frequency
3.2
The Price Signal
The price signal in each period t is received after the review signal, and after the realization of the
transitory shock,
t.
As above, suppose that the rm conducts a policy review in an arbitrary state !
e t in
period t. The pricing policy applies starting in period t. For any
0, let ! t+ = fe
! t+ ; rt+ ;
t+
g denote
the state of the world at the time of the receipt of the price signal in period t + .
De nition 3 The price signal, implemented following a policy review in an arbitrary state !
e t in period t,
is de ned by a signalling mechanism that speci es
1. St , the set of possible price signals;
2. fft+ (sj! t+ )g , the sequence of conditional probabilities of receiving the price signal s for all
0,
all s 2 St , and all ! t+ that follow the policy review in state !
e t , period t, until the next review;
3. f t (s), the overall frequency with which each price signal is received, until the next review, for all s 2 St ;
4.
t
(pjs) : St
R+ ! [0; 1], the decision rule for price-setting, which speci es the probability of charging
price p 2 R+ when the price signal s is received, for all s 2 St .
As in the case of the review signal, discussed above, both f t (s) and the decision rule for price-setting
can depend only on the state at the time of the policy review in period t.
The quantity of information required to implement a particular signalling mechanism for the
rm's
pricing decision is de ned below:
De nition 4 For any state ! in which a price signal is received, the amount of information that is required
in order to implement a signalling mechanism de ned by the conditional probability f (sj!) is de ned as
I p f; f
X
f (sj!) log f (sj!)
log f (s)
(10)
s2S
For expository purposes, S is a countable set, although the de nition can be extended to allow for
continuous signal distributions.
17
The quantity of information expected at the time of the review in period t to be required by the
implementation of the price signal in each period t + ,
p
It+
0, is then given by
Et I p ft+ (sj! t+ ) ; f t (s)
(11)
As in the case of the review signal, the rm's choices for the optimal pricing policy are simpli ed by
showing that the optimal price signal directly speci es the price that the rm should charge in each period.
In order to obtain this result, I begin by noting that the optimal decision rule assigns a deterministic
price to each signal s, and hence can be represented by a function pt (s) : St ! R+ . Otherwise, the rm
could economize on its information expenditure in (10) by acquiring a less informative (more random) signal,
and arranging for the action, namely the price to be charged, to be a deterministic function of the signal.7
Second, the rm always uses the price signal to decide which price to charge, such that the function pt (s) is
onto, pt (s) : St ! Pt , where Pt is the subset of R+ of prices that are charged with positive probability. Third,
the optimal signal does not di erentiate between states in which the same action is taken, since doing so
would increase the quantity of information that the rm must acquire, without improving the rm's decision.
Hence the function pt (s) is one-to-one. Finally, receiving a signal directly on the price p requires the same
quantity of information as does receiving a signal s that is subsequently transformed into a price using the
bijection pt (s).
This result ensures that the rm implements the cheapest signal structure in order to make its pricing
decision. The rm's pricing policy can be more succinctly described by the set of prices Pt , the sequence of
conditional probabilities fft+ (pj! t+ )g , and the anticipated frequency with which each price is charged,
f t (p), for all p 2 Pt , all
0, and all states ! t+ that follow !
e t .8 Equations (10) and (11) are replaced by:
I p f; f =
X
f (pj!) log f (pj!)
log f (p)
(12)
p2P
p
It+
= Et I p ft+ (pj! t+ ) ; f t (p)
(13)
7 The state, !, the signal, s, and the price p, form a Markov chain, in that order, so that prices are distributed independently
of states conditional on signals. As a result, the mutual information between prices and states is less than or equal to the
mutual information between signals and states. If prices are a random function of signals, then the inequality is strict.
8 With a slight abuse of notation, I continue to use \f ," which from now on denotes the distribution of the prices.
18
The following lemma summarizes the results of this sub-section:
Lemma 2 The optimal price signal, implemented following a policy review in an arbitrary state !
e t in period
t, is described by a signalling mechanism that speci es
1. Pt , the set of prices charged with positive probability;
2. fft+ (pj! t+ )g , the sequence of conditional probabilities of charging price p for all
0, all p 2 Pt ,
and all ! t+ that follow the policy review in state !
e t , period t, until the next review;
3. f t (p), the anticipated frequency with which each price is charged over all states and periods until the
next review, for all p 2 Pt .
p
required by this signalling mechanism in each period t + ,
The quantity of information It+
0, is
given by equation (13).
Following a policy review in period t, the
rm's choices when solving for the optimal pricing pol-
icy are therefore reduced to choosing three objects: the set of prices Pt , the sequence of conditional
probabilities,fft+ (pj! t+ )g, and the overall frequency f t (p). Any other signal structure would require
a quantity of information weakly greater than (13).
