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Rational Inattention, Misallocation, and Asset Prices

Naveen Gondhi †Kellogg School of Management

November 2015Download the latest version by clicking here

Abstract

I study the implications of rational inattention of rm managers for asset prices and macroe-conomic quantities. Firms face aggregate and idiosyncratic productivity shocks, the un-certainty of which varies over time. My model delivers endogenous movements in outputand measured aggregate productivity in response to exogenous changes in uncertainty. Anincrease in aggregate uncertainty leads managers to allocate less capacity to learn abouttheir idiosyncratic productivity, leading to higher misallocation of resources across rmsand lower output. An increase in idiosyncratic uncertainty has the opposite effect andresults in an economic expansion. This rationalizes the empirical nding that risk price of aggregate uncertainty is negative, whereas risk price of idiosyncratic uncertainty is positive.My model delivers novel testable predictions regarding the degree of resource misallocation,the relation between output and both types of uncertainty, the comovement of production

inputs and market betas in the cross section of rms. I conrm these predictions in thedata.

JEL Classication: G12, G14, D83, E23

Keywords: Uncertainty, Rational inattention, Misallocation, Excess comovement, Conditionalmarket betas, Efficiency.

This paper is part of my Ph.D. dissertation developed at Northwestern university and I am deeply indebted tomy advisors, Dimitris Papanikolaou, Snehal Banerjee, Sergio Rebelo and Ian Dew-Becker for all their helpful guidanceand support. I would also like to thank Nicolas Crouzet, Jesse Davis, Sebastian Di Tella, Michael Fishman, KathleenHagerty, Benjamin Iverson, Ravi Jagannathan, Guido Lorenzoni, Konstantin Milbradt, Charles Nathanson, AlessandroPavan, Brian Weller and seminar participants at Kellogg School of Management for helpful comments and suggestions.All errors are my own.

†Contact Information – Kellogg School of Management, 2001 Sheridan Rd, Evanston, IL 60208; Email: [email protected] , site: http://www.kellogg.northwestern.edu/faculty/gondhi/index.htm

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1 Introduction

Economic theory suggests that, in order to maximize output, scarce resources need to be deployedefficiently. Notably, recent work has documented substantial variation in the degree of resource mis-allocation over time (e.g., Hsieh and Klenow (2009)). As can be seen in Figure 1, the elasticity of

rm investment to its TFP – my proxy for the degree of resource misallocation – varies substantiallyover the business cycle and, furthermore, is strongly correlated with the level of aggregate produc-tivity. In this paper, I build a theory of time-varying resource misallocation based on the rationalinattention of rm managers. Firm managers rst choose how much information to acquire and thenmake input decisions conditional on the realization of their signals. In particular, I allow rm man-agers to acquire two types of information: data about the aggregate economy and information abouttheir own rm. Importantly, information processing ability is just like any other economic resource:it is in nite supply. 1 As managers pay more attention to aggregate news, they devote less time toacquiring information about their own rm. 2 I embed this feature in a tractable general equilibrium

model with shocks to the level of both aggregate and idiosyncratic uncertainty and explore its testableimplications.

My model delivers endogenous uctuations in measured aggregate productivity and output inresponse to movements in both aggregate and idiosyncratic uncertainty. In particular, the level of aggregate productivity depends on the correlation between rm input choices and their own produc-tivity. An aggregate uncertainty shock induces managers to acquire more information about the stateof the aggregate economy and hence less information about rm-specic shocks. Consequently, thedegree of resource misallocation increases leading to a drop in aggregate productivity and output. 3

This pattern is a direct consequence of the fact that the acquisition of information is endogenous.

Note that, in the US economy, reallocation of resources is a key factor driving aggregate productivity.4

Conversely, a rise in idiosyncratic uncertainty induces managers to acquire more information abouttheir own rm-specic productivity, leading to improved allocation of resources and a rise in aggregateoutput. In sum, exogenous changes in uncertainty lead to endogenous business cycles. The modeldelivers several novel predictions that are in line with the data.

On the asset pricing side, my model generates endogenous risk prices for uctuations in aggregate aswell as idiosyncratic uncertainty. In particular, the risk price of aggregate uncertainty is negative. Thatis, high aggregate uncertainty is associated with an increase in marginal utility of consumption, andtherefore households are willing to pay a higher price for securities that are positively correlated withthe shocks to aggregate uncertainty. This feature is consistent with empirical evidence in Bali, Brown,

1 In an extension, I show that my main results hold under more general cost function specications.2 Specically, rm managers acquire signals about aggregate and rm specic shocks subject to an entropy constraint.

Hence, rms face a trade-off: paying more attention to aggregate conditions requires paying less attention to idiosyncraticconditions. Aggregate signals can be interpreted as macroeconomic data about aggregate shocks that affect future cash-ows of all rms, and idiosyncratic rm level signals as rm-level data that forecasts the future protability of rms andis independent of aggregate shocks.

3 As rms invest less time to learn about their idiosyncratic shocks, the inputs decisions of high and low productiverms are similar and, hence, the degree of misallocation will be higher.

4 See for example Foster, Haltiwanger, and Krizan (2000).

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Figure 1: Time varying misallocation

This gure plots the elasticity of rm investment to its TFP (estimated every year) and the aggregate produc-tivity in the economy. I estimate rm level TFP using the semi-parametric method initiated by Olley and Pakes(1992) and construct a panel of TFP levels for publicly traded rms in the U.S. I then estimate the elasticityof rms’ investment to its TFP every year. Aggregate TFP data is constructed by Basu, Fernald, and Kimball(2006). Plotted data are ltered and then scaled to have unit variance. The correlation between these two series

is statistically signicant 0.6.

1960 1970 1980 1990 2000 2010 2020−3

−2

−1

0

1

2

3

year

Elasticity of firm I/K to firm TFPAggregate TFP

and Tang (2014); Schürhoff and Ziegler (2011) and others. 5 Perhaps surprisingly, my model impliesthat the risk price of shocks to common idiosyncratic uncertainty is positive, since it is positivelycorrelated with output growth .6 This result is in fact consistent with the evidence provided by Driessen,Maenhout, and Vilkov (2009), who show that an option trading strategy that replicates a payoff proportional to the rise in the average idiosyncratic volatility across rms earns high risk-adjustedreturns .7 In addition, it is consistent with several novel empirical facts in the real economy that Idocument: the level of idiosyncratic uncertainty is positively correlated with both the level of outputas well as the degree of resource reallocation in the economy. These observations provide support for

the proposed mechanism.My model provides a potential explanation for “excess comovement puzzle” documented in Chris-

tiano and Fitzgerald (1998) and Rebelo (2005): sectoral inputs (investment and labor) comove highlywith each other, even though the comovement in sectoral productivity (TFP) is very weak. TraditionalRBC models have difficulty explaining this fact. My model generates excess comovement in inputs dueto the fact that, when rm managers acquire information about the aggregate economy, they effec-tively learn from a noisy public signal – and hence make correlated decisions. 8 Furthermore, Increases

5 Campbell, Giglio, Polk, and Turley (2012), extending the earlier work of Campbell (1992, 1993), estimate marketvariance innovations based on a vector auto-regressive approach, and nd a negative market variance risk premium inthe cross-section of equity portfolios.

6 Previous researchers ( Herskovic, Kelly, Lustig, and Van Nieuwerburgh (2014); Schürhoff and Ziegler (2011)) docu-mented that idiosyncratic uncertainty has a strong factor structure. Given this evidence, I assume that idiosyncraticuncertainty across rms is driven by an underlying state variable.

