Strategic Inattention, Inflation Dynamics, andthe Non-Neutrality of Money*
Hassan Afrouzi†
Columbia University and NBER
February 14, 2024

Abstract
This paper studies how competition affects firms’ expectations in a new dynamic general equilibrium
model with rational inattention and oligopolistic competition where firms acquire information about
their competitors’ beliefs. In the model, firms with fewer competitors are less attentive to aggregate
variables—a novel prediction supported by survey evidence. A calibrated version of the model matches
the relationship between firms’ numbers of competitors and their uncertainty about aggregate inflation
as a non-targeted moment. A quantitative exercise reveals that firms’ strategic inattention to aggregatessignificantly amplifies monetary non-neutrality and shifts output response disproportionately towards
less competitive oligopolies by distorting relative prices.
JEL Codes: E31; E32; E71
Key Words: rational inattention, inflation expectations, oligopolistic competition, inflation dynamics,
monetary non-neutrality.

* I thank the editor and anonymous referees for their valuable comments and suggestions, which have significantly improved
this paper. I also thank Olivier Coibion, Saroj Bhattarai, and Yuriy Gorodnichenko for their invaluable and continuous support
during the development of this paper. In addition, I also would like to express my appreciation to Mark Dean, Andres Drenik,
Chen Lian, Filip Matějka, Ricardo Reis, Raphael Schoenle, Mirko Wiederholt, Michael Woodford, and seminar participants
at numerous institutions and conferences for helpful comments. Edited by Andrew Atkeson.
†
Email: hassan.afrouzi@columbia.edu. Address: 1105A IAB, 420 West 118th Street, New York, NY 10027.

1 Introduction
Almost every modern monetary model relates price changes to firms’ expectations about aggregate inflation.1
However, recent literature documents that firms’ inflation expectations are inaccurate and disconnected fromaggregate inflation (Candia, Coibion, & Gorodnichenko, 2021).2 Furthermore, the accuracy of firms’ expectations about aggregate variables correlates with the number of their competitors (Coibion, Gorodnichenko,& Kumar, 2018). These facts are inconsistent with our standard models and raise two questions: (1) How
does competition affect firms’ expectations? (2) What are the macroeconomic implications of the interaction
between competition and expectation formation?
This paper develops a new dynamic general equilibrium model with rational inattention and oligopolisticcompetition to study these questions. The interaction of these two model ingredients generates an endogenous
relationship between the number of firms’ competitors and their expectations about aggregate variables.
While both rational inattention and oligopolistic competition are necessary for this relationship—hereafter,
referred to as strategic inattention—neither one is sufficient on its own. To examine the quantitative fit of themodel, I calibrate it to firm-level survey data and find that it matches the relationship between firms’ beliefs
and the number of their competitors as a non-targeted moment. Finally, I find that strategic inattention
has quantitatively significant implications for output and inflation responses to monetary policy shocks. It
amplifies monetary non-neutrality by up to 48% and shifts the output response disproportionately towards
less competitive firms.
The basic model of this paper in Section 2 provides a closed-form characterization of oligopolistic
firms’ optimal beliefs under rational inattention. Rationally inattentive firms make mistakes in perceiving
fundamental shocks. Thus, with a finite number of competitors, the average prices of firms’ competitors
exhibit non-fundamental volatility, which is costly to the firms themselves as well as to their competitors
through strategic complementarities in pricing. Accordingly, when information acquisition is endogenous,
oligopolistic firms are strategically inattentive: they have the incentive to pay direct attention to the mistakesof their competitors, even at the expense of paying less attention to the fundamental shocks. Thus, the model
1

With sticky prices, inflation increases with expected future inflation (Woodford, 2003b). In models of information rigidity,
it increases with past expectations of current inflation (Lucas, 1972; Mankiw & Reis, 2002; Reis, 2006).
2
See, also, Kumar, Afrouzi, Coibion, and Gorodnichenko (2015).

1

predicts an endogenous relationship between competition and firms’ beliefs about aggregate variables: firms
with fewer competitors and higher strategic complementarities in pricing are less informed about aggregate
variables and have more uncertain beliefs.
Strategic inattention also implies that less competitive firms’ price changes covary less with their aggregate inflation expectations than with expectations of their competitors’ price changes. Firms that competewith only a few others do not optimize their prices relative to an aggregate price index but rather relative to theprices of their direct competitors, a feature that is reflected in their beliefs under rational inattention. As firms
pay direct attention to the beliefs of their competitors, prices are, on average, closer to firms’ expectationsof their competitors’ prices than to the aggregate price. Accordingly, expectations about aggregates are nolonger the relevant index for firms’ pricing decisions. Instead, a more appropriate index for aggregate prices
is firms’ aggregated expectations of their own competitors’ prices. Importantly, strategic inattention createsa wedge between the relevant expectations for prices and aggregate inflation expectations.
Direct motivating evidence from firm-level survey data in Section 3 supports the presence of strategic
inattention among firms. First, to assess whether the conditions required by the basic model hold in thedata, a novel question is included in a survey of New Zealand firms that measures significant strategic
complementarities in pricing. Furthermore, when asked how many direct competitors they face in their main
product market, firms report an average of 8 competitors. Second, as predicted by the model, firms withfewer competitors are more uncertain about aggregate prices.3 Third, firms are more aware of their own
industry prices than aggregate prices, which is also consistent with the model’s prediction that firms should
pay direct attention to the beliefs of their competitors.
To study the quantitative implications of strategic inattention, Section 4 extends the basic static modelof the paper to a micro-founded dynamic general equilibrium model to explore its macroeconomic implications. Oligopolistic competition is modeled through households’ preferences over different varieties, which
generates many small oligopolies with heterogeneity in the number of firms operating within them. Firms
are rationally inattentive and acquire information about their competitors’ beliefs and fundamental shocks
over time. On the methodological front, the model requires solving for the equilibrium strategy of a dynamic
3

Coibion et al. (2018) document a similar result for the size of forecast errors. The model in this paper also delivers a precise
prediction in terms of the variance of beliefs which is tested in Section 3.

2

rational inattention game within every oligopoly, which, to the best of my knowledge, is novel to this paper.
I solve these equilibrium strategies by extending recent methods for solving single-agent dynamic rational
inattention models.4
To validate the model, I calibrate it to the firm-level survey data and find that the model matches firms’
strategic inattention to inflation—i.e., the relationship between firms’ beliefs about aggregate inflation and thenumber of their competitors—as non-targeted moments. In the calibrated model, firms in more competitive
oligopolies acquire more information and allocate a larger amount of attention toward aggregate shocks,
consistent with the empirical evidence that more competitive firms are more informed about aggregates(Coibion et al., 2018, and the analysis in Section 3).
The remainder of the paper in Section 5 studies the aggregate and reallocative implications of strategic
inattention for the propagation of monetary policy shocks to output and inflation. Since firms in less competitive oligopolies acquire less information about aggregate shocks, their price responses to these shocks are
smaller and more persistent, both of which amplify monetary non-neutrality. I find that strategic inattention
has quantitatively significant aggregate effects: it increases the volatility of output due to monetary shocks
by up to 48% and increases its half-life by up to 22%. Moreover, it lowers the volatility of inflation causedby monetary shocks by up to 13% and increases its half-life by up to 9%. The fact that inflation responds
more persistently to shocks among firms with fewer competitors is consistent with evidence documented
by Schoenle (2018).
In addition to affecting the response of aggregate prices and output, strategic inattention also distorts the
response of relative prices and concentrates output response towards oligopolies with fewer firms. Since sucholigopolies are more strategically inattentive, their prices respond more sluggishly to expansionary monetaryshocks, attracting demand towards more concentrated oligopolies. Thus, more oligopolistic firms contribute
more to the output response of the economy relative to their steady-state market share. To examine theseeffects, I define the concentration multiplier of oligopolies with K competitors as the ratio of the cumulative
response of outputs in those oligopolies relative to the aggregate output response. These multipliers aredefined such that they are equal to one in a model without heterogeneity in output response. However, with
4

In particular, I use the method developed by Afrouzi and Yang (2019) which builds on and generalizes the first-ordercondition methods developed in Maćkowiak, Matějka, and Wiederholt (2018).

3

the heterogeneity caused by strategic inattention, more concentrated oligopolies drive a higher share of theoutput response. For instance, the cumulative output response in duopolies is 17% larger than the average
cumulative output response in the model.
The final step in Section 5 is a conceptual decomposition that inspects the mechanisms that are at workin the quantitative model. It is well-known that real rigidities significantly amplify monetary non-neutrality(Woodford, 2003a). Since strategic complementarities in the dynamic model are endogenous to the environment of firms and vary with competition, it may as well be that all the quantitative results are driven bydifferences in real rigidities across oligopolies rather than by strategic inattention. But this is not the case.
In fact, real rigidities work against strategic inattention in the calibrated model because firms with morecompetitors have higher strategic complementarities.
Therefore, oligopolistic competition has two opposing effects on monetary non-neutrality. Firms withfewer competitors pay less attention to monetary shocks due to strategic incentives, which amplifies monetary non-neutrality (the strategic inattention channel). However, firms with fewer competitors also havelower strategic complementarities, which attenuates monetary non-neutrality (the real rigidities mechanism).
While both effects are significant, the strategic inattention mechanism dominates and amplifies monetary
non-neutrality in oligopolies with fewer competitors.
To further investigate these mechanisms, I also derive an analytical decomposition in the static model andshow that monetary non-neutrality decreases with the number of competitors as long as demand elasticities
increase with the number of competitors. In a complementary exercise, I also solve the dynamic model
under strategic complementarities that decrease with the number of competitors. In this model, the strategic
inattention channel is mitigated because lower strategic complementarities of more competitive firms attenuate their incentives to acquire more information but are not enough to overturn the effect of larger demand
elasticities on information acquisition. Accordingly, firms with more competitors acquire more informationin the model as well and strategic inattention continues to amplify monetary non-neutrality when the numberof competitors is smaller.

Related Literature. This paper is motivated by the recent literature that investigates how firms’ expectations are related to their environment. The most related work in this area is Coibion et al. (2018) which4provides direct evidence for the relationship between firms’ number of competitors and their expectations. To
the best of my knowledge, the model in this paper is the first to provide an explanation for this relationshipand to investigate its implications. Most notably, in the model, inflation responds more persistently to shocks
among firms with fewer competitors. Schoenle (2018) documents a similar relationship in the U.S. PPI dataand provides evidence for this mechanism.
The model proposed in this paper is mainly related to the vast literature on rational inattention (Sims,
1998, 2003) and, especially, its applications to pricing models and business cycles dynamics (most notably,Maćkowiak & Wiederholt, 2009, 2015; Matějka, 2016).5 The previous work in this literature has mainly
focused on monopolistic competition models. The main contribution of this paper is to study the consequences of rational inattention in oligopolistic competition models, which is essential to the main objectiveof this study which aims to understand the effects of competition on firms’ expectations, and, through that,on inflation dynamics and monetary non-neutrality.
The oligopolistic structure of competition studied here is related to the literature that has focused on
its macroeconomic implications (Atkeson & Burstein, 2008; Rotemberg & Saloner, 1986; Rotemberg &Woodford, 1992). While this paper’s main focus is to understand the interaction of oligopolistic competition
with rational inattention, the implications of the model for monetary non-neutrality complement concurrent work by Mongey (2021) and Wang and Werning (2022), which focus on the interactions of nominal
rigidities with oligopolies. These three models provide a unified view of how competition affects output andinflation dynamics but under different mechanisms.6 In particular, the mechanism of interest here is strategic
inattention, which affects aggregate dynamics through firms’ expectations in a micro-founded model withendogenous information acquisition.
The model’s implications for inflation dynamics and monetary non-neutrality are also of particularinterest given the recent evidence on the rise of concentration (see, for instance, Autor, Dorn, Katz, Patterson,
5See, also, Pasten and Schoenle (2016); Stevens (2019); Yang (2022) for recent discussions and Maćkowiak, Matějka, andWiederholt (2023) for a detailed review of this literature. More broadly, the paper is also related to the literature on the effects
of information rigidities and monetary policy (e.g., Angeletos & La’O, 2009; Angeletos & Lian, 2016; Baley & Blanco, 2019;
Lucas, 1972; Mankiw & Reis, 2002; Melosi, 2016; Nimark, 2008; Reis, 2006; Woodford, 2003b).
6
Studying monetary non-neutrality with monopolistic competition under each of these frictions has a long history. For
information friction models, see Lucas (1972); Woodford (2003a). For random price adjustments in New Keynesian models,see Woodford (2003b)’s review of that literature. For price adjustment under menu costs see, for instance, Caplin and Spulber(1987); Golosov and Lucas (2007); Nakamura and Steinsson (2010).