3.3
The Firm's Problem
The choices de ned in Lemma 1 and Lemma 2 provide a mathematical representation of the
rm's
policy. Using these choices, the continuation value of the rm's objective (5), looking forward from the time
of a policy review in an arbitrary state !
e t in period t, is given by
Et
Here,
t+
,
8
>
>
1
<X
>
>
:
t+
=0
2
6
6
4
P
p2Pt+
r r
I
t+ +1
(p
(e
! t+
xt+ )ft+ (pj! t+ )
+1 ) ;
t+ +1
p p
I
39
>
>
! t+ +1 )
7=
t+ +1 (e
7
5>
>
;
ft+ (pj! t+ ) ; f t+ (p)
(14)
(e
! t+ ), Pt+ , f t+ (p), and ft+ (pj! t+ ) are the policy choices that are in e ect in each
period t + and in each state of the world, regardless of whether they were adopted at the time of the review
19
in period t or in some subsequent policy review. Hence, in this equation, I make no explicit reference to the
period and state in which the policy that applies in each t +
was chosen.
The rm's continuation value can be written more compactly by collecting all of the terms in the objective
that depend on the pricing policy in e ect in a particular period. Using (12), I de ne
t+
(! t+ ) as the
rm's per-period pro t in state ! t+ , period t + , expected under the pricing policy in e ect in that state,
net of the cost of the price signal only:
t+
(! t+ )
X
ft+ (pj! t+ )
(p
xt+ )
p
log ft+ (pj! t+ )
log f t+ (p)
(15)
p2Pt+
As in (14), here I continue to make no explicit reference to the period and state in which the pricing policy
that speci es Pt+ , ft+ (pj! t+ ) and f t+ (p) was chosen.
The continuation value de ned in (14) can then be written as
Et
(
t
(! t ) +
1
X
t+
(! t+ )
t+
r r
(e
! t+ )
I
t+
(e
! t+ ) ;
)
t+
=1
(16)
As above, this equation makes no explicit reference to the period and state in which the policy that applies
in each t +
was chosen.
In order to formulate the rm's objective in terms of its choices at the time of a particular policy review,
it is convenient to de ne a survival probability,
t+
(e
! t+
1 ),
which denotes the probability, expected at
the time of the review, that the review policy chosen in state !
e t , period t, continues to apply
as a function of the history of states. Since there can only be one review per period,
and for
> 1,
t+
(e
! t+
1)
Y1
[1
t+k
(e
! t+k )]
t+1
(e
!t )
periods later,
1 for all !
et;
(17)
k=1
Let V t (e
! t ) denote the maximum attainable value of the
rm's continuation value (16). Under the
assumption that an optimal policy will be chosen in all future policy reviews, (16), written in terms of the
20
rm's choices at the time of the review in period t, becomes:
Et
8
>
>
<
>
>
:
t
(! t ) +
1
X
t+
(e
! t+
=1
2
6 (1
6
1) 4
t+
(e
! t+ ))
t+
(! t+ ) +
r r
I
t+
t+
! t+ )
(e
! t+ ) V t+ (e
(e
! t+ ) ;
current policy (which is given by the survival probability), the
I
t+
(e
! t+ ) ;
t
t
(18)
This objective states that conditional on reaching a particular period t +
r r
and state !
e t+
which case it expects a per-period pro t equal to
t+
under the
rm pays the cost of the review signal,
. It then continues to apply the current policy with probability 1
review and pays the review cost
39
>
>
7=
7
5>
>
;
t+
(e
! t+ ), in
(e
! t+ ). On the other hand, if it undertakes a policy
, which occurs with probability
t+
(e
! t+ ), it expects the maximum
! t+ ).
attainable value from that state onward, V t+ (e
I can now formally state the rm's problem at the time of a particular policy review:
Problem 1 If the rm undertakes a policy review in an arbitrary state !
e t and period t, it chooses:
1. a review policy that speci es
t
and f
t+
(e
! t+ )g for all
review in state !
e t , period t, until the next review;
> 0 and all !
e t+ that follow the policy
2. a pricing policy that speci es Pt , f t (p), and fft+ (pj! t+ )g for all p 2 Pt , all
0, and all ! t+
that follow the policy review in state !
e t , period t, until the next review.
The two policies are chosen to maximize (18) subject to the de nitions (17), (15), and (8).
3.4
The Stationary Formulation
Employing a normalization allows me to rewrite Problem 1 in a stationary form. The normalization
involves introducing a set Q of normalized prices, and a normalization of the state variables !
e t+ and ! t+ .
Using this simpli cation, I derive the optimal policy independent of the time and state in which a policy
review is conducted. The problem is reformulated in terms of the following normalized variables: q, ye , y ,
21
w
e , and w . First, I de ne
q
p
ye
x
et+
(19)
x
et
ye +
y
As a result, the rm's pro t function, (p
x
et
(20)
(21)
xt+ ), is replaced by the normalized function (q
y ). Second,
for any state !
e t+ , w
e is the part of !
e t+ that represents news since the state of the world at the time of the
last review, !
e t , for all
> 0; similarly, w represents the part of ! t+ that is news since !
e t , for all
0.