7 In particular, the strategy involves selling index straddles and buying individual straddles and stocks in order to hedgeindividual variance risk and stock market risk. This portfolio earns large excess returns. The result is also consistentwith the evidence in Schürhoff and Ziegler (2011) who constructs model-free variance swaps and nd that systematicvariance risk exhibits a negative price of risk, whereas common shocks to the variances of idiosyncratic returns carry alarge positive risk premium.

8 The common error in the public signals increases the comovement beyond what is justied by productivity shocks.The error in the public signal can be interpreted as measurement error in macro-economic statistics or noise in commoninformation sources such as the nancial press. For alternative justication of these noise shocks, refer to Lorenzoni (2009)

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in aggregate uncertainty lead to higher input comovement across sectors; an increase in idiosyncratic uncertainty decreases comovement .9 For intuition, consider an increase in idiosyncratic uncertainty.As idiosyncratic uncertainty increases, rm managers shift their attention to learn more about theiridiosyncratic shocks and, hence, learn less about aggregate shocks. As all rms learn less about aggre-gate shocks, the co-movement of their inputs decreases. I show that this prediction is indeed consistent

with the data .10 First, I document that, consistent with the model, sectoral comovement of inputs ishighest in recessions. Second, I use several proxies for either aggregate or idiosyncratic uncertaintyand nd that the comovement of each sector’s inputs with aggregate inputs increases with aggregateuncertainty but decreases with each sector’s idiosyncratic uncertainty .11

The same mechanism that generates comovement in inputs also delivers substantial time-variationin comovement of market betas across rms. Empirically, there is substantial evidence that equitybetas uctuate over time .12 Yet, little is known about the source of this variation, either theoreticallyor empirically. My model’s prediction is that conditional betas should display a large time variationand that their cross-sectional dispersion decreases with aggregate uncertainty and increases with id-

iosyncratic uncertainty. 13 My model can thus explain the empirical nding of Fama and French (1997)who document that market risk of industry portfolios uctuates considerably over time. I complementtheir nding by also documenting that one standard deviation increase in aggregate uncertainty isassociated with 0.4 standard deviation reduction in the dispersion in market betas; whereas, one stan-dard deviation increase in idiosyncratic uncertainty is associated with 1.6 standard deviation increasein the dispersion of betas.

One of the assumptions of the baseline model is that rm managers do not learn from prices in thestock market. A large literature in corporate nance and macroeconomics focuses on the sensitivityof rms’ investment to a mispricing in the stock market .14 One conclusion from this literature is that

investment responds only moderately to mispricing in the stock market or that the stock market is a

and Angeletos and La’O (2013). Further, another source that impacts the expectations held by all market participants isthe noise in stock prices. In appendix C, I introduce a stock market in my setup. I allow investors and rms to learn fromequilibrium prices and update their beliefs accordingly. Non-fundamental shocks that affect stock prices – for instance’endowment’ or ’noise-trader’ shocks, in the spirit of NREE models – will impact the expectations held by all marketparticipants.

9 This result is a direct consequence of endogenous learning. In an economy with exogenous information (i.e., signalswith constant precisions), the co-movement of inputs do not change with idiosyncratic uncertainty.

10 I test the prediction using sectoral level-KLEM data compiled by Dale Jorgenson. The database combines industrydata from the US bureau of Labor Statistics (BLS) and the US Bureau of Economic Analysis (BEA). For each sector,the data-set contains information on the value and the price of four inputs (capital, labor, energy and material) and thevalue and price of output.

11 As a proxy for aggregate uncertainty, I used the variable constructed by Jurado, Ludvigson, and Ng (2013) andused VIX for robustness. As a proxy for idiosyncratic uncertainty, I used the variable constructed by Bloom, Floetotto,Jaimovich, Saporta-Eksten, and Terry (2012) and used the idiosyncratic industry uncertainty constructed using stockreturns data for robustness (see Campbell, Lettau, Malkiel, and Xu (2001)).

12 Direct evidence is provided by Bollerslev, Engle, and Wooldridge (1988); Jagannathan and Wang (1996); Lewellenand Nagel (2006); Bali and Engle (2010); Fama and French (1997) and Engle (2014) who nd signicant time-seriesvariation in the conditional betas of equity portfolios.

13 In an economy with exogenous information (i.e., signals with constant precisions), dispersion in betas increases withaggregate uncertainty.

14 Some representative papers in this area are Baker, Stein, and Wurgler (2002), Gilchrist, Himmelberg, and Huberman(2005), Polk and Sapienza (2009) and Bond, Edmans, and Goldstein (2011).

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sideshow with respect to the real economy ( Morck, Shleifer, Vishny, Shapiro, and Poterba (1990)). 15

In Section 5, I relax this constraint and introduce trading in an aggregate index where rm managerscan trade based on their aggregate information and learn from the equilibrium price. Since all rmmanagers learn from the stock market, any noise (non-fundamentals) that moves stock prices impactsthe expectations held by all rm managers. This merely adds to the common error and strengthen my

effects. This is because, my results mainly depend on the correlation of beliefs across rm managersand not on any information asymmetry between rm managers and investors.

Last, I explore the welfare implications of information acquisition. In my model, the informationprocessing capacity of a rm manager is effectively a factor of production just like labor and capital. Ithus examine the degree to which it is efficiently allocated. I nd that, as long as the cost of acquiringinformation is convex, the equilibrium information capacity chosen will be lower than the sociallyefficient capacity. Two factors drive this result. First, since rms are local monopolists, they under-invest in both physical resources as well as information .16 Second, if managers can learn from nancialmarkets, this leads to a free rider problem in acquiring information about aggregate shocks.

My paper connects to several strands of the literature. First, my paper relates to a vast theoryliterature which studies the mechanisms through which uncertainty shocks impacts the aggregateeconomy (see Bloom (2013) for an extensive survey). In contrast to the papers in this literature,I allow managers to affect the posterior uncertainty facing their rm through learning. My modeldelivers an endogenous response of output to changes in uncertainty that operates through changes inthe degree of resource misallocation. Motivated by the assumption that the aggregate and idiosyncraticuncertainties are positively correlated, previous researchers (e.g., Bloom et al. (2012)) assume that onestate variable drives both uncertainties and nd that shock to the state variable can generate drop inoutput. However, in this paper, I document that the two uncertainties have opposite business cycleimplications.

Second, it is closely related to the literature on endogenous information acquisition and rationalinattention ( Sims (2003)). Close to my paper, Mackowiak and Wiederholt (2009) study optimal inat-tention in rms pricing decisions. They assume that information frictions only have nominal biteand all real decisions are allowed to adjust under the true state of nature. By contrast, I study themore realistic scenario in which information friction has a real bite. Endogenous information acqui-sition has been studied in various scenarios. Van Nieuwerburgh and Veldkamp (2009); Van Nieuwer-burgh and Veldkamp (2010) study nancial investor’s information acquisition problem and show thatportfolio under-diversication might arise endogenously with information acquisition. Kacperczyk,Van Nieuwerburgh, and Veldkamp (2014) study information acquisition problem of a fund manager

15 To the extent that prices reect information not otherwise available from rms’ internal sources, stock marketsprovide rms with valuable information and guide real activity. However, in a recent paper, David, Hopenhayn, andVenkateswaran (2014) nd that-learning from stock prices is at best only a small part of total learning at the rm level,even in a well-functioning nancial market like US. Thus, the contribution of nancial markets to overall allocativeefficiency and aggregate performance through this channel is quite limited. This is primarily due to the high levels of noise in market prices, making them relatively poor signals of fundamentals. In contrast, a signicant amount of learningoccurs from private sources, i.e., those internal to the rm.