5

& Van Reenen, 2020; Covarrubias, Gutiérrez, & Philippon, 2020; Kwon, Ma, & Zimmermann, 2023).
My results suggest that these trends are also changing the landscape of monetary policy by affecting the
propagation of these shocks to real and nominal variables.
This paper is also related to the literature on incentives to learn about others’ beliefs in strategic environments (Hellwig & Veldkamp, 2009; Myatt & Wallace, 2012). I depart from this literature by focusing
on an unrestricted set of available information and examining how the number of players affects informationacquisition incentives in a dynamic general equilibrium model. In that sense, the paper is also related to
Denti (2023), which studies unrestricted information acquisition with a finite set of actions and states, andHébert and La’O (2023), which studies large static games with more general information cost functions.

2 Static Model
This section studies the effect of oligopolistic competition on expectations in a static model with analyticalsolutions. It shows oligopolistic firms pay attention to their competitors’ beliefs, leading to correlated
non-fundamental mistakes in equilibrium. While the main text focuses on the economic implications, Online
Appendix C provides a rigorous treatment with proofs of propositions presented in Online Appendix C.9.

2.1 The Environment
There are a large number of sectors in the economy, indexed by j ∈J, each with K price-setting firms thatcompete with one another. Firms’ profits depend on a fundamental shock, q ∼N (0,1). Given q and prices,(pl,m)(l,m)∈J×K , firm j,k experiences quadratic profit losses from charging a price pj,k :1
Lj,k (q,(pj,k )(j,k)∈J×K )=(pj,k −(1−α)q−α K−12l̸=k pj,l ) ,P

where α ∈ [0,1) denotes the degree of within sector strategic complementarity.7 In Section 4, I derive thisloss function as a second-order approximation to firms’ profits and relate α to demand parameters.
Firms are rationally inattentive. They acquire information subject to a finite attention capacity and7Here, q and (pj,k )j∈J,k∈K can be interpreted as log deviations from a steady-state symmetric equilibrium.

6choose a pricing strategy that maps their information set to a price. To understand firms’ incentives in
information acquisition, I model the information choice set such that firms can directly choose the joint
distribution of their price with q and others’ prices. While this is a well-known feature of single-agent
rational inattention problems, it is not immediately clear how this would work in a game-theoretic setting.
How can a firm directly acquire information about other firms’ beliefs in a simultaneous game? Online
Appendix C.1 formalizes the answer by constructing a rich set of available signals, denoted by S, as thevector space generated by the fundamental q and a set of countably infinite independent normal randomvariables. As shown formally in the proof of Lemma C.2 in Online Appendix C.1, any deviation in joint
Gaussian distributions can then be generated by a random variable in this vector space. 8
Therefore, a pure strategy for firm j,k is to choose a set of signals, Sj,k ⊆S, and a pricing strategy thatis measurable with respect to the σ-algebra generated by its signals, pj,k :Sj,k →R. Given a strategy profilefor others, (Sl,m ⊆S)(l,m)̸=(j,k), firm j,k solves:
ii
h
h
2
P
1
minSj,k ⊆SE minpj,k :Sj,k →RE pj,k (Sj,k )−(1−α)q−α K−1
p
(S
)
S
j,k
l̸=k j,l j,l(1)s.t. I(Sj,k ;q,(pl,m(Sl,m))(l,m)̸=(j,k))≤κ

Here I(Sj,k ;q,(pl,m(Sl,m))(l,m)̸=(j,k)) is Shannon’s mutual information function and measures the amountof information that firm’s signals contain about q and others’ prices. Moreover, the constraint requires thata firm cannot acquire more than κ nats of information.9 I start by assuming κ is exogenous to study how
firms allocate a fixed κ across q and other firms’ prices. I relax this assumption starting in Section 2.4 toalso study how oligopoly parameters affect the choice of κ itself.
Definition 1. A pure strategy Gaussian equilibrium for this economy is a strategy profile (Sj,k ⊆ S,pj,k :Sj,k →R)(j,k)∈J×K from which no firm has an incentive to deviate and (q,(pj,k )(j,k)∈J×K ) has a multivariate
Gaussian distribution.
It can be shown that in all equilibria each firm observes only one signal, collinear with their price.108My definition of a rich information set corresponds to the concept of flexible information acquisition in Denti (2023).
Every nat is equal to log2(e)≈1.443 bits. For details about Shannon’s mutual information function see Online Appendix B.
10This extends the equivalent result in single-agent rational inattention problems to our game-theoretic setting (See, e.g.,
Afrouzi & Yang, 2019; Maćkowiak et al., 2018; Steiner, Stewart, & Matějka, 2017).
9

7

Let us denote strategies with this property, where the signal recommends the optimal price generated byits σ-algebra: (Sj,k ∈S,pj,k =Sj,k ), as recommendation strategies. Proposition C.1 in Online Appendix C.1
shows that all strategies are weakly dominated by feasible recommendation strategies. Thus, we can focus
on recommendation strategies without loss of generality.
It then follows that all equilibria are unique in the joint distribution that they imply for firms’ prices andq, which is done in Online Appendix C.2. The optimality of recommendation strategies combined with theuniqueness of the equilibrium in the joint distribution of prices and q allows us to directly focus on how
firms’ prices are related to one another. Let pj,k =Sj,k be the price that firm j,k charges in the equilibrium.
Proposition C.1 in Online Appendix C.1 further characterizes these equilibrium prices as:

1
pj,k =λ×((1−α)q+α K−1

Pzj,k ⊥(q,Sm,l)(m,l)̸=(j,k)
P
1E[zj,k ]=0, Var(zj,k )=λ(1−λ)Var((1−α)q+α K−1l̸=k pj,l )l̸=k pj,l )+zj,k ,(2)

Where λ≡1−e−2κ is a change of variables and has the interpretation of firms’ optimal Kalman gains on
their equilibrium signals. Moreover, zj,k is noise in prices introduced by rational inattention. We see thata larger capacity, κ, increases the covariance of prices with q and decreases the variance of the rational
inattention noise in the signal if the signal was normalized to be of the form “ideal price plus noise.” However,note that κ has a non-monotonic effect on the variance of the rational inattention noise in the price itself,
as seen in the expression for Var(zj,k ) above. This is because higher κ decreases the noise variance in thenormalized signals but also increases how much weight firms put on these signals in setting their prices.
Thus, the variance of the noise in prices is a U-shaped function of κ.

2.2 Economics of Attention Allocation
Rationally inattentive firms make mistakes in observing q—captured by zj,k above—which affects theirprices and the profits of their competitors. This section shows that firms choose to have correlated mistakes,
which creates a wedge between their expectations of industry vs. aggregate prices.
Define a mistake as the part of a firm’s price that is orthogonal to q. Then, any firm’s price can be8decomposed into its projection on q and its mistake:

pj,k =δq+vj,k ,vj,k ⊥q, δ ∈R.

The vector (vj,k )j,k∈J×K contains the mistakes of firms in the equilibrium, whose joint distribution is determined endogenously. Importantly, these mistakes may not be independent across firms as managers ofcompeting firms optimally attend to others’ mistakes.
With Gaussian distributions, a firm’s attention to a shock—i.e., the mutual information between its information set and the shock—increases with the absolute correlation of the firm’s price with that shock.11 BuildP1
ing on this, Online Appendix C.3 shows that when others play a strategy in which K−1l̸=k pj,l =δq+vj,−k ,firm j,k’s problem can be recast into choosing two separate correlations:s.t. ρ2q +ρ2v ≤λ≡1−e−2κασv
ρv
maxρq ≥0,ρv ≥0 ρq + 1−α(1−δ)

Here σv ≡p
Var(vj,−k ) is the standard deviation of the average mistakes of j,k’s competitors, ρq is thecorrelation of the firm’s signal with the fundamental, and ρv is its correlation with the average mistake ofits competitors. The following proposition states the properties of the equilibrium.
Proposition 1. In equilibrium, (1) Firms pay strictly positive attention to the mistakes of their competitors(ρ∗v > 0) if α > 0 and K is finite. (2) Firms do not pay attention to mistakes of those in other industries:∀(j,k),(l,m), if j ≠ l, pj,k ⊥pl,m|q. (3) Firms’ knowledge of the fundamental increases in the number of theircompetitors and decreases in the degree of strategic complementarity:∂ ∗∂ ∗ρ >0, ∂αρq <0.
∂K q(3)

Firms pay strictly positive attention to the mistakes of their own competitors because they are affectedby them, but not to the mistakes of firms in other industries. Since mistakes are orthogonal to q, fixing κ, any
attention to others’ mistakes has to be traded off with attention to the fundamental. With a larger α, a firm’s
profits depend more on competitors’ mistakes and the payoff of attending to these mistakes is higher. Also,11

For two normal random variables X and Y with correlation ρ, I(X,Y )=− 21 ln(1−ρ2), which is increasing in ρ2.

9with a larger K, the average mistake of a firm’s competitors is less variable (σv is smaller), which implies
more competitive firms substitute their attention towards the fundamental q.12

2.3 Comovement of Prices and Expectations
Conventional models relate firms’ prices to their expectations of aggregate prices. However, empiricalevidence on firms’ expectations shows that there is a disconnect between firms’ prices and their expectationsof aggregate inflation (Coibion et al., 2018). The model in this section provides an explanation for this
disconnect by showing that, at least with high strategic complementarities, firms’ prices are related mainly
to their expectations of their competitors’ prices:p=(1−α)Ej,k [q]+αEj,k [pj,−k ]The key notion here, as formalized in the proposition below, is that rational inattention with oligopolies
predicts a wedge between aggregate prices and firms’ average expectations of the aggregate price.
Proposition 2. In equilibrium, the realized price is closer in absolute value to the average expectations fromown-industry prices than the average expectation of the aggregate price itself.

|p−Ej,k [pj,−k ]|<|p−Ej,k [p]|This result relies on firms’ incentives to pay attention to the mistakes of their competitors. In equilibrium,
the signals that firms observe are more informative of their own sector’s prices:=δq. covaries with aggregate price

Sj,k =
|

z}|{
p
{z

+uj +ej,k}(4)

covaries with industry prices

where we have decomposed the mistake of firm j,k as vj,k =uj +ej,k , where uj ⊥p is the common mistakein sector j and ej,k is the independent part of firm j,k’s mistake. Since Var(uj )̸= 0 (by Proposition 1), thissignal reveals more information about industry prices. Thus, oligopolistic firms are more informed about
12

The proof of the Proposition 1 deals with the subtlety that σv is an equilibrium object and formalizes this argument.