Given the laws of motion (1)-(2), all of these variables are distributed independently of the state !
e t at
the time of the policy review in period t. The optimal pricing policy chosen in any review can be written as
the choice of the set of normalized prices q 2 Q, anticipated to occur with frequencies f (q), and the sequence
of conditional distributions ff (qjw )g . The expected value,
by its normalized counterpart
(w )
t+
(! t+ ), de ned in (15), can be replaced
(w ),
X
f (qjw )
(q
p
y )
log f (qjw )
log f (q)
(22)
q2Q
Finally, the
rm's optimal review policy can also be written in normalized terms as the choice of a
sequence of hazard functions f
(e
w )g and an anticipated frequency of reviews . The survival probability
(17) becomes
(e
w
1)
Y1
[1
k
(e
wk )] ;
k=1
with
1
(e
w0 )
8 >1
(23)
1, since there is only one policy review per period.
Therefore, the rm's choices can be written without any reference to either the date t or the state !
e t in
which a policy review takes place. In normalized terms, the rm's continuation value conditional on a policy
review in an arbitrary state and period is
E
(
0
(w0 ) +
1
X
(e
w
1)
(1
(e
w ))
(w ) +
=1
22
(e
w ) V
r r
I
(e
w );
)
(24)
where V is the maximum attainable value of the rm's continuation value at the time of a policy review
in any state and period. The expectations operator E [ ] integrates over all future news states following
the policy review in any arbitrary state and period. Below I state the stationary formulation of the rm's
problem:
Problem 2 If the rm undertakes a policy review in any arbitrary state and period, it chooses:
1. a review policy that speci es
and f
(e
w )g for all
review, and
> 0 and all news states w
e until the next
2. a pricing policy that speci es Q, f (q), and ff (qjw )g for all normalized prices q 2 Q, all
0,
and all news states w until the next review.
The two policies are chosen to maximize (24) subject to the de nitions in (23), (22), and (8). Moreover,
V in (24) is the maximized value of the quantity de ned by (24).
4
The Optimal Policy
I obtain the solution to Problem 2 in steps, deriving each element of the optimal policy taking the other
elements as given. I rst solve for the sequence of conditional price distributions, ff (qjw )g . Using this
result, I then nd the optimal sequence of hazard functions, f
simplify the
(e
w )g . The set of optimality conditions
rm's policy further: the optimal policy will be shown to involve a single conditional price
distribution, f (qj ) for all q 2 Q, and a single hazard function,
( ), which are de ned for all periods and all
states until the next policy review. I then solve for the anticipated frequency of each normalized price, f (q),
for each q 2 Q, the optimal set of normalized prices, Q, and the anticipated frequency of policy reviews,
. Although I allow the rm to condition its policy on the complete state, including the number of periods
elapsed since the last review and the history of past signals, the optimal signalling mechanisms for both the
review decision and the pricing decision will be shown to allocate the entire information capacity to learning
about the change in market conditions since the last review directly, rather than paying any attention to
past events or to the passage of time directly.
23
4.1
The Sequence of Conditional Price Distributions
The rm's choice of an optimal pricing policy for a given review policy is reduced to the maximization
of the term that directly depends on the pricing policy in the rm's objective (24). Speci cally, the pricing
policy is chosen to maximize
E
where
(
1
X
+1
(e
w )
)
(w )
=0
(w ) is given by (22), and the survival probability,
+1
(25)
(e
w ), is determined by the review policy.
Consider the subproblem of choosing the optimal sequence of conditional price distributions, ff (qjw )g ,
for a given review policy, and further taking the set of normalized prices, Q, and the anticipated frequency
with which each price is charged, f (q) for all q 2 Q, as given. Objective (25) is additively separable across
dates and state, hence these distributions can be chosen independently for each
and each possible news
state w that follows a state w
e in which there is no policy review. Hence, the maximization of (25) given
f (q) and Q is a static problem: for each
and each w , the rm chooses the conditional distribution of
normalized prices f (qjw ) that solves:
max
(w )
f (qjw )
s:t:
X
f (qjw ) = 1
(26)
(27)
q2Q
f (qjw )
This subproblem only depends on the values of
0; 8q 2 Q
and w through the dependence of
(28)
(w ) on y in (22).