16 Market power introduces a constant distortion to the average level of economic activity. It affects the wedge betweenprivate and social value of information, and hence rms invest less in information acquisition compared to the sociallyefficient choice.

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who can learn about several assets and show that fund managers optimally choose to process infor-mation about aggregate shocks in recessions and idiosyncratic shocks in booms. I contribute to thisliterature by showing how the interaction between endogenous information acquisition and uncertaintyshocks leads to endogenous business cycles.

Third, my work is related to the literature on the pricing of aggregate and idiosyncratic volatility.Representative agent models have explored the role of aggregate consumption growth volatility forexplaining a host of asset pricing stylized facts. In such models, the representative agent is willingto sacrice a portion of her expected returns for insurance against a rise in aggregate volatility, butshe does not seek to hedge against idiosyncratic volatility which is fully diversiable .17 Empirically,there is mixed evidence whether idiosyncratic volatility is priced positively or negatively. Herskovicet al. (2014) argue that idiosyncratic volatility is priced negatively because of incomplete markets andhouseholds face more labor income risk when idiosyncratic volatility increases. In my model, I abstractaway from this effect and show that idiosyncratic volatility is priced positively, which is consistent withevidence in Schürhoff and Ziegler (2011) and others.

Fourth, my work is related to the recent literature in macroeconomics on ’animal spirits’ as thedrivers of the business cycle ( Angeletos and La’O (2013)). Angeletos and La’O (2010) show thatintroducing the common error shocks in the RBC model can generate zero (or even negative) correlationbetween output and employment, a moment which RBC models have tough time matching. Lorenzoni(2009) claim that the common error shocks can drive business cycles. On the empirical side, a recentwork by Angeletos, Collard, and Dellas (2014) show that condence shocks can account for the bulkof the observed business-cycle uctuations. My contribution is to show that common error shocks alsohelp us explain the “excess comovement puzzle”.

Last, my paper also relates to the growing literature on the aggregate implications of misallocated

resources, for example, Hsieh and Klenow (2009) and David et al. (2014). Close to my paper, Davidet al. (2014) study resource misallocation in an economy with imperfect (an exogenous) information.However, they do so in a deterministic model. In addition to endogenizing the information acquisitiondecision, my model allows for stochastic uctuations in uncertainty that leads to endogenous uctua-tions in aggregate productivity. I also propose a new measure of resource misallocation that is a uniqueimplication of my model, that is, the elasticity of rms’ investment to their TFP. On the welfare front,this paper is related to Colombo, Femminis, and Pavan (2014) which relates the (in)efficiency in theacquisition of information to the (in)efficiency in the use of information and explain why efficiency inthe use is no guarantee of efficiency in the acquisition. My model highlights two potential channels

which lead to the inefficiency in the acquisition of information.Layout: Section 2 introduces the model and solves the information acquisition problem. Section 3

studies the implications of endogenous learning. Section 4 provides the empirical evidence. In Section5, I discuss some extensions. Section 6 studies efficiency and Section 7 concludes. Proofs for the mainresults and some extensions can be found in the Appendix.

17 As explained by Campbell (1993), aggregate volatility is a priced state variable provided that the agent has apreference for early or late resolution of uncertainty.

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2 Model

In this section, I build a general equilibrium RBC model to investigate the link between uncertaintyshocks, rms’ information acquisition, and their real decisions. In order to focus on the particularrole of learning, I work with a simple framework in which costly information acquisition is the only

friction. My point of departure is a standard general equilibrium model of dispersed informationalong the lines of Angeletos and La’O (2010). The model adds three features to this benchmark.First, rm managers have imperfect information not only about aggregate shocks, but also about theirrm-specic (idiosyncratic) shocks. Second, uncertainty is time varying, so the model includes shocksto both level of technology (rst moment) and its variance (second moment) at both aggregate andidiosyncratic levels. Third and most importantly, rm managers can either learn about aggregate orabout rm specic shocks subject to a constraint on their information processing capability.

2.1 Setup

Time is discrete and periods are indexed by t { 0, 1, 2, · · · , }. A continuum of rms of xed measureone, indexed by i, produce intermediate goods using only labor according to 18

Y it = At Z it N αit , α ≤ 1 (1)

where N it denotes labor employed by the rm. Each rm’s productivity is a product of two separateprocesses: an aggregate component ( At ) and an idiosyncratic component ( Z it ). The aggregate andidiosyncratic components of the rm’s productivity follow AR(1) processes:

log At = ρa logAt−1 + at

log Z it = ρz log Z it −1 + zit

where the I assume zit is independent across rms and time and z

it N − 12τ z,t −1

, 1τ z,t −1

. I assumeat is independent over time and a

t N − 12τ a,t −1

, 1τ a,t −1

. I allow for the variance of innovations,

σat−1 dened as 1

τ a,t −1 and σz

t−1 dened as 1τ z,t −1

, to move over time stochastically generating peri-ods of low and high aggregate and idiosyncratic uncertainty. I assume that rms learn in advance thatthe distribution of shocks from which they will draw in the next period is changing. This captures thenotion of uncertainty that rms face about future business conditions.

The intermediate goods are bundled to produce the single nal good using a standard CES aggre-

gatorY t = Y

θ −1θ

it di θθ−1

where θ (1, ∞ ) is the elasticity of substitution.I assume that the underlying shocks become common knowledge within a period. Each time period

t has 3 stages. The time line of the economy is given below:18 For simplicity, I considered one input. Model will remain similar if rms have to choose both capital and labor as

long as both inputs can be adjusted freely every period. With adjustment costs, the model becomes intractble.

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Stage 1: Managers allocate their attention knowing I 1it = {σat−1, σz

t−1, At−1, Z it −1}

Stage 2: Managers receive signals. Workers go the respective rms, labor decisions are made andwages adjust so that the labor market clears. At this point, workers and rm managers have the sameinformation regarding future TFP shocks. Denote their information set I 2it . Workers return home andthe economy transitions into stage 3.

Stage 3: Production occurs. All information is revealed. Commodities and asset markets open.Prices clear these markets. Consumption takes place.

Figure 2: The time line of the economy

This gure illustrates the time-line of the economy:

timet − 1 t

Stage 1 Stage 2 Stage 3

I 1

it =

{σ a

t−

1, σ z

t−

1, A t − 1 , Z i,t − 1

} I 2

it =

{s i

at, s i

zt } I 1

it

Assumptions: Asset markets and good markets operate only in stage 3, when information ishomogenous. This guarantees that asset prices do not convey any information. Moreover, becausemy economy admits a representative consumer, allowing households to trade risky assets in stage 3would not affect any of the results. In Section 5, I allow rm managers to trade on their aggregateinformation and learn from equilibrium prices in stage 2 and show that the analysis remains similar.

Households: There is a representative household, consisting of a consumer and a continuum of workers with preferences:

U =∞

t=0β t E 0 U (C t ) − i V (N it ) di

where i [0, 1] indexes rm i, C t represents consumption of nal good in period t, N it is the laboreffort of the worker who works for rm i. I assume

U (C ) = C 1−γ

1 − γ and V (N ) =

N 1+ ν

1 + ν

where γ > 0 parametrizes the income elasticity of labor supply and also the coefficient of relative risk

aversion; and ν parametrizes the Frisch elasticity of labor supply. Assume that all idiosyncratic risk isinsurable. The representative household owns all the rms in the economy and its budget constraintis given by

P it C it di + B t +1 ≤ πit di + i W it N it di + R t B t

where πit denotes the prots from rm i, W it denotes the period-t wage of rm i, Rt is the period-tnominal gross rate of return on the risk-less bond, and B t is the amount of bonds held in period t .