10their industry prices than the aggregate price, even without any idiosyncratic shocks.
Moreover, these optimal signals generate correlated posteriors such that firms cannot distinguish betweenchanges in fundamental q and their competitors’ beliefs. Specifically, realizations of firms’ optimal signals
inform them of changes to their desired prices but do not reveal the source of those changes. Therefore, while
an increase in q causes an increase in firms’ prices, the strength of a firm’s response to such a change depends
on how strongly its signal covaries with both q and its competitors’ prices. This is similar to the mechanism
discussed in Hellwig and Venkateswaran (2009), where firms in a setting with exogenous information anda continuum of firms respond quickly to aggregate changes, even though they are not as well-informed about
aggregates. As such, it can be shown that in spite of lower attention to q when K is smaller, the numberof firms does not directly affect the covariance of aggregate price with the fundamental. Equation (C.6) inOnline Appendix C.2 derives the aggregate price as p=δq in the symmetric equilibrium, whereδ = λ−αλ1−αλ(5)which increases with capacity through λ and decreases with α but does not directly depend on K.
The independence of δ from K is a direct consequence of firms’ correlated posteriors discussed above.
With more competitors, firms pay more attention to the fundamental q, but with a fixed κ, this increased
attention to q comes at the expense of reduced attention to competitors’ mistakes, which lowers the covariance of their price with their expectation of their competitors’ prices. This is formalized in the following
proposition, which is proved and discussed in more detail in Online Appendix C.9.
Proposition 3. Fixing the information capacity κ, higher attention to the fundamental q is compensatedby lower attention to competitors’ mistakes, so much so that the covariance of aggregate price with thefundamental is independent of the number of firms K.
The substitution channel in Proposition 3 is a general force, but the independence of δ from the numberof competitors also relies on the fixed capacity assumption and the static environment. We investigate therole of these assumptions by endogenizing capacity first and postponing dynamics to Section 4.

11

2.4 Endogenous Choice of Information Processing Capacity
So far, we have studied how oligopolistic firms allocate a fixed amount of capacity, κ, as a function of theoligopoly parameters α and K. In this section, I endogenize κ and analyze how the oligopoly structure
affects firms’ optimal capacity, as well as the degree of monetary non-neutrality in the economy.

Optimal Information Capacity. Consider this extension of the firms’ problem in Equation (1):
2
ii
o
n
h
h 
1
∗
minκj,k ≥0 minSj,k ⊆SE minpj,k :Sj,k →RE 2 B pj,k (Sj,k )−pj,k (Sj,−k ) Sj,k +ωκj,k1
s.t. I(Sj,k ;q,(pl,m(Sl,m))(l,m)̸=(j,k))≤κj,k , p∗j,k (Sj,−k )≡(1−α)q+α K−1(6)Pl̸=k pj,l (Sj,l )where, now, in addition to choosing Sj,k , firm j,k also chooses the capacity κj,k ≥0, subject to a cost ωκj,k ,where ω >0 is a new parameter that captures the cost of producing capacity. Moreover, the new parameter
B >0 captures the curvature of the firm’s profit function and is micro-founded in Section 4.
Since we have already solved what the optimal attention strategy of firms is for a given κ, we can plugin this optimal allocation and re-write the problem just in terms of κj,k , derived in Online Appendix C.4:
1∗+ωκj,k }min { e−2κj,k BVj,−k2κj,k ≥0(7)∗
where Vj,−kis the unconditional variance of firm j,k’s ideal price given others’ strategies, p∗j,k (Sj,−k ). The

coefficient e−2κj,k captures the notion that by choosing a higher capacity firms can reduce their expected
losses from mispricing, in proportion to the curvature of their profit function B. Given the equilibrium∗
strategy of other firms, which determines Vj,−k, the optimal capacity κ∗j,k is:∗κ∗j,k = 12 max{0,ln(BVj,−k
/ω)}

(8)

Here, the max operator captures the possibility of the constraint κj,k ≥0 binding, which happens if the cost
ω is too high relative to the expected losses from mispricing. Moreover, holding ω fixed, the optimal capacity∗κ∗j,k is increasing in the curvature of the firm’s profit function B and the volatility of the ideal price Vj,−k,both of which increase the firms’ expected losses from mispricing.
12Equilibrium Capacity. To understand how κ∗j,k depends on the oligopoly parameters, we need to character∗∗ize Vj,−kand κ∗j,k jointly. As before, the change of variable λ∗ =1−e−2κ is useful, where λ∗ is the optimal

Kalman gain of firms on their equilibrium signals and increases with κ∗. Then, a symmetric equilibrium
is characterized by the following two equations, as derived in Online Appendix C.5:∗

K−1+αλ1−α 2V ∗ =( 1−αλ∗)
K−1+αλ∗ 1−α

(9)

1−αλ∗

λ∗ =max{0,1− BVω ∗ }(10)where we have dropped indexed j,k and j,−k due to symmetry. Here, the first equation gives the varianceof firms’ ideal prices as a function of their optimal capacities in a symmetric equilibrium. Moreover, thesecond equation is a reformulation of Equation (8) in terms of λ∗, where again the max operator captures∗

the possibility of the constraint κ≥0 binding—in which case λ∗ =1−e−2κ =0.
Importantly, endogenous capacity can lead to a multiplicity of symmetric equilibria with either λ∗ =0 or
λ∗ >0, as discussed in Online Appendix C.6. This is because strategic complementarity in pricing introduces
strategic complementarity in information acquisition, as discussed in Hellwig and Veldkamp (2009), albeit
in a setting with a continuum of agents.13 With high α or ω, if a firm’s competitors choose λ∗ =0, then thevalue of information could fall enough that the firm itself has no incentive to deviate from such a strategy.
However, if ω or α are small enough—precisely, if ω < B(1 − α)2Var(q), with Var(q) normalized to 1
here—this does not happen and there is a unique equilibrium with λ∗ >0. In the main text, I focus on thisequilibrium because it is the closest to the one studied in the dynamic model.
Proposition 4. When ω <B(1−α)2Var(q), there is a unique symmetric equilibrium where λ∗ decreases
with ω,α and K and increases with B.
∗

A higher ω increases the cost of information, leading to lower capacity κ∗ and λ∗ = 1−e−2κ . A higher
B increases the value of information due to higher curvature of the profit function, leading to higher λ∗.
Moreover, holding other parameters fixed, a higher α reduces the direct weight firms put on the fundamental
13

While the setting in Hellwig and Veldkamp (2009) is different enough that making exact comparisons would require alengthier discussion, the information acquisition incentives that arise under strategic complementarities are similar. However,
the main focus here is to study how these incentives vary with K and affect the propagation of shocks.

13and decreases the variance of firms’ ideal prices, lowering λ∗. Similarly, all else equal, a higher K decreases
the variance of firms’ ideal prices due to the law of large numbers and leads to lower λ∗.
Attention to the Fundamental. The relationship between λ∗ and K affects how firms pay attention to thefundamental q. Holding capacity constant, Proposition 1 shows that firms with more competitors pay more
attention to the fundamental because the law of large numbers reduces the average size of their competitors’
mistakes, reducing the value of information on the margin. With endogenous capacity, firms internalize thiseffect and opt for a smaller capacity, which reduces their total attention to shocks, including q. However,it can be shown that this secondary effect is small for small values of ω/B, and attention to the fundamental,∗2
ρ∗2
q , remains increasing with K in such as case. A more detailed discussion of how ρq varies with K canbe found in Online Appendix C.7, and an intuitive representation is provided in Online Appendix C.8 using
the following approximation of ρ∗2
q in the unique equilibrium:α+(1−α)(K−1) ω
ω
ω 2
ρ∗2
q ( B ,α,K)=1− (K−1+α)(1−α) B +O(∥ B ∥ )(11)This expansion approximates ρ∗2
q in the unique symmetric equilibrium around the full-information benchmark,i.e., ω =0. It is an appropriate approximation because the unique equilibrium requires Bω <(1−α)2Var(q)<Var(q)=1. Importantly, it shows that for small ω/B, attention to the fundamental increases with K as wellas B, and decreases with α and ω.

Covariance of Prices with the Fundamental. Let us conclude this section by revisiting the covarianceof prices and the fundamental, δ in Equation (5), now under endogenous capacity. This equation shows that,
all else equal, the covariance decreases with α both directly and indirectly through λ∗, and decreases with
K indirectly through λ∗. While we can use the predictions of Proposition 4 to perform these comparative
statics, a more intuitive way is to do an approximation of δ around the full-information benchmark, ω =0,
similar to the one used for ρ∗2
q . As derived in Online Appendix C.8:
(K−1)αωωδ∗( Bω ,α,K)=1− B(1−α)− K−1+α( B(1−α))2 +O(∥ Bω ∥3)

14

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This approximation accurately captures the comparative statics discussed, but it is also an appropriate endpoint
for our analysis of the static model. My approach so far has been to separately study the effects of α, K, andω/B, which has been helpful in isolating the mechanisms at work. However, this approach ignores the microfoundations of B and α. As we will see in the dynamic model, both α and B also depend on K, creating acascade of interactions through which δ and monetary non-neutrality vary with K. Armed with the intuition
from the static model, I first provide some motivating evidence and then turn to the dynamic model to study
the implications of strategic inattention for the transmission of monetary policy in a micro-founded setting.

3 Motivating Facts from Survey Data
Using the survey of firms’ expectations from New Zealand conducted by Coibion, Gorodnichenko, andKumar (2018); Coibion, Gorodnichenko, Kumar, and Ryngaert (2021), this section provides motivating
evidence for the predictions of the model in the previous section.14 Relative to previously documented facts,(1) I implement a new survey question that identifies the degree of strategic complementarity for firms, and(2) document that firms with more competitors are less uncertain about aggregate inflation.

Number of Competitors and Strategic Complementarity. Two of the key parameters of the model are
the number of a firm’s direct competitors and the degree of strategic complementarity. Two questions inthe survey measure these within a representative sample.
The first question asks firms “How many direct competitors does this firm face in its main product line?”
Columns (2) in Table A.1 in Online Appendix A presents a breakdown of firms’ answers from the sixth andeighth waves of the survey based on their industries. The average response in the sample is 8, which is also
fairly uniform across different industries. Moreover, Figure A.1 in Online Appendix A shows the distributionof firms’ responses in the sixth wave, with 45% of firms reporting six or fewer direct competitors.
As for the degree of strategic complementarity, I rely on the following survey question:15
14

This survey was conducted in a random sample of firms with broad sectoral coverage. The data I use here is describedin detail Coibion, Gorodnichenko, and Kumar (2018); Coibion, Gorodnichenko, Kumar, and Ryngaert (2021) and is publicly
available in the published replication packages of those articles.
15
The challenge for estimating this parameter using price data is that it is hard to find exogenous variations in the prices ofa firm’s competitors that are not correlated with aggregates or the firm’s own costs. There has been some recent progress inthis area: Amiti, Itskhoki, and Konings (2019) use international shocks as instruments for shocks that only move competitors’

15

“Suppose that you get news that the general level of prices went up by 10% in the economy:(a) By what percentage do you think your competitors would raise their prices on average?
(b) By what percentage would your firm raise its price on average?
(c) By what percentage would your firm raise its price if your competitors did not change their price at all in
response to this news?”

The question proposes a change in the firms’ environment due to aggregate variables, which affects boththeir costs and those of their competitors.16 The question then measures three different quantities that allow
me to disentangle the degree of strategic complementarity:
Answer to b.

z
}|
{
j,k
j,k
pj,k = (1−α)E [q] +αE [pj,−k ].
|{z} | {z }Answer to c.

(13)

Answer to a.