Therefore, although I allow the rm to condition its pricing policy on the complete state w , the solution
depends only on the accumulated shocks, y . The resulting conditional distribution, f (qjy ), indicates the
probability of charging a normalized price q in state y , and is independent of the number of periods elapsed
since the last review and of the history of past price signals. Since the rm faces the same unit cost of
processing information about all aspects of the complete state w , it chooses to allocate its entire attention
to monitoring changes in its target price directly, y . The complete problem and the resulting optimality
condition are stated in the following lemma:
24
Lemma 3 Let the review policy, the set of normalized prices Q, and the frequency with which the
anticipates charging each price until the next review, f (q) for all q 2 Q, be xed. If
conditional price distribution f (qjw ) solves (26) subject to (27)-(28), for each
p
rm
> 0, the optimal
0 and each w that
follows a state w
e in which there is no policy review. This conditional distribution depends only on the
normalized state y de ned in (21), and is otherwise independent of
and of all other aspects of w . Letting
Y denote the set of all possible values of the normalized state y , for any
0, occurring prior to the next
review, the optimal conditional distribution of normalized prices is given by9
f (qjy) = f (q) P
exp
qb2Q
1
(q
p
f (b
q ) exp
y)
1
p
(b
q
(29)
y)
for all y 2 Y and q 2 Q.
Condition (29) illustrates the sense in which the price signal is optimally designed, given the rm's objective function: for a given state of the world, the conditional probability of charging a particular normalized
price q is higher the higher is the pro t relative to the average pro t that the rm can expect in this state
across all normalized prices in the set Q. However, the relationship between the state and the price signal
is noisy: for any state y 2 Y, f (qjy) > 0 for all q 2 Q. This randomness re ects the need to economize on
the information cost associated with receiving the price signal in each period.
Substituting this solution into the de nition (22) of
(w ) =
where
(w ) yields
(y )
(30)
(y) is a time-invariant function.
If, before receiving the price signal in each period, the rm had independent knowledge of the number of
periods elapsed since the last review, it would have more precise information about the states of the world
that are more likely in a particular period. For example, it would have less uncertainty about the state
soon after a review. It would use this knowledge to design a signalling mechanism that speci ed di erent
9 The formula for the optimal conditional distribution is of the same form as that
static rate-distortion function.
25
rst derived by Shannon (1959) for a
anticipated frequencies for the normalized prices for each period, f (q), and hence di erent conditional
distributions, f (qjy). Similarly, if the
rm had access to the sequence of past price signals for free, a
separate signalling mechanism would also be chosen for each history of prior signals. Here, instead, the only
information that the rm has, prior to the receipt of the price signal, is the information obtained at the last
policy review. Hence, the optimal pricing policy is characterized by a single conditional distribution that is
optimal across all states and periods in which the current policy is expected to apply.
4.2
The Sequence of Hazard Functions
I now consider the rm's choice of an optimal sequence of hazard functions for a given pricing policy,
and further taking
as given. This subproblem can be given a recursive form by noting that the problem
of choosing the optimal sequence of hazard functions f
(e
w 0 )g
0
0
for all
0
, looking forward from an
arbitrary state w
e , can be formulated independently of the choices made for periods prior to , or for news
states w
e
E
(
0
that are not successors of w
e . Let V (e
w ) be the maximum attainable value of
(w ) +
1
X
0=
0
;
0
(e
w
0
1)
(1
0
(e
w 0 ))
0
(w 0 ) +
0
(e
w 0) V
r r
I
0
(e
w 0) ;
+1
(31)
where E [ ] integrates over all possible histories for dates
where the survival probability
;
0
(e
w
0
1)
0
, conditional on reaching state w
e , and
is de ned by
0
;
0
(e
w
0
1)
Y1
[1
k
(e
wk )] ;
k= +1
by analogy with (23). The optimal sequence of hazard functions f
the pricing policy. Using (30), and de ning
e (e
y )
E [ (e
y +
26
)]
8
0
0
>
(e
w 0 )g
(32)
0
maximizes (31) given
and
)
(31) can alternatively be written as
e (e
y )+
1
X
0=
0
n
E
+1
;
0
(e
w
0
1)
h
(1
0
(e
w 0 )) e (e
y 0) +
0
r r
(e
w 0) V
I
0
(e
w 0) ;
io
(33)
Hence V (e
w ), the maximum attainable value of (33) depends only on the value of ye and can be written as
V (e
w ) = V (e
y )
(34)
where V (e
y ) is a time-invariant function.
The problem of maximizing (33) can then be seen to have a recursive form:
V (e
w )=
f
max
w
+1 (e
8
>
>
<
2
6 (1
e (e
y )+ E 6
4
+1 )g >
>
:
+1
(e
w
+1 )) V +1
r r
I
where E [ ] integrates over all possible news states w
e
+1
(e
w
+1 )
+1
(e
w
+
+1
(e
w
+1 )
39
>
>
7=
7
5> (35)
>
;
V
+1 ) ;
that are possible successors to w
e . This is the
Bellman equation for a dynamic programming approach to the optimal choice of the sequence of hazard
functions f
+1
(e
w
(1
+1 )g
. For each state w
e
+1
(e
w
+1 )) V
(e
y
+1 )
+1 ,
+
the hazard function must maximize
+1
(e
w
+1 )
r r
V
I
This problem, and hence its solution, depends only on the value of V (e
y
+1 )
+1
(e
w
+1 ) ;
(36)
and is otherwise independent of
the time elapsed since the last review, + 1, and of the particular history of past signals in w
e
+1 .