Firms: Since rms choose inputs in stage 2 of every time period without any adjustment costs,

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the problem of each rm is essentially static. The rm i’s realized prot is given by

πit = P it Y it − W it N it .

In stage 2, each rm’s objective is to maximize the representative household’s valuation of its prots

i.e., E it U (C t )π it .

Market clearing: The representative household consumes the total output produced in theeconomy: C = Y , since there is no capital. In the labor market, wages adjust to equate labordemanded and supplied. In the product market, prices adjust to equate demand and supply of goodsi.e., C it = Y it .

Stochasticity and information: There are two types of shocks: aggregate TFP shocks and id-iosyncratic TFP shocks. I represent the aggregate state of the economy by the history ωt ≡ (ω0, · · · , ωt )of exogenous random variable ωt which includes the aggregate shock, aggregate and idiosyncratic un-

certainties. As dened earlier, I 1it and I 2it denote the information set of rm i at the beginning of stage1 and stage 2 of period t respectively.

2.2 Equilibrium

Denition 1. An equilibrium consists of an employment strategy N ( I 2it ), a production strategy Y it ( I 2it , ωt ),a wage function W I 2it , an aggregate output function Y (ωt ), a price function P ωt , a consumption strategy C (ωt ), signal precisions {τ sa , τ sz } I 1it such that the following are true:

(i) Stage 1: Firms acquire information to maximize the net present value of al l future cashows,

subject to an information processing constraint/cost.(ii) Stage 2: Representative household and all rms are at their respective optima given their

information set.

(iii) Stage 3: Commodity and asset prices are determined such that the respective markets clear.

I solve for the equilibrium backwards. In stage 3, the optimal demand for intermediate good i isgiven by

Y it = Y tP itP t

−θ=

P itP t

=Y itY t

−1θ

where P it denotes the price of good i and P t denotes the price of nal good. The nal good is ournumeraire, and so P = 1 . I normalize the price of nal good to be 1. Revenue of rm i is given by

P it Y it = Y 1−1

θit Y

1θ

t = Y 1θ

t A1−1

θt Z

1−1θ

it (N it,d )α 1

where α1 = 1 − 1θ α and subscript ’ d’ indicates labor demanded by each rm given wages. This

implies rm prots are given by

π it = P it Y it − W it N it,d = Y 1θ

t A1−1θ

t Z 1−1θ

it (N it,d )α 1 − W it N it,d .

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So, in stage 2, rm i’s objective is to choose labor N it,d to maximize

E it U (C t ) Y 1θ

t A1−1

θt Z

1−1θ

it (N it,d )α 1 − W it N it,d (2)

where the expectation is taken with respect to rm manager’s information set I 2it . The rst order

condition for rms’ demand of labor is given by

E it U (C t )Y 1θ

t A1−1θ

t Z 1−1θ

it α1 (N it,d )α 1−1 = E it U (C t )W it .

In stage 2, since workers who work for rm i, share the same information set as the rm, theoptimal labor supply ( N it,s ) solves the rst order condition:

E it U (C t )W it = V (N it,s ) .

Imposing market clearing in the labor market ( N it,d = N it,s = N it ) implies that the equilibrium ispinned down by the following condition

α 1E it Y 1θ −γ

t A1−1

θt Z

1−1θ

it N α 1−1it

Marginal utility of Consumption ×Marginal Product of Labor

= N ν it

Marginal Disutility

. (3)

This condition equates the private cost and benet of effort for each rm. The right hand sideis simply the marginal dis-utility of an extra unit of labor and the left hand side is the product of marginal utility of consuming extra unit of the good i and the marginal product of labor.

Lemma 1. In stage 2, the equilibrium level of labor is the solution to the following xed-point problem:

log N it I 2it = const. +1 − 1

θ E it (a t + zit )

ν + 1 − α 1+

1θ − γ E it logY t

ν + 1 − α1. (4)

The above condition is just the log-linear transformation of ( 3). Each rm’s input decision de-termines their output, which in-turn determines the aggregate output. The expectation of aggregateoutput enters into rm’s labor decision. This best-response condition is similar to the best responsein the abstract class of models (beauty-contest games) studied in Morris and Shin (2002); Angeletosand Pavan (2007). An agent’s best response is the linear function of expectation of fundamentalsand an aggregate variable. The economy features strategic complementarity when 1

θ > γ and featuresstrategic substuitability when 1

θ < γ . My results hold regardless of whether the economy featuresstrategic substuitability or complementarity.

In order to solve ( 4), I now turn to the aggregate economy, and in particular, solve for output.Note that aggregate revenue equals aggregate output which implies,

Y t = P it Y itP t

di = Y 1θ

t A1−1

θt Z

1−1θ

it α 1E it C 1θ −γ

t A1−1

θt Z

1−1θ

it

α 1ν +1 −α 1 di. (5)

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To complete the characterization of the rm’s problem and therefore the production-side equilib-rium in the economy, I need to spell out the rm’s information set. I defer the discussion to the nextsubsection where I endogenize rm’s information and, for now, conjecture that all rms have samesignal precisions, which I will later show to be true.

Specically, I assume that the signals about fundamentals are of the form:

Assumption 1. The signals available to rm manager i are of the form:

s iat = at + ηt + ν iat , where ηt N 0, τ −1

η (6)

s izt = zit + ν izt

where ν iat N 0, 1τ sa

and ν izt N 0, 1τ sz

are purely idiosyncratic shocks and they are byproduct of rational inattention.

This assumption formalizes the idea that paying attention to the aggregate conditions and payingattention to the idiosyncratic conditions are separate activities. For example, attending to the currentstate of monetary policy, is a separate activity from attending to the rm-specic productivity. Thesignal about aggregate shocks can be interpreted as public sources of information about the state of the economy, in which case the correlated error ηt may reect, for example, measurement error inmacroeconomic statistics while the idiosyncratic error ( ν iat ) may be interpreted as the byproduct of limited attention. Alternatively, ηt can be interpreted as a proxy of noise shocks studied in Lorenzoni(2009) or as condence shocks studied in Angeletos and La’O (2013).

I will look for a symmetric equilibrium in which all the rms acquire the same signal precisions.I next conjecture that the equilibrium labor chosen, nit , is linear in rm’s information set I 2it =

a t−1, z t−1i , siat , s izt and aggregate output yt is linear in state vector ωt = a t−1, t , ηt , τ ta , τ tz andverify that there is always a unique linear equilibrium.

Proposition 1. In stage 2, the labor market has unique log-linear equilibrium:1) Employment chosen by rm i given I 2i,t is given by

n it = log N it = 0 + a−1a t−1 + z

−1zi,t −1 + a s iat + z s izt . (7)

2) Aggregate output is linear in aggregate state variable ωt :

yt = log Y t = ψ0 + ψ−1a t−1 + ψa at + ψηηt . (8)

All coefficients are reported in the appendix.

Proposition 1 shows that each rm’s input decision is a log-linear function of its private signals,while the aggregate output is a log-linear function of aggregate state variable. The aggregate outputdoes not depend on any rms idiosyncratic signal noise as a result of the aggregation across a largenumber of rms with independent noise. This feature is similar to Hellwig (1980). The aggregate

10



output depends on aggregate TFP shock ( at ) as well as common noise term ηt . A key feature con-

tributing to this tractability is that the aggregate output remains log-normal as a result of the Lawof Large Numbers.