The average α implied by the responses of firms to this question is 0.82 and fairly uniform across different
industries, as reported in Column (4) of Table A.1.17 Coibion et al. (2021) follow my approach here andestimate similar strategic complementarities. Online Appendix D also examines the relationship between
firms’ number of competitors and the degree of strategic complementarity and shows that while varying
slightly and non-monotonically with K, the average α within equal quantiles and deciles of K remains, on
average, in the interval [0.8,0.9].

Uncertainty about Inflation versus Number of Competitors. We can also directly test the prediction of
the model that firms with more competitors should pay more attention to the aggregates. In the sixth wave of
the survey in 2016, firms were asked to report the distribution of their beliefs for aggregate inflation: “Pleaseassign probabilities (from 0-100) to the following ranges of overall price changes in the economy over the
next 12 months for New Zealand.” Firms were then asked to assign probabilities to a set of equally sizedprices and provide estimates of strategic complementarities for Belgian manufacturing firms. More recently, Burya and Mishra(2023) use the ACNeilsen Barcode Scanner data to estimate this object for the retail sector in the U.S.
16
I am grateful to anonymous referees for pointing out the following caveats with the framing of this question. First, thequestion takes it for granted that firms partially associate a change in the general level of prices with a change in their nominal
costs. This is a model-consistent assumption but might not hold in reality. A more accurate framing would be to propose ahypothetical scenario for an increase in nominal costs directly. Second, similar to the question about the number of competitors,
a more precise framing of this question should refer to firms’ direct competitors.
17
For reference, the usual calibration for the strategic complementarity in the U.S. in monopolistic competition models isaround 0.9 (see, e.g., Mankiw & Reis, 2002; Woodford, 2003b) which is slightly larger than what I estimate here.

16bins.18 To test the model’s prediction, I run the following regression:log(σiπ )=β0 +β1log(Ki)+ϵi,(14)where σiπ is firm i’s subjective uncertainty about the aggregate inflation—i.e., the standard deviation of theirreported distribution for inflation—and Ki is the firm’s reported number of competitors.
The model’s prediction translates to the null hypothesis that β1 <0.19 Table 1 reports the result of thisregression and finds β1 < 0 and significant. This result is robust to including firm controls such as firms’
age and size (measured by employment in main product line) as well as industry fixed effects.
This relationship is not reconcilable with full information rational expectation models or, to the best of
my knowledge, other macroeconomic models of information rigidity prior to this paper, and indicates theimportance of strategic incentives in how much firms pay attention to aggregate variables.

Knowledge about Industry versus Aggregate Inflation. The model also predicts that firms are moreaware of their competitors’ prices than the aggregate price. In the fourth wave of the survey conducted in2014, firms were asked to provide their nowcasts of industry and aggregate yearly inflation. Consistent withthis prediction, Table 2 shows that the average absolute nowcast error for industry inflation (1.16 percentage
points) is lower than the average absolute nowcast error for aggregate inflation (3.50 percentage points).
Additionally, in Figure A.2 in Online Appendix A, we see that these distributions are oppositely skewed: fornearly two-thirds of firms, their nowcast error for aggregate inflation is larger than the average error, while
the reverse is true for industry inflation.

4 A Micro-founded Dynamic Model
This section micro-founds and extends the static model of Section 2 to a dynamic general equilibrium model
to quantitatively analyze the effects of strategic inattention for the propagation of monetary policy shocks.
18

Firms were asked to assign probabilities to bins ranging from -25 percent to 25 percent with 5 percent increments. The
wide range is to avoid priming concerns, especially that firms assign positive probabilities to high inflation rates.
19
In the model σiπ2 ≡ Var(q|Sj,k ) = (1 − ρ∗2q )Var(q), which is strictly increasing in firms’ attention to the fundamental,∗2πmeasured by ρ∗2
q . Thus, the predictions of the static model for how ρq should vary with K translate to predictions about σi .

17Derivations, and proofs of propositions in this section, are included in Online Appendices F and H.

4.1 Environment
Households. The economy consists of a large number of sectors, j ∈J ≡{1,...,J}. Each sector j consistsof Kj ≥ 2 firms that produce weakly substitutable goods, where Kj is drawn from a distribution K. The
representative household takes the prices of these goods as given and decides how much to demand fromeach firm’s product. The aggregate time t consumption of the household isCt ≡
η−1  η
P
η−1
η
−1
J −1
C
,
C
≡
K
C
j,t
j
j∈J j,tk∈Kj j,k,t

Q

(15)

where Cj,t is the composite demand of the household for sector j, determined by a CES aggregator with theelasticity of substitution η >1.20 Moreover, the aggregate consumption, Ct, is a Cobb-Douglas aggregationof the composite goods across sectors. Therefore, the representative household’s problem isPtmax((Cj,k,t)(j,k)∈J×K ,Ct,Lt,Bt)∞Ef0 ∞t=0 β [log(Ct )−Lt ]
t=0
P
P
s.t.
j,k Pj,k,t Cj,k,t +Bt ≤Wt Lt +(1+it−1 )Bt−1 +j,k Πj,k,t −Tt(16)where Eft [.] is the full information rational expectations operator at time t,21 Ct is the aggregate consumption,
Lt is the labor supply of the household, Bt is their demand for nominal bonds, Wt is the nominal wage, itis the net nominal interest rate, Πj,k,t denotes the profit of firm j,k at time t, and Tt is a lump sum transfer
that is used to eliminate long-run inefficiencies of imperfect competition.
The within-sector CES aggregator leads to the following demand function for firm j,k:

Cj,k,t =QtD(Pj,k,t;Pj,−k,t), D(Pj,k,t;Pj,−k,t)≡J −1 P−ηPj,k,t1−ηl∈Kj Pj,l,t(17)

where Qt ≡PtCt is the nominal aggregate demand, with Pt denoting the price of the bundle Ct. Moreover,20

A more general aggregator can be considered here (e.g. Rotemberg & Woodford, 1992). I derive the implied demand under ageneric aggregator in Online Appendix F. Another specific case is the Kimball aggregator, which I discuss in Online Appendix G.
21
To study the effects of rational inattention under imperfect competition among firms, I assume households are fully informed
about prices and wages, which is a common assumption in the literature (see, e.g., Melosi, 2016).

18

Pj,k,t is firm j,k’s price at t, and Pj,−k,t is the vector of other firms’ prices in sector j. Furthermore, thehousehold’s intertemporal Euler and labor supply equations are given by:t]
Wt =Qt, 1=β(1+it)Eft [ QQt+1

Firms. Firms are rationally inattentive. At each period t, given their information set from the previous
period, they choose which signals to observe from a rich set of available signals, St, subject to an information
processing constraint.22 At each t, firm j, k can choose its information processing with a cost that isdenominated in labor, where the real cost of producing every unit of capacity is ωrsj units of labor. Thus, if
Lij,k,t denotes the amount of labor that the firm j,k uses for producing capacity, then κj,k,t =(ωrsj )−1Lij,k,t.
Here, ω >0 is the parameter that governs the cost of information, and rsj =(JKj )−1 is the revenue share (orrelative size) of the firm in the full-information symmetric equilibrium. This implies that the nominal cost of
producing capacity κj,k is WtLij,k,t =Wtωrsj κj,k,t, where Wt is the nominal wage. Moreover, the assumptionthat the labor cost of information is proportional to the firms’ relative size in the full-information benchmark(rsj ) hinges on three reasons. First, it makes the analysis consistent with the empirical evidence, since all theregressions presented in this paper about strategic inattention and the references to the literature control forfirms’ relative size. Second, from a theoretical perspective, it makes firms’ rational inattention problems sizeindependent so that as we take the monopolistic competition limit, information acquisition does not become
infinitely costly for firms (I will revisit this in more detail later when I derive a second-order approximation to
the firms’ problem). Finally, in the absence of this assumption, information would be relatively more costly forsmaller firms to acquire, which is inconsistent with the evidence on how firm size correlates with attention—ifanything, larger firms are more inattentive to aggregate variables (Candia et al., 2021; Coibion et al., 2018).23
After firms make their information choices, all new shocks and signals are drawn, and each firm observes
the realization of its signals. Firms then choose their prices conditional on their information sets,24 after
which demand for each variety is realized. Firms then hire enough labor to produce with a production
See Online Appendix E for the formal specification of St.
Similar assumptions are common in menu cost models. See, e.g., Gertler and Leahy (2008) where menu costs are assumed
to be proportional to firms’ relative size so that pricing decisions are size-invariant.
24
Since my main objective is to examine the real effects of monetary policy through endogenous information acquisition,
I abstract away from other sources of monetary non-neutrality, and in particular, assume that prices are perfectly flexible.
22

23

19

1

function that has decreasing returns in labor; Yj,k,t =(Lpj,k,t) 1+γ and meet their demand.
Formally, a strategy for firm j,k at t is to choose an information processing capacity conditional on
t−1
their initial information set, κj,k,t :Sj,k→R+, a set of signals to observe, Sj,k,t ⊂St, and a pricing strategyttthat maps its information set to their optimal actions, Pj,k,t :Sj,k→R, where Sj,k={Sj,k,τ }tτ=0 is the firm’s

information set at time t. Given a strategy for all the other firms in the economy, firm j,k maximizes thenet present value of their profits given their information set from the previous period:max{Sj,k,t⊂St,Pj,k,t(St ),κj,k,t(St−1)}t≥0
j,k

(18)

j,k

P
−1
d
t −1
d
)1+γ −(1−s̄j )Wt ×ωrsj ×κj,k,t)|Sj,kE[ ∞−(1−s̄j )Wt(Yj,k,t(Pj,k,tYj,k,t]t=0 β Qt
| {z } | {z } |
{z
}
{z
} |
discount factor

revenue

production cost

cost of attention

ds.t. Yj,k,t=QtD(Pj,k,t;Pj,−k,t)(demand)(l,m)̸=(j,k)τ
I(Sj,k,t;(Qτ ,Pl,m,τ (Sl,m
))τ≤t

t−1
|Sj,k
)≤κj,k,t

t−1
−1
t
Sj,k
=Sj,k
∪Sj,k,t, Sj,kgiven.

(information processing constraint)(evolution of the information set)

where I(.;.) is Shannon’s mutual information function as before and the information processing constraint
bounds the amount of information that the firm can acquire at time t by its chosen capacity κj,k,t. Moreover,s̄j is a constant hiring subsidy to firms in sector j that eliminates the steady-state inefficiencies from imperfectcompetition (see Galí, 2015, p. 73).25

Monetary Policy and General Equilibrium. Following the literature, I assume monetary policy controls
the growth of nominal aggregate demand and model it as an AR(1) process with persistence ρ:26

∆log(Qt)=ρ∆log(Qt−1)+ut.
25

(19)

Here, the presence of s̄j makes solving the model convenient by ensuring that all relative prices are the same in the
full-information economy but is not necessary, nor does it alter the economic forces at work.
26
See, e.g., Golosov and Lucas (2007); Mankiw and Reis (2002); Nakamura and Steinsson (2010); Woodford (2003a).

20

Equilibrium. A general equilibrium is an allocation for the household, ΩH ≡{(Cj,k,t)j∈J,k∈Kj ,Lst,Bt}∞
t=0 ,a strategy profile for firms given an initial set of signals−1dΩF ≡{(Sj,k,t ⊂St,Pj,k,t,κj,k,t,Lpj,k,t,Yj,k,t)∞t=0 }j∈J,k∈Kj ∪{Sj,k }j∈J,k∈Kj ,F
Hand a set of prices {it,Pt,Wt}∞
t=0 such that (a) given prices and Ω , Ω solves the household’s problem inEquation (16); (b) given prices and ΩH , no firm has an incentive to deviate from ΩF ; (c) {Qt ≡ PtCt}∞
t=0satisfies the monetary policy rule in Equation (19); (d) labor and goods markets clear.

