Acta všfs, 1/2012, vol. 6 007
Information Frictions and Monetary Policy
Informační frikce a měnová politika
FILIP MATĚJKA
CERGE-EI1
Abstract
Real effects of monetary policy depend crucially on the nature of nominal rigidities. These
rigidities are typically modelled as sticky prices with explicit assumptions on either frequency
of price adjustments (Calvo-style models) or on the cost of adjustment (menu cost
models). However, recent empirical work cast doubts on these workhorses of standard
New Keynesian models. This paper discusses another approach to nominal frictions, which
is based on the assumption that agents face difficulties processing information. If, for
instance, price-setters learn about an interest rate cut with a delay, then their price also
responds sluggishly. This rigidity implies positive temporary effects on output and unemployment.
We conclude that models based on information frictions can account for several
empirical facts other model have difficulties reconciling with, such as sluggish responses
of both real and nominal variables, frequent but staggered price changes or a steeper
Phillips curve and higher profit losses with more volatile environments. Moreover, rational
inattention provides important implications for policy.
Keywords
nominal rigidity, information frictions, monetary economics
JEL Codes
D21, D83, E31, E52
Introduction
Models that are used to assess the optimal monetary policy are typically built around
certain assumptions about the nature of nominal rigidities. Different assumptions generate
different results and thus also potentially drastically different prescriptions for optimal
policies. This paper argues that there are several good reasons to built such models
around information frictions in the form of agents’ inattention. We also discuss the basic
policy implications of such models.
The most common nominal friction are driven by explicit assumptions of price stickiness
using Calvo-style adjustments or some form of menu cost. Bils and Klenow (2002), however,
cast doubts on these assumptions by finding that individual prices do not stay fixed
for long periods of time. When the models are calibrated to fit the observed frequency of
price changes, then the implied real effects of nominal shocks are very small. Bils and Klenow
(2002) thus motivated macroeconomists to focus on prices at the micro level, too.
1	 A joint workplace of the Center for Economic Research and Graduate Education, Charles University, and
the Economics Institute of the Academy of Sciences of the Czech Republic.
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An alternative line of the modeling of nominal rigidities is based on the assumption that
agents cannot attend to all the available information about new shocks. This idea was proposed
by Christopher Sims, formulated in a framework called “rational inattention”(Sims,
1998, 2003). Simply put: if price-setters do not pay attention to new shocks, they can not
respond to them and their prices thus stay rigid.
Mackowiak and Wiederholt (2007) showed that rational inattention can generate real effects
of monetary policy. While nominal aggregates, e.g. the price level, respond to monetary
shocks with a delay, individual prices change all the time, which is, however, counterfactual.
This problem was resolved in Matějka and Sims (2010) and Matějka (2010a). These
papers expand the approach of Mackowiak and Wiederholt by not only modeling of how
much information price-setters process, but also what they process information about.
Rationally inattentive agents in these models actively seek those pieces of information
that carry the most value. It turns that the implied price dynamics of such price-setters
corresponds very well with the data: prices stay rigid for a while, but not for too long, and
aggregates respond with a delay to aggregate shocks, which generates real effects of
monetary policy.
Rational inattention describes the humans’limited ability to process information. Most of
policy-related information is accessible with very little cost. We could, in principle, find out
what the current federal funds rate is at every single moment, we could find out the last
reported unemployment rate or the GDP growth. These pieces of informational are easily
attainable at most major business magazines or on the internet. Few of us, however, do
so very frequently.
Intuitively, the less the information is important for one’s business the less likely it is that
the individual bothers to devote time to acquire it. For example, most bankers know the
current level of the funds rate at least approximately, lower number of academics know
about it, and some local businessmen would not even know whether the rate moved in
the last two years at all.
Although the papers above show that rational inattention can account for realistic dynamics
of prices, it has been criticized on the grounds that the required amount of information
price-setters process is too low. This criticism was addressed in Matějka (2010b), where
the author shows that inattentiveness on the consumers’side suffices to generate rigidity
of prices. The intuition for this result goes as follows: consumers who dislike processing
information prefer stable prices. If prices are not stable, then the consumers have to pay
extra cost finding out what the current prices are. Price-setters consider effects of their
dynamic pricing strategies and choose to keep their prices relatively stable, in order to
attract more consumers and induce higher sales.
This paper expands on Matějka (2010a) and Matějka (2010b). It does so by studying dynamic
effects of aggregate shocks and by exploring implications of inattention in various
environments. We find that, for instance, that nominal rigidity is weaker in more competitive
industries or under more volatile aggregate conditions. This is particularly important
at times of crises, such as in 2008 and 2009, when the resulting responses to monetary
Acta všfs, 1/2012, vol. 6 009
shocks can be quite different from those during normal times. When we account for endogenous
choice of attention level, prices are more flexible during times of high volatility,
since agents choose to process more information. This implies that monetary policy has
little real effects.
These findings have numerous implications for policy, which we briefly discuss in the final
section.
1 	 Nominal Rigidity
Rational inattention allows for endogenizing what pieces of information to process. Decisions
of rationally inattentive agents are not a priori biased by a specific form of any mechanism
by which they process information. All pieces of information are freely available and
agents can select which of them to process. It sounds intuitive that agents want to be more
aware of variables that are important to them, such as income to a household or input cost
to a producer. Or for instance, a price-setter in a highly competitive industry could devote
more attention to prices of his competitors than if the competition were low.
Agents not only differentiate between variables (e.g. attention to inflation or GDP, etc.), they
can also pay different amounts of attention to different levels of the same variable and also
different amounts of attention to the same variable under different conditions. If an agent
uses a credit card, he may not pay full attention to the level of his debt and may consume
some typical amount. However, once his debt approaches the credit limit, he needs to be
more aware of exactly how much more he can charge on the card. Furthermore, economic
actors do not need to pay much attention to slowly moving variables such as to most of
the aggregate indices.The more predictable variables are, the less information needs to be
processed about them. In case volatility of these variables increases, agents might check on
their current values more often, to keep being informed reasonably precisely.
This section focuses on nominal rigidity. It first presents two examples of static problems
studying effects of competition on what sources trigger swifter responses, on the
informational content of prices and on the rate of responses. Next, we present a dynamic
model studying trade-offs between attention devoted to stable aggregate variable, such
as a price level, and volatile input cost, such as commodity price. We also discuss implications
of varied levels of the respective volatilities, which provides us with intuition about
monetary policy effects on different industries. The section is concluded with a dynamic
model of an inattentive consumer.
1.1 	 Allocation across Variables: When Sellers Respond to Cost Shocks and
When to Changes of Competitor’s Prices
Let us study a model of a seller allocating his attention between unit input cost of his
product and competitor’s price. The seller faces a consumer, who has nominal endowment
e = 1, she desires to consume two different products, whose prices are p1 and p2,
to maximize the CES utility aggregate
Acta všfs, 1/2012, vol. 6010
U = cr
+ cr
1/r
,
1 2 
(1)
subject to a budget constraint: c1
p1
+ c2
p2
≤ e. θ = 1/(1 − r) is the elasticity of substitution,
θ ∈ (1, ∞). The consumer is assumed to have unlimited abilities to process information
about the two prices, her demand for the product 1 is:
1
c1(p1 , p2) =
p2 p1/p2 + (p1/p2)θ
.

(2)
Figure 1: Distribution of priceFigure 1: Distribution of price
A seller of the product 1 maximizes the expectation of his profit,
Π(p1, p2, µ) = c1(p1 , p2)(p1 − µ). (3)
He first processes information about the competitor’s price, p2, and his unit input cost, µ.