Therefore,
the solution is of the form
+1
where
(e
w
+1 )
=
(e
y
+1 )
(37)
(e
y ) is a time-invariant function. Moreover, the function V (e
y ) is a xed point of the equation
V (e
y ) = e (e
y ) + E (1
(e
y 0 )) V (e
y0 ) +
27
(e
y0 ) V
r r
I
(e
y0 ) ;
(38)
and
V = V (0)
(39)
Collecting results, and solving for the hazard function that maximizes (36) given the de nition of I
;
,
yields the following result:
Lemma 4 Let the pricing policy and the anticipated frequency of reviews,
hazard function solves (31) for each
, be xed. If
r
> 0, the optimal
> 0 and each w
e for which there is no policy review. Given f (qjy),
the optimal conditional distribution of normalized prices (29), the hazard function is the same for all
and
for all w
e , and only depends on the normalized state variable ye de ned in (20). Letting Ye denote the set of
all states ye ,
> 0, occurring with positive probability until the next review, the optimal hazard function is
given by
1
(e
y)
=
(e
y)
1
exp
1
r
V
e where V (e
for all ye 2 Y,
y ) is the continuation value for the
V (e
y)
(40)
rm's choice of an optimal hazard function,
de ned by (38), and V is continuation value upon conducting a policy review, given by (39).
The expression (40) de nes the optimal hazard as a function of the di erence between the value of
updating to a new policy, V
, and the value of continuing with the existing policy, V (e
y ). This expression
is of the same form as the optimal hazard function derived in Woodford (2008) for the case in which the
pricing policy is reduced to a single price.
The hazard function is monotonically increasing in the value of the exponent: a higher value of adjustment relative to keeping the policy unchanged is associated with a higher probability of receiving a signal that
the policy should be reviewed. For any overall frequency of policy reviews,
2 (0; 1), the hazard function
(e
y ) 2 (0; 1): in order to economize on information costs, the optimal review signal is a random function of
the value of adjusting the policy, namely, the review signal never indicates a review with certainty.
Next, I characterize the three remaining elements of the rm's optimal policy: the unconditional distribution of normalized prices f (q), the set of normalized prices Q, and the unconditional hazard rate
28
.
4.3
The Unconditional Distribution of Prices
The frequency f (q) with which each normalized price q is anticipated to occur over the life of the
policy, for a given support, Q, and a given review policy, is determined from (25). Using the optimal hazard
function,
(e
y ), determined by (40), the survival probability becomes a function of the history of normalized
states, ye
fe
y1 ; ::; ye g.These steps transform that objective into the following optimization problem for the
choice of f (q):
E
(
1
X
+1
(e
y )
)
(y )
=0
s:t:
X
f (q) = 1
(41)
(42)
q2Q
where E [ ] integrates over all the possible histories ye and over the possible values of y conditional on ye .
This subproblem and its solution are summarized in the following lemma:
Lemma 5 Let the review policy, the set of normalized prices Q, and the conditional probability of charging
each price, f (qjy), be xed. The optimal anticipated frequency with which each price is charged over the life
of the policy solves (41) subject to (42). The optimal frequency is given by
P1
Ef
=0
P
1
[E
f
qb2Q
=0
f (q) = P
+1
(e
y ) f (qjy )g
y ) f (b
q jy )g]
+1 (e
(43)
for each q 2 Q, where E [ ] integrates over all the possible histories ye and over the possible values of y
conditional on ye .
The unconditional distribution f (q) can be written in the more familiar form, as
f (q) =
Z
G (y) f (qjy) dy
(44)
where the distribution G (y) de nes the probabilities of encountering all the possible states y 2 Y until the
next review. Hence, the rm's pricing policy problem has been transformed into a static rational inattention
problem - for instance, as in Matejka (2011) - with the prior G (y).
29
Condition (43) shows that the optimal anticipated frequency is equal to the (discounted) weighted average
of the conditional price distribution over all states that the rm expects to encounter until the next review.
Before the receipt of the price signal in any period and state of the world, the rm anticipates receiving a
signal from this "default" distribution, which takes into account all possible states that the rm is likely to
encounter under the current review policy. The probability of reaching a particular state is discounted by
the probability that the rm conducts a policy review in any of the states that lead up to that particular
state, and by the discount factor .
4.4
The Support of the Price Distribution
The rst order conditions (29) and (43), for the optimality of the conditional price distribution, f (qjy),
and the unconditional distribution f (q), given the set Q, can be shown to be both necessary and su cient
for the pricing policy de ned by f (qjy) and f (q) to represent an optimal policy among all pricing policies
with support Q:
Lemma 6 Let the review policy, the value of V , and the set Q be
xed. Then if f (qjy) and f (q) are
probability distributions with support Q, such that f (q) > 0 for any q 2 Q, and such that (29) and (43) are
satis ed for all y 2 Y and q 2 Q, these distributions determine the unique optimal pricing policy among the
set of pricing policies with support Q.