4.2 Sources of Strategic Complementarity
Strategic complementarities are key for understanding how firms allocate their attention. Therefore, it isuseful to briefly discuss the sources of strategic complementarities in the model.
The first source of strategic complementarity is the sensitivity of optimal markups to prices in oligopolies.
It is well-known that CES demand with monopolistic competition implies constant demand elasticities andmarkups. With oligopolies, however, the granularity of firms implies that any change in a single firm’s pricealters the distribution of demand across its competitors and affects demand elasticities. The best responseof a firm shows this relationship:
εD (P ∗

,Pj,−k,t )∗∗Pj,k,t= εD (P ∗j,k,t,Pj,−k,t)−1 ×(1−s̄j )(1+γ)Q1+γ
D(Pj,k,t;Pj,−k,t)γ
t
j,k,t
{z
}
|
{z
}
|

(20)

marginal cost

optimal markup∗
where Pj,k,t
is the implied optimal price given Qt and Pj,−k,t, and the optimal markup has the familiar
∂Y

P

j,k,t
expression in terms of the elasticity of a firm’s demand, εD (Pj,k,t,Pj,−k,t)≡− ∂Pj,k,t. As in Atkeson andj,k,t Yj,k,t

Burstein (2008), it is informative to write these elasticities in terms of firms’ market shares:

εD (Pj,k,t,Pj,−k,t)=η−(η−1)mj,k,t, mj,k,t ≡ P

dPj,k,t Yj,k,tdl∈Kj Pj,l,t Yj,l,t

21

(21)

An immediate observation is that level of optimal markups increase in a firm’s market share:
εD (P ∗

,Pj,−k,t )

m

η
j,k,t
1
∗
µ(Pj,k,t,Pj,−k,t)≡ εD (P ∗j,k,t,Pj,−k,t)−1 = η−1+ η−1
1−mj,k,t

(22)

j,k,t

Moreover, one can derive the degree of strategic complementarity for a given set of prices by differentiating
the firm’s best response. To build intuition, let us start with the case of γ =0:∗
dPj,k,tt|γ=0 = dQ+∗
Pj,k,tQt(1−η−1)mj,k,t
|
{z
}

P

l̸=k mj,l,t dPj,l,t /Pj,l,tP

l̸=k mj,l,tstrategic complementarity

|

{z

}

average price-change of others

−

dQt
Qt



(23)

|{z}

change in wage

γ=0
An important observation is that strategic complementarity αj,k,t≡(1−η−1)mj,k,t increases with the firm’s

own market share. But why should a firm’s price be more sensitive to competitors’ prices when thosecompetitors hold lower market share? This becomes more puzzling in an extreme case when a single firmholds almost all the market with its market share approaching 1. Shouldn’t a firm that holds almost all of
the market simply disregard its competitors and act as a monopoly?
The answer relies on the structure of demand implied by CES preferences, where consumers reduce ahigher share of their demand with respect to a one percent change in the prices of a firm’s competitors whenthat firm holds a higher market share. Thus, while a monopolistic firm enjoys the sheer lack of competition,
the mere existence of small competitors shatters the autonomy of a firm in responding to their marginal costs,
especially at higher levels of market share. Therefore, while a monopolistic firm with CES demand would
charge a constant markup over its marginal cost, an almost monopolistic firm chooses to match the average
price change of their competitors with weight 1−η−1.
In the other extreme, strategic complementarity disappears as mj,k,t →0. This is not consistent with myfindings in the empirical section of the paper, where firms with a large number of competitors, and hence
potentially lower market share, still report high levels of strategic complementarity. This suggests that thesensitivity of markups is not the sole determinant of complementarities across firms, and other forces might
be at work. I capture this in the model by introducing decreasing returns to scale in labor (γ >0) as a secondsource of strategic complementarity.
Decreasing returns to scale (γ > 0) creates complementarities because relative prices affect a firm’s
22

production through demand in the equilibrium and higher production leads to higher marginal costs when
γ >0.27 Repeating differentiation of best response but now with γ >0, we obtain:1+γ
1− 1+γη(1−(1−η−1)mj,k,t)2

γ>0
αj,k,t=(1−η−1)mj,k,t +(1−(1−η−1)mj,k,t)(24)

Equation (24) shows that at high levels of market share, the strategic complementarity is mainly driven
by the sensitivity of the markup as in the case of γ = 0. However, now when mj,k,t is small, strategic
complementarity remains positive and converges to γ(η−1)
when mj,k,t →0.
1+γη

4.3 Solution Method and Incentives in Information Acquisition
An Approximate Problem. I use a second-order approximation to the firms’ problem to solve the model,
which is a usual approach to remedy the curse of dimensionality in rational inattention models.28 I derive
this second-order approximation around the symmetric full-information equilibrium. Due to symmetry, allfirms within a given sector j have the same market share under full information and charge the same markupµj over their marginal cost, (1−s̄j )Qt, given by Equation (22):full
Pj,k,t=µj (1−s̄j )Qt =Qt,∀j ∈J,k ∈Kj ,t≥0(25)where the second equality follows from s̄j =1−µ−1
j to eliminate steady-state distortions from market power.
Online Appendix F.2 derives a firm’s approximate problem under a general demand structure as:
∞
hX

i
−1
t
max t
−rsj Eβ t 12 Bj (pj,k,t(Sj,k)−p∗j,k,t)2 + ωκj,k,t |Sj,k| {z }{κj,k,t ,Sj,k,t ,pj,k,t (Sj,k )}t≥0
|
{z
}
t=0
loss from mispricing

cost of capacity

s.t. p∗j,k,t ≡(1−αj )qt +αj pj,−k,t(Sj,−k,t)(l,m)̸=(j,k) t−1
−1
t−1
τ
t
∪Sj,k,t, Sj,kgiven.
I Sj,k,t,(qτ ,pl,m,τ (Sl,m))τ≤tSj,k ≤κj,k,t, Sj,k
=Sj,k
27
28

This is a common approach in monetary models to generate strategic complementarities (see, e.g., Woodford, 2003b).
See, e.g., Maćkowiak et al. (2018) or Afrouzi and Yang (2019) for a discussion.

23

(26)

(27)

where pj,k,t ≡ log(Pj,k,t) and pj,−k,t ≡ Kj1−1P

l̸=k log(Pj,l,t ). Moreover, Bj is the curvature of firms’ profitεjDfunction in sector j around their optimal price. For a general demand structure, it has the form Bj = 1−α,jwhich in the case of the demand function assumed here is given byεjD
Bj ≡ 1−α=jη+γ(η−(η−1)Kj−1 )2
1+γ

(28)

Note that firms’ losses from mispricing are proportional to their relative size, captured by rsj . Thus, byassuming that the cost of capacity is also proportional to these revenue shares, the attention problem becomeshomogeneous in firms’ steady-state relative size, rsj . This allows the model to consistently relate to theempirical evidence on strategic inattention, which controls for firm size.
Moreover, Equation (28) shows that the attention problem of firms depends on the oligopoly parameters
only through the demand elasticity and strategic complementarity. This is similar to Wang and Werning(2022), who find that these objects are sufficient statistics for how oligopolistic prices respond to shocks
under nominal rigidities.29 My results show that, up to a second-order approximation, these objects are alsosufficient statistics for the optimal information structure of oligopolistic firms.

Information Acquisition Incentives. This approximate problem captures the trade-offs that a firm faces in
information acquisition. The quadratic term models the benefits of information acquisition: More information
allows firms to charge prices that are closer on average to their full-information best responses. This benefit
is traded off with the cost of information processing capacity, the second term.
This cost-benefit analysis depends on the number of firms in an oligopoly through two channels. First,
the extent of losses from mispricing depends on the curvature of firms’ profit functions, Bj . A larger Bj
amplifies losses from mispricing and increases the benefits of information acquisition.30 As Bj itself depends
partly on Kj , the extent of losses from mispricing also changes with the number of firms. Second, fixing
Bj , a larger αj amplifies firms’ incentives to attend to their competitors’ mistakes as discussed in the static
29

Wang and Werning (2022)’s sufficient statistics are in terms of elasticities and super-elasticities of demand. I have derived
my approximation in terms of demand elasticity and strategic complementarity, which can be written as a function of the othertwo, as derived in Online Appendix F.2.
30
There is evidence that supports this level effect. Coibion et al. (2018) document that firms with a higher slope in their profit
function around their optimal price have more accurate expectations about inflation.

24

model. Since αj also varies with Kj , the number of firms also varies the strength of strategic incentives
through this channel. Therefore, how strategic inattention varies with Kj depends on the relative importanceof these two channels, which I will discuss further in Section 5.
Finally, the persistence of qt over time introduces a dynamic force as firms rely on their past signals toinfer the current value of qt. In the oligopoly, this leads to endogenously persistent mistakes, as firms’ past
mistakes feed into their current prices and motivate their competitors to pay attention to the time series of
their mistakes. Thus, dynamic incentives have two potential effects on information acquisition. First, theyaffect the level of capacity production as firms internalize the continuation value of information. Second,
they affect how firms allocate their capacity between the fundamental and others’ mistakes. If mistakes areendogenously less persistent than fundamental qt, then more patient firms will allocate a higher portion of
their attention to qt, since the continuation value of doing so would be larger.31
Solving for a Symmetric Stationary Equilibrium. Here, I briefly discuss the outline of the algorithm
for solving the model. A detailed explanation is included in Online Appendix J, which contains the following four subsections. Online Appendix J.1 extends the notion of a pure strategy Gaussian equilibrium in
Definition 1 to the dynamic model. It also outlines the conditions that should hold in a symmetric stationary
equilibrium, where we require pricing strategies of firms to be stationary over time and symmetric within
sectors with the same number of competitors. Online Appendix J.2 then shows that characterizing such anequilibrium is equivalent to finding a fixed point for the coefficients of lag polynomials that map monetaryand mistake shocks to firms’ equilibrium prices. Online Appendix J.3 then outlines the main algorithm
that I use to solve for this fixed point based on integrated moving average approximations of equilibrium
prices. Finally, Online Appendix J.4 outlines an alternative algorithm that uses an ARMA approximation
as in Maćkowiak et al. (2018) and shows that the two algorithms yield numerically identical solutions.
To briefly outline the solution method, the model’s solution is a joint Gaussian stochastic process for
all firms’ prices and the nominal demand that satisfies the equilibrium conditions. Given a guess for thejoint process of prices and nominal demand, I derive the implied strategy for a firm’s competitors in a
31

See, e.g., Afrouzi and Yang (2019); Maćkowiak et al. (2018); Miao, Wu, and Young (2022); Steiner et al. (2017) for an
extensive discussion of dynamic incentives of a rationally inattentive agent.