Finally, he selects his own price, p1. For the purposes of this example, we study a
partial equilibrium only, taking distributions of p2 and µ as given. Let µ be uniformly
distributed in (1, 1.1) and p2 uniformly in (2, 2.2). We need to solve (11) − (15) in
Appendix. A source variable is a vector (p2, µ), while the response variable is the seller’s
price, p1.
A seller of the product 1 maximizes the expectation of his profit,
Π(p1, p2, μ) = c1(p1 , p2)(p1 − μ).  (3)
He first processes information about the competitor’s price, p2
, and his unit input cost, µ.
Finally, he selects his own price, p1
. For the purposes of this example, we study a partial
equilibrium only, taking distributions of p2
and µ as given. Let µ be uniformly distributed
in (1, 1.1) and p2
uniformly in (2, 2.2). We need to solve (11) − (15) in Appendix. A source
variable is a vector (p2
, µ), while the response variable is the seller’s price, p1
.
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Figure 1 summarizes responsiveness of the seller’s price to unit input cost and to com-
petitor’s price, both in less and more competitive markets. When θ = 3, i.e. goods are
relatively poor substitutes, price p1
varies relatively more with changes in input cost than
with shocks to competitor’s price. A distribution of prices for flexible input cost and a fixed
competitors price (upper left graph) is more spread out than when input cost is fixed and
competitor’s price is varied (upper right).
On the other hand, when θ = 20, i.e. goods are much better substitutes - the market is more
competitive, price p1 responds more or less to changes in the competitors price only.
When θ = 3, the seller possesses tighter knowledge about µ, while if θ = 20, then almost
all information capacity is spent on tracking p2.
1.2 	 Choice of Information Amount:
Competitive Industries Generate Flexible Prices
In the first example, we addressed the question of attention distribution, while the total
amount of information was kept fixed. However, we could assume that agents also choose
how much information to process. Let RIκ stand for the original model of a rationally inattentive
seller with a fixed information and RIλ denote a model with fixed unit cost of
information. In RIλ, sellers find the processing somewhat unpleasant and process more
information only as long as its cost is lower than the marginal benefits from it. Agents
maximize expectation of profit,

(4)Π(μ, p,κ) = p−θ
(p − μ) − λκ,
∼
where λ is the cost of processing 1 bit of information about µ, and κ is the amount of
information the price-setter chooses to process.
It turns out that sellers in highly competitive industries decide to process more information
about input cost, because their profits are more sensitive to suboptimal pricing. We
compare sellers of the same size that face demands with different elasticities. Elasticity of
demand is a measure of the degree of competition in an industry. Magnitude of demand
(size of the firm) determines shadow price of information and thus influences choices
of how much information to process. Larger sellers decide to process more information.
Normalizing the magnitude thus allows for unbiased comparisons of information choices
across markets.
The seller maximizes

(5)Π(μ, p,κ) =
p
popt(μ0)
−
(p − μ) − λκ.∼ θ
Figure 2 shows computational results of the dependence of a selected κ on demand elasticity,
θ, for unit input cost uniformly distributed in (0.8, 1.2) and λ = 0.5 · 10−3 . Optimal
information capacity is an increasing function of θ.
To justify this result analytically, we can apply the approach developed in Matějka (2010a).
The corresponding approximate loss factor at µ0
is
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0
θ
L(μ0 , θ) =
2μ
.

(6)
It is a decreasing function of θ. The higher elasticity of demand, e.g. degree of competition,
the bigger the loss from imperfect information about the input cost. Given the same
levels of input cost and the same firms’ sizes, agents process more information in more
competitive industries. Moreover, since the price is tracked more closely, more information
leads to more flexible prices.
Mutual information between two random variables is a measure of how much about one
variable can be inferred from learning about the other. Even for an outside observer, the
seller’s higher information capacity implies that prices carry more information about the
input cost.
Figure 2: Comparisons across industries
Demand Elasticity
InformationCapacity
0 5 10 15 20 25 30
3,5
3
2,5
2
1,5
1
0,5
This finding relates to Hayek’s famous defense of free markets, Hayek (1945), specifically
on the grounds of markets’ability to convey information. Rational inattention implies that
the more competitive a market is the more information can be extracted by observing its
prices.
1.3 	 Dynamics and Aggregate Trade-Offs
In this section, we will use a model introduced in Matějka (2010a). It shows that unlike
sticky-price models, rational inattention can generate frequent price changes together
with delayed aggregate responses. The model’s results agree very well with the empirical
findings in Eichenbaum, Jaimovich, and Rebelo (2008).
In the following model, there are two stochastic variables to which the seller responds.
Let the unit input cost be composed of two parts: an i.i.d. real unit input cost denoted by
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µ and the second one be a serially correlated nominal variable A. µ is supposed to be an
idiosyncratic volatile part of the input cost specific to the seller, while A plays the role of
a slowly moving aggregate variable, e.g. a price level. The profit function takes the following
form.
Π(A,μ, p) = p−θ
(p − Aμ).  (7)
A is a price index shifting the distribution of the nominal input cost Aµ. The aggregate
variable takes two different values only and that it is Markov. Let the Markov process be
symmetric with a probability of transition to the other state equal to t. A is binary, its
distribution is determined by the probability of either one of the two states. Let the state
variable be x = P rob(AL
), where AL
stands for the lower value of A. The model’s equations
are formulated in Appendix.
For computations, I used κ = 1, θ = 3, µ uniformly distributed over (0.8, 1.2), AL
= 1, AH
= 1.1,
t = 0.002 and β = 0.9992. One period is supposed to be one week. t = 0.002 implies that
the probability of changing a state (a 10% shock to the aggregate variable) at least once
during a year is about 10%. The annual discount factor is 0.96.
Figure 3 shows the results of simulations over 120 periods. There is a shock to A in the
period denoted as 1, when A switches from AL
to AH
. The top series in the figure presents
one realization of a price series and the second one shows a time-series of knowledge
about A in the same simulation.
The price setter processes information about Aµ and responds to it, trying to target the
optimal price θ/(θ − 1)Aµ. Although Aµ is distributed over a continuous range in every period,
prices again exhibit lots of rigidity of the values as well as in the i.i.d. case. Given prior
knowledge about A, together with its true value, the distribution of prices as responses
to realizations of µ is discrete. However, when knowledge about A changes, the distribution
of prices changes too. For the used values of parameters (κ, t, AH
, etc.), knowledge
adjustment is rather abrupt. The second series in Figure 3 shows a knowledge adjustment
that is quite typical for all realizations of single simulations with these parameters. What
varies from one simulation to another is the period in which the seller finds out that A has
probably switched to a new value. The sudden change of knowledge is, however, not
inherent to all solutions under rational inattention. The next subsection discusses this
point in a little more detail.
The bottom two series in Figure 3 are prices and knowledge averaged over 10 000 runs.The
average knowledge about A shifts slowly, while the average price does actually change
abruptly in period 1. The variable of interest to the seller is in fact Aµ, not values of A and
µ separately. Due to different dynamical properties of A and µ, and a non-uniform prior
on Aµ, the agent does not process information exactly about Aµ only. Although, the seller
does pay special attention to Aµ, he also refines knowledge about other regions in the
whole A × µ space. In period one, after a positive shock to A, the seller is likely to find out
that the value of Aµ is high and thus the probability that a distribution’s top price is realized
increases. Due to the prior knowledge that A is probably at the lower state, the agent
underestimates the true value of Aµ. Expected price adjusts abruptly, but still less than
optimally. Since A stays at the higher level, the agent obtains signals on a high Aµ several
Acta všfs, 1/2012, vol. 6014
periods in a row and slowly learns that it is not due to a streak of high µ, but rather due
to a jump in A. The average price further increases towards the new optimal level. Prices
change frequently, but responses to shocks to the aggregate variable are delayed.