The proof of this lemma results from the strict concavity of (41).10 However, if f (q) = 0 for some q 2 Q,
then one can have a solution to (29) and (43) that is not a local maximum. Nevertheless, one can derive
conditions that check the optimality of the hypothesized support Q by checking that f is a locally stable
xed point of the mapping obtained by integrating (29) over all states y 2 Y:
f (q) = Z q; f f (q)
(45)
where
Z q; f
10 See
Z
G (y) P
exp
qb2Q
1
p
(q
f (b
q ) exp
also Csiszar and Korner (1982) in the information theory literature.
30
y)
1
p
(b
q
y)
dy
(46)
The conditions for f to be a locally stable xed point are
Z q; f
Z q; f
=
1 8q
(47)
1 8q 2 suppf
(48)
Hence, necessary conditions for a nite set Q = fq1 ; :::; qn g to be the optimal support of the price signalling mechanism are, for i = 1; :::; n,
Z qi ; f
=
1
(49)
Z 0 qi ; f
=
0
(50)
0
(51)
Z 00 qi ; f
where the rst condition holds if the probability distributions satisfy (29) and (43). The following lemma
provides the explicit conditions for the points of support:
Lemma 7 Let the review policy be xed, and let f (qjy) and f (q) be probability distributions that satisfy the
optimal conditional distribution of normalized prices (29) and (43). All points of support q 2 Q must satisfy
Z
G (yjq)
"
Z
G (yjq)
@2
(q
@q 2
@
(q
@q
y) +
1
p
y) dy = 0
@
(q
@q
(52)
2
y)
#
dy
0
(53)
These two equations give conditions for the optimal support of the signalling mechanism. The optimality
condition (52) ensures that each normalized price is optimally chosen to maximize the rm's pro t over its
image in the space of the state. All prices in the support must also satisfy the second order condition de ned
in (53). These equations pin down the set of normalized prices that the rm charges between policy reviews.
Together with equations (29) and (43), they provide a complete characterization of the rm's pricing policy,
for a given review policy.
31
If the information cost associated with the price signal,
p
, is high enough, relative to the volatility
of market conditions, and given the shape of the rm's objective function, the rm chooses to acquire no
information through its price signal between reviews. In this case, the optimal single-price policy is to charge
the normalized price q , implicitly de ned by
Z
G (y)
@
(q
@q
y) dy = 0
(54)
Hence the model endogenously generates the single-price policy assumed in Woodford (2009). Equation (53)
provides a threshold of how high
p
has to be in order for the rm to forgo learning about market conditions
for its pricing decision:
Lemma 8 The threshold
p
above which the rm acquires no information through its price signal is given
by
p
For
p
>
p
=
R
2
@
G (y) @q
(q y) jq=q
dy
R
@2
G (y) @q2 (q y) jq=q dy
(55)
, the rm pursues a single-price policy with the no-information optimal price implicitly de ned
by (54). For
p
p
, the rm acquires information through its price signal and pursues a multiple-price
policy.
For information costs below the threshold, generating a discrete pricing policy is of particular interest,
given the empirical evidence that pricing regimes are characterized by discreteness. Conditions (49)-(51)
provide a way to verify the optimality of a pricing policy with a nite number of prices. Existing work has
established that the optimal signals are discrete if the support of the state is continuous but bounded. In the
literature on information theory, Fix (1978) provides a proof, while in economics, Matejka and Sims (2010)
independently derive a di erent proof. However, the literature on information theory stresses that discrete
solutions may arise in other contexts. The derivation above is based on the work of Fix (1978) and Rose
(1994).
32
4.5
The Optimal Frequency of Policy Reviews
Given the optimal hazard function (40) and the optimal pricing policy, the optimal
is chosen to
maximize (24).
Lemma 9 The optimal anticipated frequency of policy reviews maximizes the rm's continuation value (24)
given the rest of the rm's policy choices determined in (29), (40), (43), (52), and (53). The optimal review
frequency is given by
=
P1
Ef
=0
P
1
Ef
=0
+1
(e
y ) (e
y )g
(e
y
)g
+1
(56)
where E [ ] integrates over all possible histories ye .
Together with equation (40), equation (56) provides a complete characterization of the rm's review
policy, for a given pricing policy.
The rm's complete policy is summarized in the following proposition:
Proposition 1 If the rm undertakes a policy review in any arbitrary state and period, it maximizes (24)
by choosing:
1. a review policy given by
and
e until the next review,
(e
y ) for all normalized state variables ye 2 Y,
which satisfy equations (56) and (40);
2. a pricing policy given by Q, f (q), and f (qjy) for all normalized prices q 2 Q, and all state variables
y 2 Y, until the next review, which satisfy equations (52), (53), (43), and (29), respectively.