25

symmetric stationary equilibrium,32 which then implies a stochastic process for the “ideal prices” of firmsin Equation (27), whose processes are inputs to firms’ rational inattention problems.
I then approximate the processes for these ideal prices with an integrated moving average projection on
monetary shocks (where the integrated part is included to account for the unit root in nominal demand) toderive a Markov state space representation. Moreover, strategic inattention implies that the process for afirm’s ideal price depends also on the non-fundamental shocks (mistakes) to their competitors’ prices. With
dynamics, these mistakes are persistent, and their auto-covariance structure is endogenous to the equilibrium.
Incorporating these requires extending the conventional solution methods for monopolistic competition
rational inattention models to allow firms to pay attention to endogenous non-fundamental shocks. To dothis, I augment the state space of a firm’s ideal price with the moving average (MA) representation of thefirm’s competitors’ mistakes and solve for the endogenous distribution of these mistakes over time as partof the fixed point problem described above.
With this approximated Markov state space representation of ideal prices at hand, I then use the methodin Afrouzi and Yang (2019) to solve the firms’ rational inattention problems, which is fast enough to makethe solution of the model with K ∈Supp(K) (a total of 43 values) and several iterations of ω for calibration
feasible. Given this solution, I then solve for the stochastic processes of the firms’ beliefs and prices. Doing
this for all K in the support of K, I then derive the new guess for the joint stochastic process of firms’ prices
and iterate until convergence to the fixed point.33
32

Here symmetry requires that all firms in sectors with K competitors have the same strategies for information acquisitionand pricing decisions. Moreover, a stationary strategy is one where a firm’s beliefs and prices depend on time only through thehistory of its signals. A stationary equilibrium is then a pair of initial information sets under which all firms’ best responses
are stationary strategies. Similar to Maćkowiak and Wiederholt (2009), one could interpret such information sets as ones where,
after solving their inattention problem under the equilibrium strategy of others, all firms receive an infinitely long sequence of
signals such that their own best responses are to use stationary strategies. Focusing on stationary equilibria allows us to avoid
dealing with time-varying impulse response functions or transition dynamics of second-order moments of beliefs. See Online
Appendix J.1 for a precise definition and discussion of a symmetric stationary equilibrium.
33
In contrast to the static model, which had multiple equilibria with both zero and positive capacity, the dynamic model cannot
have any equilibria with zero capacity. This is because the process for qt has a unit root, meaning that given a strategy withzero capacity, the variance of any given firm’s ideal price grows unboundedly as long as α < 1 (see the proof of Proposition 6
for a more formal argument in that case). This implies that at some point, this variance would be large enough for the firm
to deviate from this strategy, regardless of what its competitors do. As a result, firms must choose a positive capacity in thestationary equilibrium of the game, which is similar to the unique equilibrium in the static model when ω/B is small relative
to the unconditional variance of q (normalized to 1).

26

A Special Case with a Closed-Form Phillips Curve. In general, the equilibrium signal structure of firms
does not admit a closed-form representation. However, we can characterize optimal signals in closed-form
when firms are myopic in information acquisition (β =0) which is useful for intuition.
Proposition 5. Given a strategy profile for all other firms in the economy, every firm prefers to see only
one signal at any given time. Moreover, if β =0, the optimal signal of firm j,k at time t istSj,k,t =(1−αj )qt +αj pj,−k,t(Sj,−k)+ej,k,tThis expression for optimal signals illustrates the main departure of this paper from models that assume
a measure of firms. Since firms are granular in an oligopoly, mistakes propagate through the inclusion ofpj,−k,t in firm j,k’s signal and result in the excess correlation of prices beyond what is implied by shocks
to qt, as discussed in the static model. We can also derive a closed-form expression for the Phillips curvewhen there is no heterogeneity in the number of competitors across sectors (these assumptions are madefor illustrative purposes and I revert to the general case in the calibrated model).
Proposition 6. Suppose β = 0 and Kj = K,∀j ∈ J for some K ∈ N. Then, αj = α,∀j ∈ J and in thestationary equilibrium κj,k,t =κ>0,∀j ∈J,k ∈K. Moreover, the Phillips curve of this economy isj,k2κ
πt =(1−α)Ej,kt−1 [∆qt ]+αEt−1 [πj,−k,t ]+(1−α)(e −1)yt ,j,kwhere Ej,kt−1 [∆qt ] is the average expected growth of nominal demand at t−1, Et−1 [πj,−k,t ] is the average

expectation across firms of their competitors’ price changes, and yt is the output gap.
This Phillips curve indicates that in economies with large strategic complementarities, the main driver of
inflation is firms’ expectations of their competitors’ prices. As Proposition 5 shows, a larger α means firms
learn more about their competitors’ prices relative to aggregate demand. Therefore, when α is large, notonly are firms’ expectations of their competitors’ prices the main driver of inflation but these expectations
are also formed under information sets that are more informative of those prices.
Additionally, the slope of the Phillips curve shows how these strategic complementarities and the capacity
for processing information interact in affecting monetary non-neutrality in this economy. The higher capacity27

of processing information makes the Phillips curve steeper, such that in the limit when κ→∞ (which arises
endogenously when ω → 0), the Phillips curve is vertical. In contrast, higher strategic complementarity
makes the Phillips curve flatter since firms’ higher-order beliefs become more important in their pricing
decisions (Woodford, 2003a). Thus, to understand how the number of competitors, K, affects the slope of
the Phillips curve, we need to investigate how α and κ jointly change with K, which I will come back toin detail in Section 5.3.

4.4 Calibration
The model is calibrated to the firm-level survey data from New Zealand at a quarterly frequency, with a
discount factor β =0.961/4. A calibration to US data might be desirable, but a main objective of quantifying
the model is to examine if it fits the relationship between competition and firms’ expectations about aggregate
inflation, the evidence for which comes from the New Zealand survey data.34 The key and new parameters
are the distribution of competitors, K, and the cost of attention ω. Other parameters are externally calibrated,
as presented in Table 3 and discussed in more detail in Online Appendix I.1. In particular, based on
Equation (24), I choose γ to match the degree of strategic complementarity measured from the survey data in
Table A.1 in Online Appendix A. Moreover, for the distribution of Kj , denoted by K, I choose it to match theempirical distribution of the number of competitors in the survey data (Figure A.1 in Online Appendix A).35
To calibrate ω, I target the weight that firms put on their priors in their inflation forecasts, as in Wiederholt(2015). This approach identifies ω because, with larger ω, firms’ signals in the model are less accurate,
leading firms to rely more on their priors in their forecasts. The fourth wave of the New Zealand survey asks
firms about their yearly inflation forecasts and their inflation nowcasts for the previous year in waves one andfour. These waves were conducted 12 months apart (2013:Q4 to 2014:Q4), allowing for the comparison of
ex-ante and ex-post beliefs for the subset of firms present in both waves. Using this data, I run the following
34
In addition, to calibrate the model to the US data, one needs microdata on firms’ expectations about inflation to calibrate
the cost of attention in the US as well as data on how many competitors firm directly face to calibrate the distribution of thenumber of competitors, none of which are available for the US to the best of my knowledge
35
As far as I know, there is no data available on how many competitors firms directly face in their market for the US. It is
important to note that the value of K in this model corresponds to direct competitors of a firm that are only a small subset of
all the firms that operate in a single SIC classification. Market segmentation, such as spatial constraints for consumers, makethe number of firms within a SIC classification not suitable for calibrating this model.

28regression for this calibration:

Ei,t[πt]=constant+δEi,t−4[πt]+error(29)

where δ is the coefficient of interest.
Column (1) of Table A.2 in Online Appendix A reports the baseline estimates for this specification,
while Column (2) controls for firms’ different beliefs about long-run inflation rates (Patton & Timmermann,
2010).36 I calibrate by targeting the coefficient in Column (2) using the same regression on simulated data,
resulting in ω = 0.037. Figure A.3 in Online Appendix A shows that ω is identified as the regression
coefficient δ increases with ω within the model.
To see how this value compares to the estimates of information rigidity in the literature, we can compare
the implied Kalman gain of firms in the model with the documented values in the literature for professional
forecasters. The average firm in this model has a Kalman gain of 0.49 as seen in Figure A.4 in Online
Appendix A, higher than the estimated value of 0.45 for Professional Forecasters in the US (Coibion & Gorodnichenko, 2015). This suggests that firms in the model are more informed about their optimal prices than professional forecasters are about aggregate inflation, but they exhibit large degrees of information rigidity in inflation forecasts because their optimal signals are less informative of inflation than it is of their optimal prices.

4.5 Examining Non-Targeted Moments: Subjective Uncertainty in the ModelCan the calibrated model replicate the strategic inattention of firms observed in the data? Table 1 shows thatfirms’ uncertainty about aggregate inflation decreases with the number of their competitors. This relationshipis not consistent with benchmark models without rational inattention and oligopolistic competition but
emerges endogenously in this model with strategic inattention incentives.
Figure 1 shows this relationship both in the model (the solid line) and the data (binned scatter plot).37
The model accurately reproduces the decrease in subjective uncertainty with the number of competitors.
36

The exact question about long-run inflation is “What annual percentage rate of change in overall prices do you think theReserve Bank of New Zealand is trying to achieve?” Matching the coefficient that controls for this response is the model
consistent approach because in the model all firms have the same long-run inflation forecast. Also, this is the more conservative
calibration of ω as matching the coefficient in Column (1) would imply a larger value for the cost of attention.
37
I have normalized average uncertainty both in the data and in the model to 1.

29

Both the heterogeneity in the number of competitors and endogenous information acquisition are key forthis relationship: the former creates the differential incentives for information acquisition, and the latter isessential for the endogenous variation in information acquisition. Figure A.4 in Online Appendix A showsthe equilibrium level of firms’ information acquisition and their implied Kalman gains as a function of
the number of firms’ competitors. More competitive firms (1) produce a higher capacity for processing
information and (2) allocate more capacity towards aggregate shocks. As a result, more competitive firms
have more accurate posteriors about aggregate variables.

5 Macroeconomic Implications
In this section, I investigate the aggregate and reallocative implications of strategic inattention for the
propagation of monetary shocks to inflation and output. To do so, I consider three measures. To measuremonetary non-neutrality, following Nakamura and Steinsson (2010), I use the variance of output (normalized
by its natural level).38 To measure the persistent effects of monetary shocks, I use the cumulative half-lifeof output and inflation responses (time until the area under the impulse response reaches half of its full
cumulative response). Finally, to compare reallocative effects of policy across sectors, I use the cumulative
response of output (see, e.g., Alvarez, Le Bihan, & Lippi, 2016) defined as the area under the output IRF
of sectors with different numbers of competitors.

5.1 Aggregate Effects: Monopolistic vs. Oligopolistic Competition
I start by comparing the calibrated model to a monopolistic competition model, nested when Kj →∞.
To define the proper monopolistic competition benchmark, it is important to ensure it has the same levelof strategic complementarity as the calibrated model so that the only difference between the two models
is firms’ strategic inattention; i.e., the only difference in impulse responses comes from the different signals
firms choose in the two models.39 This is because we know from previous research that higher strategic38

Up to a second-order approximation to the household’s utility, the variance of output normalized by its natural level is
proportional to her welfare loss in consumption equivalent units (Lucas, 2003): E log(Yt/Ȳ ) ≈− 12 var YȲt .
39
Firms in the oligopolistic model pay direct attention to the mistakes of their competitors, but firms in the monopolisticcompetition model, similar to Woodford (2003a), only pay attention to the fundamental shocks. There is, however, adifference between the shape of signals in the monopolistic competition here with those in Woodford (2003a), which assumes

30complementarities amplify monetary non-neutrality (see, e.g., Ball & Romer, 1990)—Section 5.3 discusses
this in more detail. To generate the same strategic complementarity in the monopolistic competition model,
I replace the within-sector CES aggregator of the oligopolistic model with a Kimball aggregator, which
introduces a new parameter that allows me to calibrate the two models to the same strategic complementarity
while keeping other parameters the same (see Online Appendix I.2).
The first two rows in Columns (1) and (2) of Table 4 report the absolute and relative variance of output
across the two models, respectively.40 Output is 28% more volatile in the benchmark model, indicating thatfirms in the monopolistic competition model are more informed about aggregates due to the lack of strategic
inattention motives. Column (4) shows output response is also 9% more persistent in the benchmark model:
as reported in Column (3), it takes 3.72 quarters for output to reach its half-life in the benchmark model as
opposed to 3.40 quarters in the monopolistic competition model.
The first two rows of Table 5 compare the behavior of inflation across these models. Inflation is smaller
and more persistent in the model with strategic inattention. Columns (1) and (2) show that inflation is 6%
less volatile compared to the model with monopolistic competition. Column (3) shows that it takes inflation
4.42 quarters to reach its cumulative half-life in the monopolistic competition model, compared to 4.66
quarters in the benchmark model, a 5% increase as reported in Column (4).
Figure A.5 in Online Appendix A also presents the impulse response functions of output and inflationin the two models along with those of a duopoly model, showing how monetary non-neutrality is amplified
and inflation response is dampened with strategic inattention.