2.3 Endogenous Information acquisition

Until now, I solved for the rms’ optimal input decisions given their signals i.e., solved for stage 2equilibrium. In this subsection, I endogenize their signals i.e., solve their stage 1 problem. I assumethat the rm manager has access to a variety of information sources and has to decide what to payattention to, subject to an information ow constraint. Processing information is modeled as receivingnoisy signals about the fundamentals. Firm managers’ chooses the precisions of their signals, τ sa andτ sz , subject to a learning constraint.

Two Learning Technologies

Formulating a problem with information choice requires a learning technology. Which learning tech-nology is appropriate depends on the type of data agents are acquiring .19 Since agents are learningabout future productivity, which is subjective, rational inattention effectively captures the learningconstraint. Recall that the agent’s optimal action (from 4) is a function of both the fundamentals andaggregate output, which, in turn, is driven by both the fundamentals and correlated error. So, an agentwants to learn not only about the fundamentals, but also about common error from the aggregatesignals. Given this, inattention constraint needs to be modied for my information structure.

There are two ways of writing the constraint,1. Approximate constraint: If agent only learns about fundamentals, his constraint can be written

asI ( a

t ; s iat )

κ 1

+ I ( zit ; s izt )

κ 2

≤ κ and κ1, κ2 ≥ 0, (9)

2. Exact constraint: Agent tries to extract a linear combination of aggregate shock and commonnoise π a

t + πηηt from his signal s iat i.e.,

I (π at + πηηt ; s iat )

κ 1

+ I ( zit ; s izt )

κ 2

≤ κ and κ1, κ2 ≥ 0, (10)

where I(x;y) denotes the mutual information between random variables x and y; π and πη are equi-librium objects dened in the appendix.

The information constraint in the second approach is endogenous, since the coefficients π and πη

are equilibrium objects. In the rst approach, the agent incurs cost in extracting information aboutfundamentals only. In both constraints, κ1 denotes the information ow allotted to learning aggregateconditions and κ2 denotes the information ow allotted to learning idiosyncratic conditions.

19 For general information technologies, refer to Hellwig, Kohls, and Veldkamp (2012)

11



In appendix, I show that information ow constraint is always binding. This implies that, rmsface a trade-off: attending more carefully to aggregate conditions requires attending less carefully toidiosyncratic conditions. This substitutability in learning is a crucial assumption and drives a lot of results in this paper.

The second constraint I impose on agents’ learning is the no-forgetting constraint, which insures

that the chosen precisions are always non-negative. It prevents an agent from erasing any priorinformation, to make room to gather new information in other dimension.

Information choice problem

I now solve the rm manager’s information acquisition problem (i.e., stage 1 problem). Substitutingthe optimal labor hiring ( 7) in (2), the net present value can be rewritten as

E it U (C t ) C 1θ

t A1−1

θt Z

1−1θ

it N α 1it − W it N it = E it [X it ]

ν +1ν +1 −α 1 α

α 1ν +1 −α 11 − α

ν +1ν +1 −α 11

where X it = C 1θ −γ t A1−1θ

t Z 1−1θit . Information choice problem is to choose attention allocated to aggregate

and idiosyncratic conditions by maximizing the expected payoff subject to the information constrainti.e.,

maxκ 1 ,κ 2

E E it [X it ]ν +1

ν +1 −α 1 |a t−1 (11)

subject to

s it =atzit

+ηt

0+

ν iat

ν izt

and either ( 9) or (10). The outer expectation operator in ( 11) is the expectation under the information

set I 1it , whereas the inner expectation is under the information set I 2it . The solution to this problemis given below:

Proposition 2. The optimal attention is given by

κ1 =

κ if x ≤ 2−2κ

κ2 − 1

4 log2 (x) if x 2−2κ , 22κ

0 if x > 22κ

(12)

Case 1. If the learning technology is given by ( 9 ), then x = π2

z

(π η τ a + π τ η )2τ 2η

Relative Importance

τ aτ z

Relative uncertainty

;

Case 2. If the learning technology is given by ( 10 ), then x = π2

zπ 2 τ η + π 2

η τ aτ η

Relative Importance

τ aτ z

Relative uncertainty

where π and πη are equilibrium objects dened in the appendix.

The attention paid to aggregate conditions, κ1, is decreasing in x. x has two factors: 1. Relative

12



Figure 3: Equilibrium attention paid to aggregate conditions

This gure plots attention paid ( κ1 ) and signal precisions ( τ sa ) as a function of aggregate uncertainty and id-iosyncratic uncertainty in an economy with endogenous learning. Left plot corresponds to equilibrium attentionpaid to aggregate conditions and the right one corresponds to corresponding precisions of aggregate signals.Solid lines and dotted lines correspond to constraints ( 10) and ( 9) respectively.

0.6 0.7 0.8 0.9

0.2

0.25

0.3

0.35

0.4

τa

κ 1

τz

=0.1

τz

=0.11

τz

=0.12

0.6 0.7 0.8 0.9

0.25

0.3

0.35

0.4

0.45

0.5

0.55

τa

τ a s

τz

=0.1

τz

=0.11

τz

=0.12

(a) Aggregate attention (b) Aggregate signal precisions

uncertainty 2. Relative importance. When aggregate conditions are more variable than idiosyncraticconditions, agents pay more attention to aggregate conditions. Relative importance is an endogenousobject and depends on the equilibrium information acquired, because π and πη are equilibrium objects.This implies that we need to solve a xed point problem to determine exact precisions acquired inequilibrium. I rst show that the xed point has a unique equilibrium.

Lemma 2. The xed point given by

κ1 = κ2

− 14

log2π2

z(π η (κ 1 )τ a + π (κ 1 )τ η )2

τ 2η

τ aτ z

has a unique equilibrium.

Figure 3 illustrates the equilibrium attention paid to the aggregate conditions, κ1 as function of aggregate uncertainty. In the gure, solid lines and dotted lines correspond to constraints ( 10) and(9) respectively. The various colors correspond to different values of idiosyncratic uncertainty levelsas indicated in the legend. As aggregate uncertainty decreases (as we move along the x axis), agentspay less attention to aggregate conditions and more attention to idiosyncratic conditions. As idiosyn-cratic uncertainty increases, agents pay less attention to aggregate conditions and more attention toaggregate conditions. Note that the results (equilibrium allocations) from two learning constraints arequalitatively very similar. So, I will focus on the second constraint since it is analytically tractable.

Theorem 1. Optimal attention paid to aggregate conditions:1. Increases with aggregate uncertainty;2. Decreases with idiosyncratic uncertainty;

13



3. Increases with risk aversion for γ > 1.

The result tells us that rm managers want to learn more about any shock that has a high priorpayoff variance. Information is most valuable about the most uncertain outcomes. In addition, thereare feedback effects because the output depends on the information acquisition of other rms. Whenother rms pay more attention to aggregate shocks, each rms nds it more optimal to pay moreattention to aggregate shocks when actions are strategic complements. However, when actions arestrategic substitutes, the opposite result holds. This result is rst noted by Hellwig and Veldkamp(2009).

Recessions: If recessions are periods of high risk aversion and high aggregate uncertainty, themodel’s prediction is that rm managers’ pay more attention to aggregate shocks in recessions. I willuse this result later to argue why output endogenously falls in recession in the model.

The following section explain the model’s key implications:

• Real side implications: comovement of inputs

• Asset pricing implications

– Idiosyncratic uncertainty is positively priced, whereas, aggregate uncertainty is negativelypriced.