5
A Model of Price Setting
I explore the implications for price adjustment of the information structure presented in section 2 in
a standard model of price-setting under monopolistic competition. I assume that all aggregate variables
evolve according to the full-information, exible price equilibrium, and focus on the price adjustment of a set
of information-constrained rms of measure zero. In this case, the problem of the information-constrained
rms will be shown to map exactly into the generic setup introduced in section 2. The treatment of price
33
adjustment in a general equilibrium framework in which all rms are information-constrained requires that
each rm track not only an exogenous target price, but also the distribution of prices in the economy. I leave
the general equilibrium results for future work.
5.1
The Agents
The economy consists of three types of agents: an in nitely-lived representative household, a continuum
of in nitely-lived monopolistically competitive producers of di erentiated goods, and a government that
follows an exogenous policy.
Household: The problem of the representative household is standard. The household is perfectly
informed and supplies di erentiated labor to all rms i in the economy. It chooses sequences of consumption,
hours, and bond holdings to maximize a discounted utility stream de ned by:
E0
1
P
t
t=0
where
2
4
1
1
1
1+
Ct1
Z1
0
3
Ht (i)1+ di5
2 (0; 1) is the discount factor, Ct is the consumption basket,
(57)
> 1 is the constant relative risk
aversion parameter, Ht (i) is the total amount of labor (in hours) supplied by the representative household
to sector i, and
0 is the inverse of the Frisch elasticity of labor supply.
Maximization of (57) is subject to a standard budget constraint in each period t:
Z1
0
Wt (i) Ht (i) di +
Z1
t
(i) di + Bt + Mt
1
+ Tt
Pt Ct + Et [Rt;t+1 Bt+1 ] + Mt
(58)
0
where Wt (i) is the nominal hourly wage of sector i,
portfolio of nominal bond holdings in the period, Mt
t
1
(i) is the dividend received from sector i, Bt is the
is the household's money balance entering period t, Tt
is the net monetary transfer received from the government, Pt is the aggregate price index for the consumption
basket Ct , and Rt;t+1 is the stochastic discount factor used to discount income streams between time t and
time t + 1.
The representative household also faces a no-Ponzi-scheme condition, and a cash-in-advance constraint
34
on consumption purchases:
Pt Ct
Mt
1
+ Tt
(59)
Finally, the consumption basket, Ct , is given by a Dixit-Stiglitz aggregator over a continuum of di erentiated products i 2 [0; 1], with elasticity of substitution " > 1 and good-speci c preference shocks, At (i),
whose law of motion is speci ed in the next subsection:
Ct
3""1
21
Z
" 1
4 [At (i) Ct (i)] " di5
(60)
0
Inter-temporal consumer optimization yields the following standard rst order conditions for the optimal
supply of labor and for the stochastic discount factor:
Ht (i)
Wt (i)
=
Pt
Ct
and
T
Rt;T =
t
Ct
CT
Pt
PT
(61)
Intra-temporal expenditure minimization yields a demand function for each variety i,
" 1
Ct (i) = At (i)
"
Pt (i)
Pt
Ct
(62)
where Pt is the aggregate price index de ned by:
Pt
21
Z
4
1 "
Pt (i)
At (i)
311"
di5
0
(63)
Firms: Each rm produces a di erentiated good i using a production function given by
1
Yt (i) = Ht (i) =Zt (i)
where
(64)
1 captures decreasing returns to scale in production, Ht (i) is the di erentiated labor input,
and, for later convenience, Zt (i) denotes the inverse of rm-speci c productivity. The evolution of Zt (i) is
described in the next subsection.
35
The rm's nominal pro t each period, excluding the resources spent to acquire information about market
conditions, is:
t (i)
= Pt (i)Yt (i)
Wt (i) Ht (i)
(65)
In every period, the rm sets its price and commits to ful ll all demand at that price.
In the absence of information costs, the rm would seek to maximize its discounted pro t stream,
E0
1
X
R0;t
t (i)
(66)
t=0
Government: For simplicity, the government pursues an exogenous policy. The net monetary transfer
in each period is equal to the change in money supply, which is assumed to evolve exogenously, as described
in the next subsection:
T t = Mt
5.2
Mt
(67)
1
The Shocks
The economy is subject to three kinds of shocks: (1)
t,
permanent monetary shocks, which are the only
source of aggregate disturbances in the economy, are generally small, and are summarized in the exogenous
evolution of money supply; (2)
t
(i), permanent idiosyncratic quality shocks, which a ect both the demand
for an individual product and the cost of producing it; and (3)
t
(i), transitory idiosyncratic productivity
shocks.
The log of money supply11 is assumed to follow a random walk process,
mt = mt
i:i:d:
t
where
11 I
t
1
+
t
g
is independent over time and from any other disturbances in the economy.
use lower-case letters to denote logs of di erent variables introduced in subsection 5.1.