The difference between RIκ and RIλ
versions of the dynamic model correspond to differences
between their static counterparts. Stochastic properties of the variables of interest
(µ and A) influence the agent’s choice of what pieces of information and potentially how
much information to process. Let us vary these properties and study their implications for
responses to shocks to the aggregate variable A.
1.3.1	 Idiosyncratic Volatility
Thus far, µ was uniformly distributed in (0.8, 1.2). Table 1 summarizes numerical results for
RIκ with κ = 1, θ = 3 and t = 0.002 for three different widths of the distribution of µ:
Figure 3: Two stochastic variables, sudden learning, t = 0.002, κ = 1
Figure 4: Average price response, 3 distributions of µ, λ = 0.003
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Figure 4: Average price response, 3 distributions of µ, λ = 0.003
for µ fixed at 1, uniformly distributed in (0.9, 1.1) and in (0.8, 1.2). Two charac
of responses to a shock to A averaged over 10 000 runs are in columns 2 and 3. “
adj.” represents a portion of the average long-term adjustment that was realized
the first period, while “90% adjustment” denotes the number of periods it ta
average price until 90% of the full adjustment is realized.
Table 1: Implications of idiosyncratic volatility for average responses, κ =
µ 1st per. adj. 90% adjustment profit loss
1
(0.9,1.1)
(0.8,1.2)
100%
83%
61%
1
2
11
0%
0.28%
1.09%
The more volatile is the seller’s idiosyncratic part of the input cost the slower he r
to aggregate shocks. When µ is fixed at 1, 1 bit of information is sufficient
innovations of the binary variable A perfectly. The column “profit loss” presents
losses in comparison with pricing under perfect information - this quantity was e
with both µ and A simulated according to their stochastic properties. As expected
losses are higher in more volatile environments.
Results of the similar experiments for RIλ, λ = 0.003, are shown in Table 2
Figure 4. Unlike RIκ, RIλ generates faster responses to aggregate shocks wh
more volatile.
Table 2: Implications of idiosyncratic volatility for average responses, λ = 0.
µ 1st per. adj. 90% adjustment profit loss mean κ
1 8% 25 0.08 0.0086
for µ fixed at 1, uniformly distributed in (0.9, 1.1) and in (0.8, 1.2). Two characteristics of
responses to a shock to A averaged over 10 000 runs are in columns 2 and 3.“1st per. adj.”
represents a portion of the average long-term adjustment that was realized during the
first period, while “90% adjustment” denotes the number of periods it takes the average
price until 90% of the full adjustment is realized.
Table 1: Implications of idiosyncratic volatility for average responses, κ = 1
µ 1st per. adj. 90% adjustment profit loss
1 100% 1 0%
(0.9,1.1) 83% 2 0.28%
(0.8,1.2) 61% 11 1.09%
The more volatile is the seller’s idiosyncratic part of the input cost the slower he responds
to aggregate shocks. When µ is fixed at 1, 1 bit of information is sufficient to track innovations
of the binary variable A perfectly. The column “profit loss” presents seller’s losses in
comparison with pricing under perfect information - this quantity was evaluated with
both µ and A simulated according to their stochastic properties. As expected for RIκ, losses
are higher in more volatile environments.
Results of the similar experiments for RIλ
, λ = 0.003, are shown in Table 2 and in Figure 4.
Unlike RIκ, RIλ
generates faster responses to aggregate shocks when µ is more volatile.
Table 2: Implications of idiosyncratic volatility for average responses, λ = 0.003
µ 1st per. adj. 90% adjustment profit loss mean κ
1 8% 25 0.08 0.0086
(0.9,1.1) 22% 17 1.34% 0.014
(0.8,1.2) 60% 12 1.34% 0.86
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The table also presents average κ that was selected by the seller during the simulations.
Agents in RIλ
choose to process more information when volatility of input cost increases,
Figure 5: Average processed information, µ ∈ (0.9, 1.1), λ = 0.003, t = 0.02
The table also presents average κ that was selected by the seller during the simula
Agents in RIλ choose to process more information when volatility of input cost incre
Figure 5: Average processed information, µ ∈ (0.9, 1.1), λ = 0.003, t = 0.02
which increases the marginal value of information. Since a rationally inattentive a
processes optimal joint signals about A × µ, then more total information also im
moreinformation about A. However, faster average responses do not always mean
precise responses - profit loss is the same for µ ∈ (0.9, 1.1) and µ ∈ (0.8, 1.2). Th
dropsdramatically when µ is fixed at 1. Input cost becomes a binary variable, ther
the intuition derived from the linear-quadratic approximation does not apply
well.
In RIλ , the amount of processed information is not constant. It varies according t
expected value of information given a prior. Figure 5 presents average inform
amount as a function of time. Immediately after the shock occurs, a large fracti
sellers realize there might have been a shock and they choose to process more. Late
average selected capacity decreases. Once the transition period is over, the
equilibrium information capacity is actually below the initial level - the new average
cost is higher and the marginal value of information is thus lower.
1.3.2 Aggregate Volatility
Aggregate volatility can be adjusted by varying the Markov parameter t, the proba
of transition between the two states. Tables 3 and 4, present characteristics of av
responses for κ = 1 and λ = 0.003 for four different levels of t.
Table 3: Implications of aggregate volatility for average responses, κ = 1
which increases the marginal value of information. Since a rationally inattentive agent
processes optimal joint signals about A × µ, then more total information also implies
moreinformation about A. However, faster average responses do not always mean more
precise responses - profit loss is the same for µ ∈ (0.9, 1.1) and µ ∈ (0.8, 1.2). The loss
dropsdramatically when µ is fixed at 1. Input cost becomes a binary variable, therefore, the
intuition derived from the linear-quadratic approximation does not apply very well.
In RIλ
, the amount of processed information is not constant. It varies according to the expected
value of information given a prior. Figure 5 presents average information amount
as a function of time. Immediately after the shock occurs, a large fraction of sellers realize
there might have been a shock and they choose to process more. Later on, average selected
capacity decreases. Once the transition period is over, the new equilibrium information
capacity is actually below the initial level - the new average input cost is higher and
the marginal value of information is thus lower.
1.3.2	 Aggregate Volatility
Aggregate volatility can be adjusted by varying the Markov parameter t, the probability
of transition between the two states. Tables 3 and 4, present characteristics of average
responses for κ = 1 and λ = 0.003 for four different levels of t.
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Table 3: Implications of aggregate volatility for average responses, κ = 1
t 1st per. adj. 90% adjustment profit loss
0.001 56% 18 1.08%
0.002 61% 11 1.09%
0.006 73% 7 1.13%
0.02 80% 4 1.20%
Table 4: Implications of aggregate volatility for average responses, λ = 0.003
t 1st per. adj. 90% adjustment profit loss mean κ
0.001 55% 19 1.34% 0.86
0.002 60% 12 1.34% 0.86
0.006 73% 5 1.35% 0.88
0.02 83% 3 1.35% 0.92
More volatile A generates faster responses to its innovations in both of the models, RIκ
and RIλ
. This is due to a higher marginal value of processing new information about A if
A is more likely to vary and due to the signal extraction of A from Aµ. When volatility of
A increases, shocks to Aµ are more likely to be attributed to A. However, unlike in the Lucas’
signal extraction of the whole Aµ, profit losses are higher when the aggregate environment
is more volatile. Less stable Aµ is more difficult to be tracked precisely.