5.2 Reallocative Effects and Concentration Multipliers
I continue my analysis by investigating the differences in inflation and output responses across sectors withdifferent numbers of competitors. To do so, I conduct two analyses. First, I compare the output volatilityof sectors with different numbers of competitors to the same monopolistic competition model as before.
Second, I compare the output response of different sectors to the response of aggregate output in the same
Sj,k,t = qt +noise. Here, signals under monopolistic competition are linear functions of innovations to qt (plus noise), but theexact linear combination is an endogenous object that is solved for as a fixed point.
40
Magnitudes in Column (1) are small since the variance of innovations to nominal GDP growth is small. The same is true
for the US (Nakamura & Steinsson, 2010).

31

model, focusing on the relative differences within the same economy.

Output Volatility Conditional on Number of Competitors. How do output and inflation responses differ
across sectors for different values of K? Table 4 reports output volatility and amplification factors relative
to the model with monopolistic competition. Monetary non-neutrality is larger, and output response is more
persistent in sectors with fewer competitors. For instance, in the duopoly model, output volatility is 48%
larger and the cumulative half-life of output is 22% longer. Table 5 reports the equivalent results for inflation.
Inflation response is more muted and its half-life is longer in sectors with fewer competitors. In the duopoly
case, for instance, the variance of inflation is 13% smaller than the model with monopolistic competition,and its cumulative half-life is 9% longer.

Concentration Multipliers. As prices are less responsive to aggregate shocks in sectors with fewer
competitors, monetary shocks have also reallocative effects across sectors. A natural exercise to measure themagnitude of these distortions is to calculate what share of the total output response is driven by the firms
with fewer competitors. Formally, let Yk denote the average cumulative impulse response of log output to a
one standard deviation monetary policy shock in sectors with k competitors, and let Y denote the cumulative
impulse response of aggregate output:

 P
P∞
∂
Yk ≡Ej ∂u∂ 0 ∞t=0 log(Yj,t ) Kj =k , Y ≡ ∂u0
t=0 log(Yt )(30)

It is then straightforward to derive the relationship between these aggregate and sectoral responses asPY≡ ∞k=2 sk Yk , where sk is the steady-state market share of sectors with k competitors. We can now define
the concentration multiplier of sectors with k competitors as the ratio Mk ≡ YYk . These concentration multipliers capture reallocative effects because they would be equal to one for all k if there was no heterogeneity
in output response. However, with heterogeneity, it measures the share of the cumulative response of output
in sectors with k competitors relative to the aggregate output response.
Figure 2 plots these multipliers for different numbers of competitors and shows that less competitive
sectors respond more strongly to monetary shocks in their output. For instance, duopolies have a 17% larger
output response to monetary policy shocks relative to the aggregate output response.
32Thus, expansionary monetary policy concentrates production among less competitive firms, increasing
the impact of such firms on the economy.41 It is important to note that more competitive firms contribute lessto output response despite having higher strategic complementarities. Conventional models with exogenous
information rigidity, such as Woodford (2003b), show that higher strategic complementarities lead to higher
monetary non-neutrality. The results here show that endogenous information acquisition reverses this resultin a calibrated model through strategic inattention. The following section explains and decomposes the rolesof each of these forces in the model.

5.3 Inspecting the Mechanism: Strategic Inattention vs. Strategic Complementarities
The number of competitors affects both the degree of strategic complementarity and the amount of capacity
produced by firms. Thus, the degree of monetary non-neutrality across sectors with different K is the sumof two separate forces: (1) The well-known real rigidity channel that alters monetary non-neutrality throughthe degree of strategic complementarity, and (2) The new strategic inattention channel that alters monetary
non-neutrality through information acquisition and utilization.
In the calibrated model, these two forces work in opposite directions. As discussed in Section 4.5,
firms with more competitors allocate a higher amount of attention to aggregates and their prices move more
swiftly in response to monetary shocks, which dampens their output response as a result. Hence, monetary
non-neutrality decreases with competition through the strategic inattention channel.
On the other hand, the degree of strategic complementarity in Equation (24) increases with the numberof competitors in the calibrated model, which is depicted in Figure A.6 in Online Appendix A. Therefore, by fixing the capacity of processing information, a larger number of competitors increases monetary
non-neutrality through higher strategic complementarities. Specifically, higher strategic complementarity
41

More competitive firms have more flexible prices so in response to expansionary (contractionary) monetary shocks, they
adjust their prices faster and their output falls (increases) relative to less competitive firms. Thus, in response to contractionary
shocks prices (markups) of more competitive firms fall relative to those of less competitive firms, which, in relative terms,
reallocates labor towards more competitive firms (who are the firms with lower steady-state markups in the model). This isconsistent with the evidence presented in Baqaee, Farhi, and Sangani (forthcoming). Using Compustat data, they find that “acontractionary shock leads high-markup firms to increase their markups relative to low-markup firms; the result ... is a reallocationof resources away from high-markup firms and toward low-markup firms,” (Baqaee et al., forthcoming, p. 41).

33

amplifies non-neutrality by putting a larger weight on firms’ higher-order beliefs, whose responses to shocks
are more rigid (See, e.g. Maćkowiak et al., 2018; Nimark, 2008; Woodford, 2003a). To verify this mechanism
within the model, Figure A.7 in Online Appendix A shows the IRFs of firms’ higher-order beliefs to a 1%
increase in nominal demand for three different values of K. With larger K, the responses of higher-orderbeliefs are smaller and more persistent, indicating that monetary non-neutrality increases with the numberof competitors through the real rigidity channel.
To better understand the separate roles of these two channels, below I present three complementary
analyses. First, I decompose the effects of these channels on the variance of output and inflation and show
that, while both channels are significant, the strategic inattention channel dominates. Second, given the
micro-foundations of this section, I revisit the static model where I can derive analytical expressions forthese channels, which provide further insight into their relative importance. Third, I redo the quantitative
analysis of the model under an alternative specification where strategic complementarities decrease with K
and find that while the strategic inattention channel is mitigated in this case, it continues to amplify monetary
non-neutrality with lower K.

Quantitative Decomposition in the Calibrated Model. To decompose the effects of these two opposing
forces in the calibrated model, let us define α(K) to be the degree of strategic complementarity in a model
where all sectors have K competitors, and all the other parameters are fixed at their calibrated values.
Moreover, let σy2(α(K),K) denote the output variance in the model where every sector has K competitors.
The first argument captures the effect of the number of competitors on the weight that higher-order beliefs
receive in the model (the real rigidity channel), and the second argument captures the effect of the numberof competitors on the attention allocation of firms (strategic inattention channel). Then, we can decompose
the difference in monetary non-neutrality of the two extreme models (K =2 versus K →∞) as
 2

 2

 2

σ (α(2),2)σ (α(2),2)σ (α(2),K)
limK→∞log σ2y(α(K),K) =limK→∞log σ2y(α(2),K) +limK→∞log σ2y(α(K),K)
y
y
y
|
{z
} |
{z
} |
{z
}
total
percentage change

percentage change due tostrategic inattention(31)

percentage change due toreal rigidities

Column (1) of Table 6 shows the results of this decomposition. Output variance is 18.6% larger with
K =2 relative to K →∞ (percentage difference here is calculated as the log-difference from Table 4). Once34

decomposed to its two contributing factors, decreasing the number of competitors from K →∞ to K =2
increases monetary non-neutrality by 78.5 percentage points due to the strategic inattention channel anddecreases it by 60.0 percentage points through the real rigidity channel. As for inflation, Column (2) ofTable 6 shows that decreasing the number of competitors from K → ∞ to K = 2 decreases the varianceof inflation by 19.8 percentage points through the strategic inattention channel and increases it by 10.1
percentage points through the real rigidity channel.

Analytical Decomposition in the Static Model. To further examine the relative importance of these twochannels, here, I revisit monetary non-neutrality in the static model of Section 2.4 using the micro-foundations
derived in this section. The detailed derivations can be found in Online Appendix K.
To begin, let us denote the average price of oligopolies with K competitors as pK , and their average
output as the difference between nominal demand and their average price, yK = q−pK . Then, it follows
from Equation (5) that the response of output to a monetary shock is given by ∂yK /∂q =1−δK . Moreover,the variance of output is also related to this object as Var(yK )=(1−δK )2Var(q). Thus, differentiating this
response with respect to K, we can formalize the role of these two channels on how monetary non-neutralitychanges with the number of competitors:∂K (∂yK /∂q)=∂K (1−δK )=(1−λK )λK
1−αK
∂K αK −∂K λK
2(1−αK λK )(1−αK λK )2
|
{z
} |
{z
}
Channel A: Real Rigidity(32)

Channel B: Strategic Inattention

Where we have indexed αK , λK with K to note that both of these objects vary with K. The first term in theexpression above captures the real rigidity channel: holding information processing capacity fixed, higher
strategic complementarity increases monetary non-neutrality. The second term captures the strategic inattention channel: holding strategic complementarity fixed, higher information processing capacity decreases
monetary non-neutrality. Since λK is itself an endogenous object, the question of how the two channels
interact condenses to how λK varies with K. To answer this question, recall from Equation (10) that in an
equilibrium with positive capacity, λK = 1− BKωV ∗ where VK∗ is the variance of firms’ desired prices andK

BK is the curvature of a firm’s profit function in an oligopoly with K competitors, respectively. Moreover,as shown in Equation (28) and derived for a generally specified profit function, the curvature BK is itself a
35

εK

D
function of firms’ demand elasticities εK
D and strategic complementarity αK : BK = 1−αK . Thus, the second

term in Equation (32) can be further decomposed as:∂K ln(εK )| {z D}∂K λK =(1−λK )(+change in BK through elasticity

1∂ α1−αK K K|{z}change in BK through αK+ ∂K ln(VK∗ ) )| {z }(33)

change in varianceTaken together, Equations (32) and (33) show that changes in αK have two effects in monetary non-neutrality.
First, directly, it increases monetary non-neutrality through the real rigidity channel. Second, indirectly,
it decreases monetary non-neutrality by increasing the curvature of firms’ profit functions, which in turnincreases the information processing capacity of firms.
To further simplify the expressions above, let us consider the first-order Taylor expansion of the equilibrium of the static model in Section 2.4 around the full-information benchmark (ω =0), as derived in Online
Appendix C.8. Re-writing Equation (32) with this approximation, we obtain:
elasticity on B

∂K (∂yK /∂q)=

ω
∂K αK
εK
D (1−αK )

|

{z

}

pass-through on BKz }| {K
z
}|
{
ω 2
ω
ω
∥)
− εK (1−αK ) ∂K αK − εK ∂K ln(εKD ) +O(∥
D
D
B
K
|
{z
}

Channel A (first-order effects of ω)(34)

Channel B (first-order effects of ω)

A key observation is that up to this first-order approximation, the direct and indirect effects of how strategic
complementarity changes with K (∂K αK ) fully offset each other. In other words, while a higher αK
increases monetary non-neutrality through the real rigidity channel, this effect is offset up to the first order ofω through the higher information acquisition of firms as αK increases the curvature of their profit functions.
Thus, the only relevant first-order factor is how εK
D changes with K:≥0z }| {
ωω 2∂K (∂yK /∂q)=− εK ∂K ln(εKD )+O(∥ BK ∥ )

(35)

D

Since more competitive firms have higher demand elasticities and lower markups, as shown in Equation (22),the total first-order effect is negative and monetary non-neutrality decreases with K.42
Finally, it is useful to note that while the sign and magnitude of ∂K αK do not matter for how K affects
42

For recent empirical evidence on how demand elasticities decrease with market share (1/K in the model), see, e.g., Burstein,
Carvalho, and Grassi (2020); Burya and Mishra (2023).