– Dispersion in market betas.

For each prediction, I state and prove a proposition. Later sections explain how I test the hypothesesin the data.

3 Implications of endogenous learning

An obvious hurdle in testing models with endogenous learning is that we cannot directly observe thevariables included in rm managers’ information set. In this section, I test the model indirectly: linkthe information choice (solved in the previous section) to testable patterns in data.

3.1 Comovement of inputs:

In the model, rm managers choose inputs to maximize the NPV of the rm conditional on theirinformation set. Note that ( 7) gives the optimal labor input chosen by the rm. By denition,idiosyncratic TFP shocks are uncorrelated across rms, which then implies that the comovement of

inputs across rms is given by:

2a

Endogenous object

1τ a

Aggregate uncertainty

+ 1

τ η

Common error variance

. (13)

Equation ( 13) implies that the forces causing inputs to co-move are aggregate uncertainty, commonerror in beliefs and an endogenous object a . It is easy to show that the endogenous object a

14



increases as agents learn more about aggregate shocks. First, observe that, if the common error termis sufficiently noisy, we can get excess comovement in inputs not warranted by comovement in TFP.

Benchmark economy: In the spirit of existing literature ( Angeletos and La’O (2010)), the bench-mark economy in my view is the one in which learning is exogenous (i.e., one in which signal precisionsare constant and independent of uncertainty faced by the agents) .20 I will refer to this specication

as the benchmark economy and come up with predictions which would help us to distinguish thisbenchmark economy from an economy with endogenous learning.

Theorem 2. 1. If τ η is sufficiently low, comovement(inputs)>comovement(TFP).

2. In an economy with endogenous learning, comovement of inputs across sectors increases with ag-gregate uncertainty and decreases with idiosyncratic uncertainty. This result doesn’t hold in benchmark economy.

Sectoral comovement of inputs can be higher than their comovement of TFP because of common

error in beliefs. This helps us explain the “excess comovement puzzle”: sectoral inputs (investment andlabor) comove highly with each other, even though the comovement in sectoral productivity (TFP) isvery weak. The common error term could be the result of measurement error in macroeconomic statis-tics or any sender specic noise shocks ( Veldkamp and Wolfers (2007)) or animal spirits/sentiments(Angeletos and La’O (2013)) or expectation shocks ( Lorenzoni (2009)).

Part 2 of the theorem states that, with endogenous learning, comovement of inputs across sectorsincreases with aggregate uncertainty and decreases with idiosyncratic uncertainty. As aggregate uncer-tainty increases, rms learn more about aggregate shocks and ψa increases. This implies comovementof inputs increases with aggregate uncertainty. This result holds even without endogenous learning

(i.e., in benchmark economy) as can be seen from expression ( 13). On the other hand, comovement of inputs decreases with idiosyncratic uncertainty. As idiosyncratic uncertainty increases, rms allocateless capacity to learn about aggregate shocks and this decreases a . As all rms learn less aboutaggregate shocks, comovement of inputs across rms decreases. This is not true in the benchmarkeconomy. In the benchmark economy, as idiosyncratic uncertainty increases, comovement of inputs donot change. This is evident from ( 13) which depends on aggregate uncertainty and aggregate signalprecisions and is independent of idiosyncratic uncertainty.

Figure 4 plots the comovement of inputs versus aggregate uncertainty (left plot) and idiosyncraticuncertainty (right plot) in the baseline economy (solid line) and in an economy with endogenouslearning (dotted line). In the left plot, note that the results are qualitatively the same for both theeconomies. In the right plot, note that the comovement of inputs do not change with idiosyncraticuncertainty in the baseline economy and decreases with idiosyncratic uncertainty in an economy withendogenous learning. This leads us to my rst testable prediction:

Prediction 1: Comovement of inputs across sectors increases with aggregate uncertainty and decreases with idiosyncratic uncertainty.

20 Other possible benchmark model is the one in which agents allocation of capacity is independent of uncertainty theyface. All my results hold for this denition of benchmark also.

15



Figure 4: Plot of comovement in inputs

This gure plots the comovement of inputs as a function of aggregate uncertainty (left) and idiosyncraticuncertainty (right) in an economy with endogenous learning (dotted line) and the benchmark economy (solidlines).

0.2 0.25 0.3 0.35 0.40.2

0.4

0.6

0.8

1

1.2

C o v a r i a n c e o

f i n p u

t s

τa

Endogenous learningBaseline

0.1 0.15 0.2 0.25

0.4

0.5

0.6

0.7

0.8

0.9

τz

C o v a r i a n c e o

f i n p u

t s

Endogenous learningBaseline

(a) Aggregate uncertainty (b) Idiosyncratic uncertainty

3.2 Asset pricing Implications

In this section, I will study the aggregate and asset pricing implications of uncertainty shocks andendogenous learning.

Recall that the preferences of the representative household are given by

U = ∞t=0

β t E U (C t ) − i V (N it ) di

and its budget constraint is given by

P t C t + B t +1 + V it φi,t +1 di ≤ (V it + π it ) φi,t + i W it N it + R t B t

where V it denote the ex-dividend value of rm i in period t, φit denote the ownership of rm i inperiod t , and πit denote the dividend paid by rm i in period t . Market clearing in the asset marketsis given by φit = 1 for all i, t . The rst order condition for the inter-temporal consumption allocation

problem yields

E t β C t +1

C t−γ

S t +1S t

π it +1 + V it +1

V it

R i,t +1

= 1 .

This is a standard expression of consumption CAPM and has to hold for each asset i, where St +1St

isthe stochastic discount factor (SDF) and R i,t +1 is the one-period return of holding rm i’s share from

16



t to t + 1 . We can write the logarithm of SDF as

logSt +1

St= log β − γ log

C t +1

C t.

Note that the consumption in the economy is log-linear and is given by

ct = log C t = ψ0(τ a , τ z ) + ψaat + ψη ηt .

I call ψ0 the unconditional aggregate productivity in this economy and is given by

ψ0 = const 0+ const 1 δ τ η π2 + π2

η τ a + τ sa (1 − γ )2

τ sa τ a + ( τ a + τ sa ) τ η+ δ 2

∆ 21

τ sa−

(1 + δ ) πz

τ a

Effect of Aggregate uncertainty

+ const 2πz δr z (κ2) − 1

θ2τ z

Effect of Idiosyncratic uncertainty(14)

where the endogenous coefficients are reported in appendix. Uncertainty shocks affect output only

through their effect on ψ0, the unconditional aggregate productivity.In my economy, there are four fundamental shocks:

• First moment macro TFP shock ( at )

• Second moment idiosyncratic uncertainty shock (τ zt )

• Second moment aggregate uncertainty shock (τ at )

• First moment common noise shock (ηt ).

Each of them affect the consumption of the representative agent. To understand their risk premia, I

investigate how much do households care about each risk.Theorem 3. Log linearizing consumption growth around the steady state, Stochastic discount factor in my economy can be approximately written as

log S t +1

S t≈ a − b1∆ t+1 − b2∆ ηt +1 − b3∆ σz,t + b4∆ σa,t (15)

where b1, b3, b4 > 0 and b2 > 0 iff γ < 1; and ∆ x denotes change in x.

This is the main result of the general equilibrium model. In equilibrium, asset risk premia aredetermined by the covariance of asset returns with the stochastic discount factor. Equation ( 15)shows that the asset risk premia are determined by the exposure to the four innovations. Here, Idene the price of risk of a random shock z as the Sharpe ratio of a security whose payoff is perfectlycorrelated with realizations of z:

rp zt = covt log

St+1

St, zt+1 .