The effect of accelerated average adjustment is slightly stronger in RIλ
, since higher volatility
provides additional motive for processing more total information.
Volatile aggregate environment generates higher losses and swifter responses and thus
potentially also a steeper Phillips curve.
1.4 	 Nominal Rigidity Driven by Consumers’ Inattention
In the previous sections, one has to use quite low information capacity, κ, or high cost
of information, λ, to be able to generate quantitatively realistic rigidities. In many cases,
low information capacity is not unappealing, consider a local businessmen selling homemade
honey, but corporations such as Microsoft, IBM or GM also set prices rigidly, while
probably having very good information about economic aggregates. In these cases, it is
perhaps less likely that nominal rigidities would be driven by information constraints of
the price-setters. This section shows that consumers’ inattention suffices to generate the
rigidity.
Consumers often do not realize what a product’s exact price is at the moment of a purchase
decision. This is inspired by the observation that some consumers just grab certain
products in a supermarket without even looking at their prices. Many of us at least read
price’s first few digits, while ignoring the cents. Typically, we implicitly assume that prices
end with .95 or .99. If the number of cents is actually 85, we may not spot it and still keep
our initial guess. Sometimes, we read just the first digit only or none at all.
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If it is unpleasant, i.e. costly, to inspect prices and if uncertainty about the true price can
discourage consumption, then sellers could try to accommodate consumers with more
predictable prices. It might be optimal for the seller not to respond to every minor change
of input cost. Such frequent price changes would require consumers to pay lots of attention
to the price, and if they did not want to, then they could rather decide to consume
less.
Rationally inattentive agents learn about new innovations slowly. If there is a shock to an
aggregate variable and if the seller’s input cost is correlated with this variable, then the
seller chooses to respond to such a shock gradually - he chooses to price in line with the
consumer’s expectations.
Imagine a consumer has some partial knowledge about shocks to energy prices. She also
knows that energy prices are the main determinant of the input cost for her favorite local
sauna club. The consumer’s expectations about the admission prices to the sauna vary
with what she knows about the current prices of energy. The sauna owner might postpone
new price changes until consumers expect them to occur.
1.4.1	 Model
The model has these features:
i)	The input cost is drawn from a binary distribution in the period 0.
ii)	The consumer’s knowledge of the seller’s cost evolves independently of the sell-
er’s actions. Knowledge is gradually refined.
iii)	The seller’s price is a function of the unit input cost and the time elapsed from the
initial shock.
Let the seller’s unit input cost be equal to an aggregate variable A. The seller is small and
has a negligible influence on the consumer’s knowledge about shocks to A - the knowledge
evolves independently from the seller’s pricing responses to A. A is drawn from
asymmetric binary distribution {AL
, AH
} in the period 0 and stays constant forever after.
Consumers know that one of the two possible shocks is realized, but need to process information
to find out which one it is. Such a setting with a one-time shock is both simpler
to solve and yet illustrative enough to document the implications of gradual knowledge
adjustment.
In fully-fledged models under rational inattention, we would specify consumers’ preferences
and allow them to choose what pieces of information to process. I will, however,
assume one specific form of information structure. The qualitative properties of the results
do not rest on this assumption.
Let us assume that the consumer’s knowledge in period t has the same form as if the
consumer acquired one signal through a binary channel with a noise level X (t). X (t)
is decreasing in t, which models knowledge refinement2
. With increasing time, there is
2	 No sequence of signals across periods is considered, just one signal, which gets tighter in latter periods.
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a higher probability that agents receive the correct signal. Posterior knowledge is thus
more concentrated.
If A = AH
, then the probability that an agent receives the correct signal, (A = AH
), is
1 − X (t). The posterior knowledge of an agent having received such a signal is {P(AL
) = X
(t), P(AH
) = 1 − X (t)}. The posterior knowledge of agents who received the corrupted signal,
(A = AL
), is {P(AL
) = 1 − X (t), P(AH
) = X (t)}.
The seller chooses his pricing strategy. Unlike in the earlier sections of this paper, the
strategy is not a function of the input cost only. The consumer’s knowledge evolves even
after period 0, when the input cost is kept fixed. Different consumer’s knowledge can imply
a different optimal pricing response to the same input cost. The pricing strategy takes
the form p = p˜(A, {g(A)}) = p˜(A, t), where {g(A)} is the distribution of knowledge3
in the
population of consumers. {g(A)} is determined by X (t), which is pinned down by time t.
The strategy can be expressed using two functions, p˜L
(t) and p˜H
(t), each corresponding
to one level of unit input cost:
pL(t) if A = AL,
p =
pH(t) if A = AH.∼
∼

(8)
Consumers are rational, they know the form of p˜L
(t) and p˜H
(t). Together with their knowledge
about the aggregate shock (A determines which one of p˜L
(t) and p˜L
(t) is to be applied),
the pricing strategy generates the consumer’s prior on price. More specifically, the pricing
strategy forms the prior’s support, while knowledge about A determines the relative probabilities
of its two points. The term prior reflects knowledge ”before” a consumer processes
information about the seller’s price, but it is ”after”she has processed information about the
aggregate shock. Since the consumer’s knowledge about A is independent of the seller’s actions,
optimal prices at different levels of X (t) can be set independently of each other.
Definition 1. Model: X (t) is given for all t ∈ {0.∞}; it is a non-increasing function. For each
t, the seller chooses p˜L
(t) and p˜H
(t), maximizing the expectation of his profit
decreasing in t, which models knowledge refinement2
. With increasing time, there is a
higher probability that agents receive the correct signal. Posterior knowledge is thus
more concentrated.
If A = AH , then the probability that an agent receives the correct signal, (A = AH ), is
1 − X (t). The posterior knowledge of an agent having received such a signal is {P(AL)
= X (t), P(AH) = 1 − X (t)}. The posterior knowledge of agents who received the
corrupted signal, (A = AL), is {P(AL) = 1 − X (t), P(AH) = X (t)}.
Definition 1. Model: X (t) is given for all t ∈ {0.∞}; it is a non-increasing function.
For each t, the seller chooses p˜L (t) and p˜H (t), maximizing the expectation of his
profit
{p˜L (t), p˜H (t)} = arg max
{p˜
0
(t),p˜
0
(t)}
L H
1 0
2
(p˜L
(t) − AL
)
(1 − X (t))E[C |t, AL ] + X (t)E[C |t, AH ]
+ 1 0
2
(p˜H
(t) − AH
)
X (t)E[C |t, AL ] + (1 − X (t))E[C |t, AH ]
o
. (9)
The expression for expected profit weights the true realizations of AL and AH and also the
consumer’s priors generated by receiving signals on AL or AH. E[C |t, AL] denotes the
consumption expectation when a consumer’s prior is determined by a signal pointing to
AL with a noise level X (t) - the corresponding prior is {g(p˜L (t)), g(p˜L (t))} = {1 −X

(9)
3	 It is actually a distribution of distributions.
Acta všfs, 1/2012, vol. 6020
The expression for expected profit weights the true realizations of AL
and AH
and also the
consumer’s priors generated by receiving signals on AL
or AH
. E[C |t, AL
] denotes the consumption
expectation when a consumer’s prior is determined by a signal pointing to AL
with a noise level X (t) - the corresponding prior is {g(p˜L
(t)), g(p˜L
(t))} = {1 −X (t), X (t)}. On
the other hand, a prior determining E[C |t, AH
] is {1 − X (t), X (t)}.