36
(68)
(69)
The inverse of rm-speci c productivity, Zt (i), contains independently distributed permanent and transitory components, where the permanent component is the same as the household preference shock. In logs,
this term evolves according to
zt (i) = at (i) +
at (i) = at
t
(i)
t
1
(i) +
i:i:d:
(i)
t
(i)
t
(70)
(i)
(71)
g
i:i:d:
(72)
g
(73)
The permanent shock at (i) is a quality shock that increases both the utility from consuming the product
and the e ort required to produce it. The assumption that this shock shifts both the household's demand
for the good and the cost of producing the good implies that the rm's pro t is shifted in the same way by
the permanent nominal shock mt and by the permanent idiosyncratic shock, at (i). This assumption enables
a reduction in the state space of the problem, thus increasing tractability. The same assumption is made
by Midrigan (2010) and Woodford (2010). The permanent quality shock will generate large and persistent
movements in both individual prices and relative prices over time, consistent with the data.
The shock
t
(i) is a purely transitory productivity shock that helps to generate large price changes, as
observed in the data.
5.3
The Equilibrium Under Full Information
In the exible-price equilibrium with no information costs and no other costs to nominal price adjust-
ment, the rm chooses its price in each period to maximize its per-period pro t in units of marginal utility.
The full-information optimal log-price,12 denoted by xt (i), is a linear combination of all the shocks in the
economy:
xt (i) = mt + at (i) +
12 The
t
(i) ,
optimal log-price is rescaled by a constant that is omitted.
37
"
"+1
<1
(74)
5.4
The Partial Equilibrium under Costly Information
Suppose that all aggregate variables evolve according to the exible price, full information equilibrium. A
set of rms of measure zero are information-constrained. When substituting the full-information equilibrium
outcomes, the pro t of an information-constrained rm is proportional to (pt (i)
xt (i))13 , where pt (i) is
the log-price charged by the information-constrained rm14 , xt (i) is the optimal full-information log-price
determined in (74), and
(p
x) = e(1
"
")(p x)
1
"
e
" (p x)
(75)
Equation (75) de nes the pro t function introduced in section 2. Note that (75) is maximized at
pt (i) = xt (i), hence xt (i) is also the current pro t-maximizing price for the information-constrained rm in
the static problem, excluding information costs. Therefore, the rationally inattentive rm would like to set
a price that is as close as possible to the target full-information price, xt (i), subject to the costs of acquiring
information about the evolution of this target.
The shocks are mapped into the notation used in section 2 by de ning
et (i)
t
t
(i)
+
t
t
(i)
(i)
(76)
(77)
To map the current model into the notation of section 3.4, which employs the normalized variables q,
y, and ye, and the pro t function
(q
y), note that the full information price in each period t can then
be written as a function of the permanent state at the time of the last review,
periods ago, and the
13 I
omit a term that does not a ect optimization.
log-price charged by the rationally inattentive
constant.
14 The
rm and the optimal log-price are rescaled by the same (omitted)
38
accumulated shocks since then,
xt (i)
y (i)
ye (i)
= mt
+ at
ye (i) +
X
j
+
j=1
(i) + y (i)
(i)
X
j
(78)
(79)
(80)
(i)
j=1
with ye0 (i) = 0.
Conditional on no review, the information-constrained price is
pt (i) = mt
The per-period pro t (pt (i)
+ at
(i) + q (i)
xt (i)) is replaced by (q (i)
(81)
y (i)), a function of the normalized price
and the normalized state, both of which are indexed by , the number of periods since the last policy review,
with (q
6
y) de ned by (75).
Numerical Results
[COMING SOON.]
7
Concluding Remarks
This paper presents a theory of price-setting in which rms design simple pricing policies that they
update infrequently. The key friction in the model is that all information that is relevant to the rm's pricing
decision is costly. In a departure from the existing literature on price setting under imperfect information,
both the decision of which price to charge from the current policy and the decision of whether or not to
conduct a review and design a new policy are based on costly, noisy signals about market conditions. The
precision of these signals is chosen endogenously, at the time of the policy review, subject to a cost per unit
of information.
The theory generates pricing patterns consistent with the evidence on discrete multiple-price regimes
39
documented in Stevens (2011). Matching salient features of the micro data pins down the quantity of
information acquired by
rms, which in turn determines the degree to which prices are tied to market
conditions. Nominal rigidity in this model depends on the quantity of information contained in prices, and
less so on the frequency with which individual prices change: prices in this model change frequently, yet they
are always only partially related to concurrent market conditions.
Future work will embed this theory in a general equilibrium framework that can explore the real e ects of
monetary shocks in the economy. This question has been an important part of the monetary research agenda
ever since nominal price rigidities were introduced as a key link in the monetary transmission mechanism.
A potential extension of the current setup is the introduction of price discrimination. The present model
abstract from any frictions other than information processing costs. Nevertheless, tpricing patterns in
the micro data suggest that at least parf of the volatility of prices is driven by
rms engaging in price
discrimination over time. This extension may further reduce the quantity of information acquired by rms,
and hence, for a given frequency of price changes, generate pricing policies that are tied to market conditions
even less.
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