The price of risk associated with z depends on how the state price of consumption S is correlated withthe shock z. If the marginal utility of wealth, and, hence, the state price of consumption, is lower

17



following an increase in z, then this shock will carry a positive risk premium. Since households attacha lower value to these states, they are willing to pay a lower price for a security that pays off in lowconsumption states, or equivalently they demand a positive risk premium. Conversely, if the stateprice of consumption is higher following a positive shock to z, households are willing to pay a higherprice for securities that are positively correlated with z, and thus, the risk premium is negative. I next

describe how each of the shocks are priced in equilibrium.Macro TFP shock: The price of risk of the productivity shock ( t ), b1, is positive which implies

that households demand a positive risk premium to invest in securities that are positively correlatedwith the aggregate productivity shock.

Common error shock: The price of risk of common error shock is positive if and only if γ < 1.This result is because of general equilibrium forces. If γ > 1, an increase in noise leads to higherexpected aggregate income which discourages labor supply (from 7) and raises real wages, which hasthe opposite effect on rm prots, production, and employment. Since all rms now invest less, thisleads to lower output and, hence, lower consumption. 21 This increases marginal utility of wealth and,

hence, priced negatively. The opposite is true when γ < 1. Macro-economists interpret common errorshock as sentiment shocks/ animal spirits as in Angeletos and La’O (2013). My model shows thatthese sentiment shocks are priced and the risk premium associated with these shocks is negative if andonly if γ >1.

Idiosyncratic uncertainty shock: From equation ( 15), changes in idiosyncratic uncertainty posi-tively effect consumption growth and, hence, the price of risk of idiosyncratic uncertainty shock ispositive. My model links rational inattention at the rm level to resource misallocation and, hence, toaggregate productivity. In the model, rm managers choose inputs under limited information abouttheir fundamentals. This informational friction leads to a misallocation of resources across rms in

an ex-post sense, reducing output. As rms learn more about their idiosyncratic shocks, this infor-mation friction shrinks: resources are allocated more efficiently, which leads to higher output. In aneconomy with endogenous learning, as idiosyncratic uncertainty increases, rms shift their attentionto learn more about rm-specic (idiosyncratic) shocks, which decreases the extent of misallocationin the economy and increases output. This increases the consumption of representative householdand, hence, decreases the marginal utility of consumption. Therefore, idiosyncratic uncertainty ispro-cyclical, and hence, has a positive price of risk. Empirically, there’s mixed evidence whether id-iosyncratic uncertainty/volatility is priced positively or negatively. Empirically, Schürhoff and Ziegler(2011) show that common shocks to idiosyncratic volatility are positively priced, consistent with mytheory. In the next section, I provide more empirical evidence consistent with the proposed channel.

Aggregate uncertainty shock: The price of risk of aggregate uncertainty shock is negative. Asaggregate uncertainty increases, rms learn more about aggregate shocks and less about idiosyncraticshocks. This leads to inefficient allocation of resources and, hence, decreases the aggregate productivity.This implies that the total output and, hence, consumption will be lower, which increases the marginalutility of wealth and, therefore aggregate uncertainty is priced negatively. This is consistent withthe evidence in Bali et al. (2014) and others. Bloom, Bond, and Van Reenen (2007) shows using

21 This is similar to effects of news shock in Jaimovich and Rebelo (2009).

18





As an econometrician, I dene market beta as:

β it = Cov(R it , R mt )

V ar(Rmt ) .

For now, assume that the idiosyncratic shocks are not persistent. This assumption implies that

price of all securities will be the same since shocks every period are independent of past shocks. Thisis an extreme assumption but made only for tractability. Since rms are solving the static problem ineach time period, betas are derived from covariance of the rm’s cash ow with the cash ow of themarket: 22

β it = Covt−1(R it , R mt )

V art−1(Rmt ) =

Cov(D it , D t )V ar( D t )

where D t denotes the market dividends. Assume that rms pay all their revenues as dividends:

D it = Y 1θ

t A1−1

θt Z

1−1θ

it N α 1it − N it W it .

Substituting dividends into the expression for market beta, I get the following result:

Proposition 3. Market beta of rm i is given by

β (ν ia , ν iz ) = ξ 1i f 1 + ξ 2i f 2

f 1 + f 2(16)

where ν ia and ν iz denote the idiosyncratic signal errors in aggregate and idiosyncratic signals of rm i;ξ 1i and ξ 2i are functions of signal errors with mean 1 and all the variables are dened in the appendix.

From the above expression, rst note that average beta across all rms is one since ξ 1i and ξ 2i arefunctions with mean one. Market beta of rm i is measured as a function of signal errors of aggregateand idiosyncratic errors. From 3 , its very easy to compute the dispersion of betas.

Denition 2. Dispersion of betas across rms is given by

Disp (β ) ≡ i (β i − 1)2 di.

Dispersion is calculated as the extent to which betas differ from their mean 1. 23

Proposition 4. In the model, dispersion of market betas across rms is given by

Disp β ≈ f 2a

τ sa+

2z

τ sz

where f(.) is a monotonic function.22 As robustness, I also computed beta of a security as beta with respect to the wealth portfolio. Results remain

qualitatively the same.23 In the empirical section, I compute it as weighted average dispersion of betas around mean. In the model, the weights

are equal because the price of all rms is the same at the beginning of every period.

20



Note that a and z denote the weights managers’ put on signals siat and sizt respectively inthe optimal input chosen (from ( 7)). Dispersion of market betas across rms is driven by dispersionof beliefs about aggregate shocks and idiosyncratic shocks. As the dispersion of beliefs increase, thedispersion of betas increases.

Theorem 4. If 1θ = γ ,

1. Without endogenous learning, dispersion of market betas increases with both aggregate and idiosyncratic uncertainty.

2. With endogenous learning and substantial common noise (i.e., τ η ≤ τ η), dispersion of market betas decreases with aggregate uncertainty and increases with idiosyncratic uncertainty.

I can analytically prove this result for 1θ = γ , but numerically veried that the result holds more

generally. This theorem states that, in an economy in which agents are endowed with signals of constantprecision (benchmark economy), dispersion of betas increase with both types of uncertainties. Thisis because, as uncertainty increases, Bayesian agents put more weight on the signals they receive.

Because the signal realizations are heterogeneous across rms, the resulting posterior beliefs becomemore different from each other which increase the dispersion of beliefs. This increases the dispersionof market betas. This is true for both aggregate and idiosyncratic uncertainty. So, in the benchmarkeconomy , dispersion of betas increase with both aggregate and idiosyncratic uncertainty.

On the other hand, in an economy with endogenous learning, dispersion of betas decreases withaggregate uncertainty and increases with idiosyncratic uncertainty if the common error is sufficientlynoisy. Recall that this condition is exactly what is required to generate excess comovement in inputs inthe data. Higher aggregate uncertainty implies that the rm managers’ priors are uninformative, andendogenous learning implies that managers’ devote more attention to learning aggregate conditions.

Combining these two effects, Bayesian learning implies that the managers’ put more weight on thesignals received. If there is sufficient common noise in the aggregate signals, the managers’ signalsbecomes increasingly similar with more learning. Because the managers’ weigh the similar signalsmore, their resulting posterior beliefs become more similar to each other. This convergence in beliefsgenerates similar real decisions and cash-ows, and hence, the dispersion of market betas across rmsdecreases. This leads us to my second testable prediction:

Prediction 2: Cross-sectional dispersion in market betas decreases with aggregate uncertainty and increases with idiosyncratic uncertainty.