For each X (t), the numerical representation of the optimal pricing strategy can be found
simply by evaluating the expected profit for all combinations of {p˜L
(t), p˜H
(t)}. Let noise
decrease at the following rate:
xpected profit weights the true realizations of AL and AH and also the
nerated by receiving signals on AL or AH. E[C |t, AL] denotes the
ion when a consumer’s prior is determined by a signal pointing to
l X (t) - the corresponding prior is {g(p˜L (t)), g(p˜L (t))} = {1 −X
her hand, a prior determining E[C |t, AH ] is {1 − X (t), X (t)}.
numerical representation of the optimal pricing strategy can be
uating the expected profit for all combinations of {p˜L (t), p˜H (t)}.
the following rate:
X (t) = 0.5 − 0.05t, ∀t ∈ {0..10}. (10)
e is A = AH, then the seller’s price gradually increases until it
rmation price in period 10. Otherwise, it gradually decreases. For
ignals across periods is considered, just one signal, which gets tighter in latter

(10)
If the realized value is A = AH
, then the seller’s price gradually increases until it reaches the
full information price in period 10. Otherwise, it gradually decreases. For simplicity,
Figure 6: Gradual price adjustment, scaled to information amount
let κ = 0. Consumers do not process any additional information about the seller’s price,
they only use their knowledge about the aggregate variable. If κ > 0, the same optimal
prices would correspond to higher levels of X.
Figure 6 shows the resulting solution. Consumers possess very little knowledge about A in
early periods. They know the seller’s pricing strategy, but have difficulties distinguishing
between the two different values of prices, p˜L
(t) and p˜H
(t), that can be realized in the
particular period. If X (t) = 0, consumers always acquire the correct signal, then the seller
sets the perfect information optimal prices, which are represented by the dash-dotted
bounds. If X (t) = 0.5, consumers can not tell at all which of the two prices was realized - in
such a case, the seller chooses to set one price only. Like in the static model, consumers
consume more when they are less uncertain about prices. With the increasing probability
of the correct signal, optimal prices p˜L
(t) and p˜H
(t) are set further and further away from
each other.
021Acta všfs, 1/2012, vol. 6
Figure 6 presents the impulse response of prices to an aggregate cost shock. Although the
price-setter is perfectly attentive, prices adjust slowly. The more information the consumer
processes, the faster the prices adjust. This implies that prices adjust faster if, for instance,
the price makes up for a relatively large portion of the consumer’s budget.
2	 Implications and Conclusions
Most importantly, this paper shows that rational inattention can generate several properties
of price dynamics that are observed in data. We find that prices can change quite
frequently and yet generate inertial of nominal aggregates, Section 2.3. In data, Bils and
Klenow (2002); Eichenbaum, Jaimovich, and Rebelo (2008), prices in retail stores stay fixed
on average for less than 4 months. On the other hand, price level fully responds to an aggregate
shock only after at least a year. The presented model reconciles with such findings.
We thus conclude that rational inattention could provide the proper microfoundations for
the models of nominal rigidities used in monetary policy.
Moreover, all of the following implications of the presented model agree with the
evidence.
1) Prices are more flexible in volatile and competitive industries, Sections 2.3.1 and 2.3.1.
2) Prices are more flexible in volatile aggregate environments, Section 2.3.2.
3) Prices of small-budget products are less flexible, Section 2.4.
What does all this imply? For monetary policy, the findings have three main implications:
The proposed modeling approach based on rational information.1.	
Monetary policy can become ineffective in stimulating output during crises. We find2.	
that prices become flexible when the aggregate environment is more volatile. In such
cases, the price-setters choose to pay more attention to new shocks. Paying more
attention then implies that prices adjust faster. If prices are flexible, then real effects
of monetary policy diminish. Unfortunately, this can occur at exactly the times when
the stimulation of output could be highly desirable.
Optimal monetary policy should focus on stabilizing the price level. The big question3.	
in monetary economics is how to balance trade-offs between stabilizing price-level
and output. Recently, Paciello and Wiederholt (2011) find that when decision- makers
in firms choose how much attention they devote to aggregate conditions, complete
price stabilization is optimal also in response to shocks that cause inefficient fluctuations
under perfect information. This finding goes in the opposite direction to what
standard sticky-price models imply. Under sticky-prices, i.e. when explicit adjustment
costs occur, pure price level stabilization is not optimal in case when mark-up shocks
or taste shocks occur. Rational inattention thus provides additional reassurance that
central banks having the price level stabilization as their primary objective, e.g. the
Czech National Bank, choose the optimal policy objective.
The presented model provides some intuition for implications for fiscal policy too. The
model we studied, and is formulated in Appendix, is a general setup of responses of inat-
Acta všfs, 1/2012, vol. 6022
tentive agents to exogenous shocks. Those shocks can be of non-monetary nature too,
while most results would still hold.
In 2008 and 2009, policy-makers around the world considered what actions to take to
stimulate the output and employment as quickly as possible. Rational inattention implies
that while the adjustment of the federal funds rate may be ineffective at turbulent
times, fiscal policy can generate desirable results. This is exactly for the reason that at such
time agents pay more attention, which weakens the effects of monetary policy, but can
strengthen effects of fiscal policy.
Lower taxes go sometimes wasted when public does not notice them. During 2008-2009,
public paid much more attention to business news than usually. This period was thus particularly
receptive to the lowering of income taxes, including the social security payments.
At the same time, higher consumption taxes were the prime candidate to fill the government
budget. When noticed, higher consumption taxes increase inflation expectations,
which only increases current consumption and thus output too.
On the other hand, increased government spending may be far from being the optimal
action. It is true that effect of such spending would be the quickest, but it might be very
weak. When agents know about the negative nature of the aggregate shock, then they
are better aware of the future budget consequences of the current spending and may
not increase their consumption. In other words, Ricardian equivalence becomes stronger
during such volatile times.
The model presented in Section 2.4 and its extended version in Matějka (2010b) also provide
a novel framework for quantification of costs of too complicated laws and tax codes.
It has become common knowledge that complicated tax codes are detrimental. We, however,
still do not know what the proper trade-off between the code’s theoretical optimality
and the optimal complexity that can be handled by real citizens. The presented model of
the rationally inattentive consumer has the features needed to tackle the problem: the
consumer (citizen) finds it complicated to understand all details of the pricing strategy
(tax code), so the seller (government) chooses to keep the prices more rigid (simpler tax
code) to accommodate the consumer.
Although, this is mainly a theoretical paper, the potential implications seem important
enough to further study and develop the theory of rational inattention.
References
BILS, M.; KLENOW, P. J. (2002). Some Evidence on the Importance of Sticky Prices. NBER
Working Papers 9069, National Bureau of Economic Research, Inc.
EICHENBAUM, M.; JAIMOVICH, N.; REBELO, S. (2008). Reference Prices and Nominal Rigidities.
NBER Working Papers 13829, National Bureau of Economic Research, Inc.
HAYEK, F. (1945). Use of Knowledge in Society. The Americal Economic Review. 35.
MACKOWIAK, B. A.; WIEDERHOLT, M. (2007). Optimal Sticky Prices under Rational Inattention.
CEPR Discussion Papers 6243. C.E.P.R. Discussion Papers.
MATĚJKA, F. (2010a). Rationally Inattentive Seller: Sales and Discrete Pricing. CERGE-EI
Working Papers wp408.
023Acta všfs, 1/2012, vol. 6
MATĚJKA, F. (2010b). Rigid Pricing and Rationally Inattentive Consumer. CERGE-EI Working
Papers Series. 20(2), pp. 1–40.
MATĚJKA, F.; SIMS, C. A. (2010). Discrete Actions in Information-Constrained Tracking Problem.
Technical report, Princeton University.