36monetary non-neutrality up to first-order, they do matter for the contribution of the strategic inattention
channel. In particular, a negative ∂K αK decreases the contribution of the strategic inattention channel to thedecline of monetary non-neutrality with K by reducing the curvature of firms’ profit functions to K anddampening the sensitivity of firms’ information acquisition to K. My next exercise is to illustrate this bysolving the dynamic model when αK decreases with K.
Alternative Specification of Strategic Complementarities. While the analytical results from the static
model give us insight into how the real rigidity and strategic inattention channels interact and change withthe sign of ∂K αK , they do not provide a quantitative assessment of the relative importance of these channels
when strategic complementarities decrease with K; especially since we only considered the first-order effectsof ω/BK . It is possible that the interactions may be more complex in the dynamic model or that higher ordereffects of ω/BK may be important. To address this concern, I solve the dynamic model when αK decreases
with K with Atkeson and Burstein (2008) preferences.
This exercise is described in detail in Online Appendix L. In summary, its results confirm the intuition
developed from the analytical decomposition of the two channels in the static model. First, Table L.2 in
Online Appendix L shows that despite the negative sign of ∂K αK , monetary non-neutrality still decreases
with K, consistent with Equation (34) and the increasing demand elasticities with K in this model. Second,as expected from Equation (33), a negative ∂K αK reduces the curvature of firms’ profit functions as K
increases, dampening the strength of the strategic inattention channel. However, this effect is not strong
enough to fully counteract the effect of changes in demand elasticity on firms’ capacity production. As
shown in Figure L.1 in Online Appendix L, capacity still increases with K, albeit with a small slope. Finally,
Table L.3 in Online Appendix L presents the decomposition of Equation (31) in this case and shows that bothchannels work in the same direction to decrease the degree of monetary non-neutrality with a negative ∂K αK .

5.4 Additional Robustness Exercises
Before concluding, I briefly mention four additional robustness exercises that are explored in more detail in
the appendices. I solved the model by approximating firms’ problems in Equation (26) around a symmetricequilibrium and my solution method relies on the symmetries implied by this approach. Online Appendix
37

M.1 discusses and speculates on the role of asymmetric market shares. In Online Appendix M.2, I examine
whether the persistence of the growth rate of nominal demand affects the results by resolving the modelfor ρ = 0.23 and find similar results to the benchmark calibration. Online Appendix M.3 investigates theinteraction of dynamic and strategic incentives in information acquisition by calibrating the model to a lower
discount factor and finding that strategic motives become stronger when firms are more myopic. Finally,
Online Appendix M.4 discusses how sector or firm-level idiosyncratic shocks may impact the results of themodel and solves a numerical example with sector-level shocks.

6 Concluding Remarks
This paper develops a new model to study how imperfect competition affects firms’ information acquisitionand expectations. The interaction of these two frictions creates an endogenous correlation between theaccuracy of firms’ beliefs and the number of their competitors. Oligopolistic firms find it optimal to acquire
more information and pay direct attention to the beliefs of their competitors, an incentive that is stronger
when they have fewer competitors or higher strategic complementarities in pricing.
The model’s implications for monetary non-neutrality and inflation dynamics speak to recently documented trends in rising concentration and market power. These results suggest that with more concentration,monetary policy is more potent and its real effects are stronger. Furthermore, the reallocative effects ofstrategic inattention imply that this change in potency is not uniform across all firms. These heterogeneous
effects introduce new distortions to relative prices that might lead to new sources of misallocation and, more
broadly, to efficiency loss, which should be of interest for future research.
Moreover, in tracking their competitors’ beliefs, firms ignore aggregate shocks, and, as a result, theirbeliefs about aggregate variables are more inaccurate and noisy than the beliefs that feed into their prices.
Thus, firms’ expectations about aggregate variables are no longer the appropriate measures for their decisions
with oligopolies. These results are informative for surveys that aim to connect firms’ expectations to theirdecisions: under oligopolistic competition, there’s a wedge between firms’ relevant expectations for theirprices and their aggregate inflation expectations. These expectations are relatively unimportant for firms
and do not have much impact on their pricing decisions.
38

Furthermore, the results in this paper have implications for policies that target expectations. In particular,
they provide a new perspective on why managing inflation expectations might be less effective than whata model with monopolistic competition would suggest. Oligopolistic firms do not directly care about
aggregate inflation and are mainly concerned with how their competitors’ prices respond to shocks. Thus,
any communication about aggregate variables will be discounted accordingly.
Nevertheless, this result does not necessarily rule out policies that target expectations but rather provides
a new view on how those policies should be framed and which expectations they should target. An importanttakeaway from this paper is that for such a policy to be successful, it has to communicate the course ofmonetary policy to price-setters not in terms of how it will steer the overall prices but in terms of how it willaffect their own industry prices. In other words, framing policy in terms of the aggregate variables will not
gain as much attention and response from firms as it would if the news about the policy were to reach firms
in terms of how their competitors would be affected. How policy can achieve these ends remains a questionthat deserves more investigation.

7 Data Availability
Code replicating the tables and figures in this article can be found in Afrouzi (2024) in the Harvard Dataverse,
https://doi.org/10.7910/DVN/AO6C85.

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Table 1: Subjective Uncertainty of Firms and the Number of Competitors.
(1)
(2)
π
log(σ ) log(σπ )log(K)-0.116 -0.115(0.012) (0.013)
Observations 1661
1661
Notes: Column (1) of the table reports the result of regressing the log standard deviation of firms’ reported distribution for theirforecast of aggregate inflation on the log of their number of competitors. Column (2) reports the same coefficient while controllingfor firm age, firm size measured by employment in the main product line, and fixed effects for construction, manufacturing,professional and financial services, and trade industries. Robust standard errors are reported in parentheses.

43

Table 2: Size of Firms’ Nowcast Errors
Observations
Industry
Construction
Manufacturing
Financial Services
Trade
Total

(1)
57
415
477
307
1256

Industry inflation
mean
std
(2)
(3)
0.62
0.51
1.46
1.92
1.33
1.45
0.59
0.91
1.16
1.54

Aggregate inflation
mean
std
(4)
(5)
4.55
2.75
2.73
2.29
4.73
2.31
2.44
2.13
3.50
2.51

Notes: The table reports summary statistics for the size of firms’ nowcast errors in perceiving aggregate inflation versus industry
inflation for the 12 months ending in December 2014 (from wave 4 of the survey). Industry (aggregate) inflation nowcast errors
are defined as the absolute difference between firms’ nowcasts and the actual industry (aggregate) inflation rate in that year.

44Table 3: Calibration Summary
ParameterKωη
1/(1+γ)ρ
σuDescription
Distribution of K
Cost of attention
Elasticity of substitution
Curvature of production
Persistence of ∆q
Std. Dev. of shock to ∆q

Value
∼ K̂
0.037
12
0.514
0.707
0.011

Moment Matched
Empirical distribution (Fig. A.1)
Weight on prior in inflation forecasts
Elasticity of markups to 1/(1−Kj−1)
Average strategic complementarity
Persistence of NGDP growth in NZ
Std. Dev. of NGDP growth in NZ

Notes: The table reports the calibrated values of the parameters for the dynamic model.

45

Table 4: Output and Monetary Non-Neutrality Across Models
Model
Monopolistic Competition
BenchmarkK ∼ K̂
2-Competitors K =2
4-Competitors K =4
8-Competitors K =8
16-Competitors K =16
32-Competitors K =32∞-Competitors K →∞Variancevar(Y)
amp. factor
(1)
(2)
3.17
1.00
4.07
1.28
4.69
1.48
4.14
1.30
3.99
1.26
3.94
1.24
3.91
1.23
3.89
1.23
×104

Persistencehalf-life qtrs amp. factor
(3)
(4)
3.40
1.00
3.72
1.09
4.14
1.22
3.78
1.11
3.65
1.07
3.60
1.06
3.57
1.05
3.55
1.04

Notes: The table presents statistics for monetary non-neutrality across models with different numbers of competitors at the
micro-level. var(Y ) denotes the variance of output conditional on monetary shocks multiplied by 104. Half-life denotes thelength of the time that it takes for output to live half of its cumulative response in quarters. Amp. factor denotes the factor bywhich the relevant statistic is larger in the corresponding model relative to the model with monopolistic competition.

46

Table 5: Inflation Across Models
Model
Monopolistic Competition
BenchmarkK ∼ K̂
2-Competitors K =2
4-Competitors K =4
8-Competitors K =8
16-Competitors K =16
32-Competitors K =32∞-Competitors K →∞Variancevar(π)
damp. factor
(1)
(2)
1.47
1.00
1.37
0.94
1.28
0.87
1.36
0.93
1.39
0.95
1.40
0.95
1.41
0.96
1.41
0.96
×104

Persistencehalf-life qtrs amp. factor
(3)
(4)
4.42
1.00
4.66
1.05
4.83
1.09
4.68
1.06
4.64
1.05
4.62
1.05
4.62
1.05
4.61
1.04

Notes: The table presents statistics for inflation response across models with different numbers of competitors at the micro-level.
var(π) denotes the variance of inflation conditional on monetary shocks multiplied by 104. Half-life denotes the length of thetime that it takes for inflation to live half of its cumulative response in quarters. Damp. factor (amp. factor) denotes the factor
by which the relevant statistic is smaller (larger) in the corresponding model relative to the model with monopolistic competition.

47

Table 6: Decomposition: Strategic Inattention vs. Real Rigidities
Percentage change
in variance of
outputinflation(1)(2)
Total Change (percent)
18.6
-9.7
Due to Str. Inattention (ppt) 78.5
-19.8
Due to Real Rigidities (ppt) -60.0
10.1
Notes: The table shows the decomposition of the effects of the strategic inattention and real rigidity channels for the change
in volatility of output (monetary non-neutrality) and inflation conditional on monetary shocks, as derived in Equation (31).

48

0.15
0.1
0.05
0
-0.05
-0.1
2

4

8

16

32

Figure 1: Subjective uncertainty about inflation: Model vs. Data.
Notes: The figure presents the fit of the model for the relationship between firms’ (log) subjective uncertainty about aggregate
inflation and the number of their competitors. The dots show the binned scatter plot of log-standard deviation of firms’ subjective
beliefs of the 12-month ahead forecast of aggregate inflation against the number of competitors in data (Table 1). The blackline depicts this relationship in the calibrated model. Subjective uncertainty in the model is calculated as the standard deviationof firms’ beliefs about the full information rational expectations 12-month ahead forecast of inflation. The average subjective
uncertainty is normalized to one in both the data and the model. This relationship was not targeted in the calibration of the model.

49

1.2
1.15
1.1
1.05
1
0.95
0.9
2

4

8

16

32

Figure 2: Concentration Multipliers
Notes: The figure shows the concentration multiplier as a function of the number of competitors. Concentration multiplier of k isdefined as the cumulative response of output coming from sectors with k competitors relative to the aggregate cumulative responseof output. Less competitive sectors are responsible for a higher share of output response relative to their steady-state market share.

50