PACIELLO, L.; WIEDERHOLT, M. (2011). Exogenous Information, Endogenous Information
and Optimal Monetary Policy.
SIMS, C. A. (1998). Stickiness. Carnegie-Rochester Conference Series on Public Policy. 49(1),
pp. 317–356.
SIMS, C. A. (2003). Implications of rational inattention. Journal of Monetary Economics.
50(3), pp. 665–690.
Contact adress / Kontaktní adresa:
Filip Matějka
CERGE-EI
(Filip.matejka@cerge-ei.cz)
Acta všfs, 1/2012, vol. 6024
Appendix: 	 Formulation of Models under Rational Inattention
Response strategy under rational inattention. Let g(Y) be the agent’s prior knowl-edge
about Y , κ be her information capacity and U (y, z) be the indirect utility function. Her
decision strategy f (Y, Z ) is a solution to the following maximization problem.
A Formulation of Models under Rational Inattention
Response strategy under rational inattention. Let g(Y ) be the agent’s prior knowledge
about Y , κ be her information capacity and U(y, z) be the indirect utility function.
Her decision strategy f(Y, Z) is a solution to the following maximization problem.
f(Y, Z) = arg max E[U(Y, Z)] = arg max
\int U(y, z)f0
(y, z)dydz, (11)
subject to
f 0 (·,·) f 0 (·,·)
\int f0
(z, y)dz = g(y) ∀y
(12)
f0
(y, z) ≥ 0, ∀y, z (13)
I(Y ; Z) ≤ κ, (14)
I(Y ; Z) is mutual information between Y and Z, defined as
I(Y ; Z) = H (Y ) − H (Y |Z) =
= \int f(y, z) log
f(y, z)
g(y)f(z)
dydz. (15)
(12) requires consistency with prior knowledge, agents can not process more information
simply by forgetting what they knew in advance, and (13) states non-negativity of a
probability distribution. (14) is the information constraint.
The recursive formulation of the seller’s dynamic problem, a RIκ version, is as follows.
V (x) = max
f
\int Π(A, µ, p) + βV
(x0
)
i
f(A, µ, p)dAdµdp, (16)
subject to
Z
x0
= f(AL |p)(1 − t) + 1 − f(AL |p) t (17)
f(A, µ, p)dp = g(A, µ) = g1(A)g2 (µ) (18)
g1(AL ) = x (19)
I(A, µ; P ) ≤ κ. (20)
f(A, µ, p) is a joint distribution summarizing the seller’s choice of signals and responses
in the given period, (17) is law of motion for knowledge, it generates a prior on A in the
following period from a posterior in the current period via a Markov process with the
transition probability t. (18) is the constraint on a prior, µ and A are assumed to be
independent. g2(µ) is fixed, g1(AL ) = x,(19), which implies g1(AH ) = 1 − x. (20) is the
traditional constraint on mutual information between the source variables µ and A, and

(11)
subject to
mulation of Models under Rational Inattention
rategy under rational inattention. Let g(Y ) be the agent’s prior knowlY
, κ be her information capacity and U(y, z) be the indirect utility function.
strategy f(Y, Z) is a solution to the following maximization problem.
(Y, Z) = arg max E[U(Y, Z)] = arg max
\int U(y, z)f0
(y, z)dydz, (11)
f 0 (·,·) f 0 (·,·)
\int f0
(z, y)dz = g(y) ∀y
(12)
f0
(y, z) ≥ 0, ∀y, z (13)
I(Y ; Z) ≤ κ, (14)
utual information between Y and Z, defined as
I(Y ; Z) = H (Y ) − H (Y |Z) =
= \int f(y, z) log
f(y, z)
g(y)f(z)
dydz. (15)
consistency with prior knowledge, agents can not process more information
orgetting what they knew in advance, and (13) states non-negativity of a
istribution. (14) is the information constraint.
formulation of the seller’s dynamic problem, a RIκ version, is as follows.
V (x) = max
f
\int Π(A, µ, p) + βV
(x0
)
i
f(A, µ, p)dAdµdp, (16)
Z
x0
= f(AL |p)(1 − t) + 1 − f(AL |p) t (17)
f(A, µ, p)dp = g(A, µ) = g1(A)g2 (µ) (18)
g1(AL ) = x (19)
I(A, µ; P ) ≤ κ. (20)
a joint distribution summarizing the seller’s choice of signals and responses
period, (17) is law of motion for knowledge, it generates a prior on A in the
iod from a posterior in the current period via a Markov process with the
obability t. (18) is the constraint on a prior, µ and A are assumed to be
g2(µ) is fixed, g1(AL ) = x,(19), which implies g1(AH ) = 1 − x. (20) is the
onstraint on mutual information between the source variables µ and A, and
 (12)
lation of Models under Rational Inattention
tegy under rational inattention. Let g(Y ) be the agent’s prior knowlκ
be her information capacity and U(y, z) be the indirect utility function.
ategy f(Y, Z) is a solution to the following maximization problem.
Z) = arg max E[U(Y, Z)] = arg max
\int U(y, z)f0
(y, z)dydz, (11)
f 0 (·,·) f 0 (·,·)
\int f0
(z, y)dz = g(y) ∀y
(12)
f0
(y, z) ≥ 0, ∀y, z (13)
I(Y ; Z) ≤ κ, (14)
al information between Y and Z, defined as
I(Y ; Z) = H (Y ) − H (Y |Z) =
= \int f(y, z) log
f(y, z)
g(y)f(z)
dydz. (15)
nsistency with prior knowledge, agents can not process more information
etting what they knew in advance, and (13) states non-negativity of a
ribution. (14) is the information constraint.
rmulation of the seller’s dynamic problem, a RIκ version, is as follows.
V (x) = max
f
\int Π(A, µ, p) + βV
(x0
)
i
f(A, µ, p)dAdµdp, (16)
Z
x0
= f(AL |p)(1 − t) + 1 − f(AL |p) t (17)
f(A, µ, p)dp = g(A, µ) = g1(A)g2 (µ) (18)
g1(AL ) = x (19)
I(A, µ; P ) ≤ κ. (20)
oint distribution summarizing the seller’s choice of signals and responses
iod, (17) is law of motion for knowledge, it generates a prior on A in the
d from a posterior in the current period via a Markov process with the
ability t. (18) is the constraint on a prior, µ and A are assumed to be
(µ) is fixed, g1(AL ) = x,(19), which implies g1(AH ) = 1 − x. (20) is the
traint on mutual information between the source variables µ and A, and
 (13)
lation of Models under Rational Inattention
tegy under rational inattention. Let g(Y ) be the agent’s prior knowlκ
be her information capacity and U(y, z) be the indirect utility function.
ategy f(Y, Z) is a solution to the following maximization problem.
, Z) = arg max E[U(Y, Z)] = arg max
\int U(y, z)f0
(y, z)dydz, (11)
f 0 (·,·) f 0 (·,·)
\int f0
(z, y)dz = g(y) ∀y
(12)
f0
(y, z) ≥ 0, ∀y, z (13)
I(Y ; Z) ≤ κ, (14)
al information between Y and Z, defined as
I(Y ; Z) = H (Y ) − H (Y |Z) =
= \int f(y, z) log
f(y, z)
g(y)f(z)
dydz. (15)
onsistency with prior knowledge, agents can not process more information
etting what they knew in advance, and (13) states non-negativity of a
ribution. (14) is the information constraint.
ormulation of the seller’s dynamic problem, a RIκ version, is as follows.
V (x) = max
f
\int Π(A, µ, p) + βV
(x0
)
i
f(A, µ, p)dAdµdp, (16)
Z
x0
= f(AL |p)(1 − t) + 1 − f(AL |p) t (17)
f(A, µ, p)dp = g(A, µ) = g1(A)g2 (µ) (18)
g1(AL ) = x (19)
I(A, µ; P ) ≤ κ. (20)
joint distribution summarizing the seller’s choice of signals and responses
riod, (17) is law of motion for knowledge, it generates a prior on A in the
d from a posterior in the current period via a Markov process with the
ability t. (18) is the constraint on a prior, µ and A are assumed to be
(µ) is fixed, g1(AL ) = x,(19), which implies g1(AH ) = 1 − x. (20) is the
straint on mutual information between the source variables µ and A, and

(14)
A Formulation of Models under Rational Inattention
Response strategy under rational inattention. Let g(Y ) be the agent’s prior knowledge
about Y , κ be her information capacity and U(y, z) be the indirect utility function.
Her decision strategy f(Y, Z) is a solution to the following maximization problem.
f(Y, Z) = arg max E[U(Y, Z)] = arg max
\int U(y, z)f0
(y, z)dydz, (11)
subject to
f 0 (·,·) f 0 (·,·)
\int f0
(z, y)dz = g(y) ∀y
(12)
f0
(y, z) ≥ 0, ∀y, z (13)
I(Y ; Z) ≤ κ, (14)
I(Y ; Z) is mutual information between Y and Z, defined as
I(Y ; Z) = H (Y ) − H (Y |Z) =
= \int f(y, z) log
f(y, z)
g(y)f(z)
dydz. (15)
(12) requires consistency with prior knowledge, agents can not process more information
simply by forgetting what they knew in advance, and (13) states non-negativity of a
probability distribution. (14) is the information constraint.
The recursive formulation of the seller’s dynamic problem, a RIκ version, is as follows.
V (x) = max
f
\int Π(A, µ, p) + βV
(x0
)
i
f(A, µ, p)dAdµdp, (16)
subject to
Z
x0
= f(AL |p)(1 − t) + 1 − f(AL |p) t (17)
f(A, µ, p)dp = g(A, µ) = g1(A)g2 (µ) (18)
g1(AL ) = x (19)
I(A, µ; P ) ≤ κ. (20)
f(A, µ, p) is a joint distribution summarizing the seller’s choice of signals and responses
in the given period, (17) is law of motion for knowledge, it generates a prior on A in the
following period from a posterior in the current period via a Markov process with the
transition probability t. (18) is the constraint on a prior, µ and A are assumed to be
independent. g2(µ) is fixed, g1(AL ) = x,(19), which implies g1(AH ) = 1 − x. (20) is the
traditional constraint on mutual information between the source variables µ and A, and
Formulation of Models under Rational Inattention
sponse strategy under rational inattention. Let g(Y ) be the agent’s prior knowlge
about Y , κ be her information capacity and U(y, z) be the indirect utility function.
r decision strategy f(Y, Z) is a solution to the following maximization problem.
f(Y, Z) = arg max E[U(Y, Z)] = arg max
\int U(y, z)f0
(y, z)dydz, (11)
bject to
f 0 (·,·) f 0 (·,·)
\int f0
(z, y)dz = g(y) ∀y
(12)
f0
(y, z) ≥ 0, ∀y, z (13)
I(Y ; Z) ≤ κ, (14)
Y ; Z) is mutual information between Y and Z, defined as
I(Y ; Z) = H (Y ) − H (Y |Z) =
= \int f(y, z) log
f(y, z)
g(y)f(z)
dydz. (15)
) requires consistency with prior knowledge, agents can not process more information
mply by forgetting what they knew in advance, and (13) states non-negativity of a
obability distribution. (14) is the information constraint.
e recursive formulation of the seller’s dynamic problem, a RIκ version, is as follows.
V (x) = max
f
\int Π(A, µ, p) + βV
(x0
)
i
f(A, µ, p)dAdµdp, (16)
bject to
Z
x0
= f(AL |p)(1 − t) + 1 − f(AL |p) t (17)
f(A, µ, p)dp = g(A, µ) = g1(A)g2 (µ) (18)
g1(AL ) = x (19)
I(A, µ; P ) ≤ κ. (20)
A, µ, p) is a joint distribution summarizing the seller’s choice of signals and responses
the given period, (17) is law of motion for knowledge, it generates a prior on A in the
owing period from a posterior in the current period via a Markov process with the
nsition probability t. (18) is the constraint on a prior, µ and A are assumed to be
ependent. g2(µ) is fixed, g1(AL ) = x,(19), which implies g1(AH ) = 1 − x. (20) is the
ditional constraint on mutual information between the source variables µ and A, and

(15)
(12) requires consistency with prior knowledge, agents can not process more information
simply by forgetting what they knew in advance, and (13) states non-negativity of a probability
distribution. (14) is the information constraint.
The recursive formulation of the seller’s dynamic problem, a RIκ version, is as follows.
A Formulation of Models under Rational Inattention
Response strategy under rational inattention. Let g(Y ) be the agent’s prior knowledge
about Y , κ be her information capacity and U(y, z) be the indirect utility function.
Her decision strategy f(Y, Z) is a solution to the following maximization problem.
f(Y, Z) = arg max E[U(Y, Z)] = arg max
\int U(y, z)f0
(y, z)dydz, (11)
subject to
f 0 (·,·) f 0 (·,·)
\int f0
(z, y)dz = g(y) ∀y
(12)
f0
(y, z) ≥ 0, ∀y, z (13)
I(Y ; Z) ≤ κ, (14)
I(Y ; Z) is mutual information between Y and Z, defined as
I(Y ; Z) = H (Y ) − H (Y |Z) =
= \int f(y, z) log
f(y, z)
g(y)f(z)
dydz. (15)
(12) requires consistency with prior knowledge, agents can not process more information
simply by forgetting what they knew in advance, and (13) states non-negativity of a
probability distribution. (14) is the information constraint.
The recursive formulation of the seller’s dynamic problem, a RIκ version, is as follows.
V (x) = max
f
\int Π(A, µ, p) + βV
(x0
)
i
f(A, µ, p)dAdµdp, (16)
subject to
Z
x0
= f(AL |p)(1 − t) + 1 − f(AL |p) t (17)
f(A, µ, p)dp = g(A, µ) = g1(A)g2 (µ) (18)
g1(AL ) = x (19)
I(A, µ; P ) ≤ κ. (20)
f(A, µ, p) is a joint distribution summarizing the seller’s choice of signals and responses
in the given period, (17) is law of motion for knowledge, it generates a prior on A in the
following period from a posterior in the current period via a Markov process with the
transition probability t. (18) is the constraint on a prior, µ and A are assumed to be
independent. g2(µ) is fixed, g1(AL ) = x,(19), which implies g1(AH ) = 1 − x. (20) is the
traditional constraint on mutual information between the source variables µ and A, and

(16)
subject to
A Formulation of Models under Rational Inattention
Response strategy under rational inattention. Let g(Y ) be the agent’s prior knowledge
about Y , κ be her information capacity and U(y, z) be the indirect utility function.
Her decision strategy f(Y, Z) is a solution to the following maximization problem.
f(Y, Z) = arg max E[U(Y, Z)] = arg max
\int U(y, z)f0
(y, z)dydz, (11)
subject to
f 0 (·,·) f 0 (·,·)
\int f0
(z, y)dz = g(y) ∀y
(12)
f0
(y, z) ≥ 0, ∀y, z (13)
I(Y ; Z) ≤ κ, (14)
I(Y ; Z) is mutual information between Y and Z, defined as
I(Y ; Z) = H (Y ) − H (Y |Z) =
= \int f(y, z) log
f(y, z)
g(y)f(z)
dydz. (15)
(12) requires consistency with prior knowledge, agents can not process more information
simply by forgetting what they knew in advance, and (13) states non-negativity of a
probability distribution. (14) is the information constraint.
The recursive formulation of the seller’s dynamic problem, a RIκ version, is as follows.
V (x) = max
f
\int Π(A, µ, p) + βV
(x0
)
i
f(A, µ, p)dAdµdp, (16)
subject to
Z
x0
= f(AL |p)(1 − t) + 1 − f(AL |p) t (17)
f(A, µ, p)dp = g(A, µ) = g1(A)g2 (µ) (18)
g1(AL ) = x (19)
I(A, µ; P ) ≤ κ. (20)
f(A, µ, p) is a joint distribution summarizing the seller’s choice of signals and responses
in the given period, (17) is law of motion for knowledge, it generates a prior on A in the
following period from a posterior in the current period via a Markov process with the
transition probability t. (18) is the constraint on a prior, µ and A are assumed to be
independent. g2(µ) is fixed, g1(AL ) = x,(19), which implies g1(AH ) = 1 − x. (20) is the
traditional constraint on mutual information between the source variables µ and A, and
 (17)
Formulation of Models under Rational Inattention
esponse strategy under rational inattention. Let g(Y ) be the agent’s prior knowlge
about Y , κ be her information capacity and U(y, z) be the indirect utility function.
er decision strategy f(Y, Z) is a solution to the following maximization problem.
f(Y, Z) = arg max E[U(Y, Z)] = arg max
\int U(y, z)f0
(y, z)dydz, (11)
bject to
f 0 (·,·) f 0 (·,·)
\int f0
(z, y)dz = g(y) ∀y
(12)
f0
(y, z) ≥ 0, ∀y, z (13)
I(Y ; Z) ≤ κ, (14)
Y ; Z) is mutual information between Y and Z, defined as
I(Y ; Z) = H (Y ) − H (Y |Z) =
= \int f(y, z) log
f(y, z)
g(y)f(z)
dydz. (15)
2) requires consistency with prior knowledge, agents can not process more information
mply by forgetting what they knew in advance, and (13) states non-negativity of a
obability distribution. (14) is the information constraint.
he recursive formulation of the seller’s dynamic problem, a RIκ version, is as follows.
V (x) = max
f
\int Π(A, µ, p) + βV
(x0
)
i
f(A, µ, p)dAdµdp, (16)
bject to
Z
x0
= f(AL |p)(1 − t) + 1 − f(AL |p) t (17)
f(A, µ, p)dp = g(A, µ) = g1(A)g2 (µ) (18)
g1(AL ) = x (19)
I(A, µ; P ) ≤ κ. (20)
A, µ, p) is a joint distribution summarizing the seller’s choice of signals and responses
the given period, (17) is law of motion for knowledge, it generates a prior on A in the
llowing period from a posterior in the current period via a Markov process with the
ansition probability t. (18) is the constraint on a prior, µ and A are assumed to be
dependent. g2(µ) is fixed, g1(AL ) = x,(19), which implies g1(AH ) = 1 − x. (20) is the
aditional constraint on mutual information between the source variables µ and A, and
 (18)
Formulation of Models under Rational Inattention
onse strategy under rational inattention. Let g(Y ) be the agent’s prior knowlbout
Y , κ be her information capacity and U(y, z) be the indirect utility function.
cision strategy f(Y, Z) is a solution to the following maximization problem.
f(Y, Z) = arg max E[U(Y, Z)] = arg max
\int U(y, z)f0
(y, z)dydz, (11)
to
f 0 (·,·) f 0 (·,·)
\int f0
(z, y)dz = g(y) ∀y
(12)
f0
(y, z) ≥ 0, ∀y, z (13)
I(Y ; Z) ≤ κ, (14)
) is mutual information between Y and Z, defined as
I(Y ; Z) = H (Y ) − H (Y |Z) =
= \int f(y, z) log
f(y, z)
g(y)f(z)
dydz. (15)
quires consistency with prior knowledge, agents can not process more information
by forgetting what they knew in advance, and (13) states non-negativity of a
bility distribution. (14) is the information constraint.
cursive formulation of the seller’s dynamic problem, a RIκ version, is as follows.
V (x) = max
f
\int Π(A, µ, p) + βV
(x0
)
i
f(A, µ, p)dAdµdp, (16)
to
Z
x0
= f(AL |p)(1 − t) + 1 − f(AL |p) t (17)
f(A, µ, p)dp = g(A, µ) = g1(A)g2 (µ) (18)
g1(AL ) = x (19)
I(A, µ; P ) ≤ κ. (20)
, p) is a joint distribution summarizing the seller’s choice of signals and responses
given period, (17) is law of motion for knowledge, it generates a prior on A in the
ng period from a posterior in the current period via a Markov process with the
ion probability t. (18) is the constraint on a prior, µ and A are assumed to be
ndent. g2(µ) is fixed, g1(AL ) = x,(19), which implies g1(AH ) = 1 − x. (20) is the
onal constraint on mutual information between the source variables µ and A, and
 (19)
Formulation of Models under Rational Inattention
sponse strategy under rational inattention. Let g(Y ) be the agent’s prior knowle
about Y , κ be her information capacity and U(y, z) be the indirect utility function.
decision strategy f(Y, Z) is a solution to the following maximization problem.
f(Y, Z) = arg max E[U(Y, Z)] = arg max
\int U(y, z)f0
(y, z)dydz, (11)
ject to
f 0 (·,·) f 0 (·,·)
\int f0
(z, y)dz = g(y) ∀y
(12)
f0
(y, z) ≥ 0, ∀y, z (13)
I(Y ; Z) ≤ κ, (14)
; Z) is mutual information between Y and Z, defined as
I(Y ; Z) = H (Y ) − H (Y |Z) =
= \int f(y, z) log
f(y, z)
g(y)f(z)
dydz. (15)
requires consistency with prior knowledge, agents can not process more information
ply by forgetting what they knew in advance, and (13) states non-negativity of a
bability distribution. (14) is the information constraint.
recursive formulation of the seller’s dynamic problem, a RIκ version, is as follows.
V (x) = max
f
\int Π(A, µ, p) + βV
(x0
)
i
f(A, µ, p)dAdµdp, (16)
ject to
Z
x0
= f(AL |p)(1 − t) + 1 − f(AL |p) t (17)
f(A, µ, p)dp = g(A, µ) = g1(A)g2 (µ) (18)
g1(AL ) = x (19)
I(A, µ; P ) ≤ κ. (20)
, µ, p) is a joint distribution summarizing the seller’s choice of signals and responses
he given period, (17) is law of motion for knowledge, it generates a prior on A in the
owing period from a posterior in the current period via a Markov process with the
nsition probability t. (18) is the constraint on a prior, µ and A are assumed to be
ependent. g2(µ) is fixed, g1(AL ) = x,(19), which implies g1(AH ) = 1 − x. (20) is the
ditional constraint on mutual information between the source variables µ and A, and
 (20)
f (A, µ, p) is a joint distribution summarizing the seller’s choice of signals and responses in
the given period, (17) is law of motion for knowledge, it generates a prior on A in the following
period from a posterior in the current period via a Markov process with the transition
probability t. (18) is the constraint on a prior, µ and A are assumed to be independent.
g2
(µ) is fixed, g1
(AL
) = x,(19), which implies g1
(AH
) = 1 − x. (20) is the traditional constraint
on mutual information between the source variables µ and A, and a response variable p.