3.3 Proxy for MisallocationIn this subsection, I develop a proxy of misallocation – the elasticity of rm investment to its TFP.Recall that rm’s optimal choice of inputs is given by

log Inputs it = 0 + a−1a t−1 + z

−1zi,t −1 + a s iat + z s izt . (17)

A strong prediction of the model is that z increases as managers’ learn more about their id-iosyncratic shocks. Higher idiosyncratic uncertainty leads to more about idiosyncratic shocks, which

21



increases z and also decreases the extent to misallocation in the economy and hence aggregate pro-ductivity increases. Hence high z is a proxy for resource misallocation.

Imagine an economist with cross-sectional rm-level data on Investment I, capital K and productiv-ity (TFP). Suppose the economist estimates the regression coefficients in log(I/K )i,t = α t + γ t tfp it +1 +

i,t by ordinary least squares (OLS). In this regression, the α and γ are unknown coefficients and it

is an error term that accounts for the fact that the right-hand side variables do not perfectly predictlog investment rate.

Suppose we estimate the regression every period. Comparing the regression equation with ( 17),I claim that γ t captures the effect of ψz . This is because, intercept at each instant, αt , captures theeffect of aggregate shocks and constants. Any time variation in γ t corresponds to time variation in ψz .In particular, higher γ t is associated with rms’ learning more about idiosyncratic shocks and henceimplies less misallocation of resources. This leads to higher aggregate productivity. Figure 1 plotsthe time series of γ and aggregate productivity. Note that the correlation between these two series isstatistically signicant 0.6, consistent with my theory.

4 Empirical Analysis

In this section, I rst describe how I empirically construct the uncertainty measures. Volatility inZ it leads to cross-sectional dispersion-based measures in rm performance (productivity shocks, sales,stock market returns, etc.), while volatility in A t induces higher variability in aggregate variables likeGDP growth and the S&P500 index. 24 I construct proxies for idiosyncratic uncertainty using cross-sectional dispersion measures, and aggregate uncertainty using volatility in aggregate variables likeGDP, VIX, etc.

I conduct all my analysis at the sector/ Industry level. Given that idiosyncratic uncertainty isdened as cross-sectional measures, it is easy to depict this at sectoral level than at the rm level.Moreover, betas estimated at the industry level will be more stable than betas estimated at the rmlevel. In principle, similar tests could be conducted at the rm level.

Idiosyncratic Uncertainty over the Business cycle

In this subsection, I give details on what I mean by (measure) idiosyncratic uncertainty. I use twodifferent methods to proxy this:

1. Using stock returns: I use cross section dispersion in realized excess returns (compared to CAPMmodel) of industry portfolios to measure idiosyncratic uncertainty at industry level. This reects thevolatility of news about industry performance. This is the method adopted by Campbell et al. (2001).For more details on the estimation, refer to their paper.

2. Using Census & ASM data: First calculate establishment-level TFP (z j,t ) using the standardapproach from Foster et al. (2000). I then dene TFP shocks ( e j,t ) as the residual from the following

24 The term Z it accommodates two interpretations: the productivity of intermediate good producer i or the rm specicdemand shifter. I will not be able to distinguish these two interpretations from my theory or using empirical work.

22



rst-order auto regressive equation for establishment level log TFP:

log(z j,t ) = ρ log(z j,t −1) + µ j + λ t + e j,t

where µ j is an establishment level xed effect and λ t is a year xed effect. I dene micro uncertainty at

the industry level as cross-sectional dispersion of e jt of establishments in each industry. More detailson the estimation procedure can be found in Bloom et al. (2012).One important caveat when using the variance of productivity shocks to measure uncertainty is

that the residual is a productivity shock only in the sense that it is unforecasted by the regressionequation, rather than unforecasted by the establishment.

Macroeconomic (Aggregate) Uncertainty over the Business cycle

In this subsection, I give details on how I measure macroeconomic uncertainty. I use two differentmethods to proxy this:

1. Using stock returns: I use VIX as a proxy for macro economic uncertainty. Since VIX is forwardlooking, this is known to agents before. Since VIX is only available from 1986, I use realized volatilityas a proxy for expected volatility for years<1986. Fore more details on how I construct this variable,refer to Bloom et al. (2007).

2. Using macro series: I use the methodology developed in Jurado et al. (2013) on measuring macrouncertainty. It is dened as the common variation in the unforecastable component of a large numberof economic indicators. By construction, this proxy ensures that these measures be comprehensive, asfree as possible from both the restrictions of theoretical models and/or dependencies on a handful of economic indicators. This series has lot less number of peaks of macro uncertainty compared to VIX.

[Table 1 about here.]

Table 1 provides the correlation matrix of the various uncertainties proxies. Note that proxies of aggregate uncertainties are positively correlated and proxies of idiosyncratic uncertainties are positivelycorrelated. Proxies of aggregate and idiosyncratic uncertainty are also correlated.

4.1 Comovement in inputs

In this subsection, I give details on how I test the real side hypothesis of my model. I use KLEM annualdata from Dale Jorgenson and collaborators from 1949 to 2005 .25 The data comprise 35 industriesthat cover the entire non-farm, non-mining private economy. This database provides factor inputs andoutputs, along with their prices. My goal in this section is to explore sectoral input dynamics andexamine how they relate to aggregate and idiosyncratic uncertainty faced by rms. I will rst showthat there is excess co-variation in the inputs used across sectors, which is not driven by co-variation inproductivity. Second, I will show that the co-variation in inputs across sectors increases with aggregate

25 The Jorgenson’s KLEM (stands for Capital, Labor, Energy and Material) database combines industry data from theUS bureau of Labor Statistics (BLS) and the US Bureau of Economic Analysis (BEA). One advantage of using KLEMdata is that it covers entire economy unlike compustat database.

23



uncertainty and decreases with idiosyncratic uncertainty, consistent with the rst prediction of themodel.

Decomposition of aggregate volatility:

I now perform a decomposition of aggregate variance of TFP and inputs of capital, labor andother materials into sectoral variances and covariances across sectors. Let γ s,t denote the particularvariable in sector s at time t, and ωsec

s,t be the share of sales for sector s in the aggregate sales inthe economy. Also, let V [Z τ ]t+5

t−4 denote the variance of {Z t−4, . . . Z t , . . . , Z t +5 } for any generic

variable Z and Cov [Z τ ]t +5t−4 , [Y τ ]t +5

t−4 be the covariance between {Z t−4, Z t−3, . . . Z t , . . . Z t+4 , Z t+5 } and{Y t−4, Y t−3, . . . Y t , . . . Y t+4 , Y t+5 }. By denition, the aggregate variable is given by

γ t =s

γ s,t ωsecs,t .

Using the denition of the variance:

V [γ τ ]t+5t−4 ≡ 1

10t+5

τ = t−4 sγ s,τ ωsec

s,τ − 110

t +5

τ = t−4 sγ s,τ ωsec

s,τ

2

.

For simplicity, suppose that ωsecs,t = ωsec

s for all sectors s and all years t. Then V [Z τ ]t+5t−4 can be

written as follows:

V [γ τ ]t+5t−4 =

s(ωsec

s )2 V [γ s,τ ]t +5t−4

Variance component

+s j = s

ωsecs ωsec

j Cov [γ s,τ ]t +5t−4 , [γ j,τ ]t+5

t−4

Covariance component

. (18)

Equation ( 18) shows how I decompose aggregate variance into sectoral variance and covarianceacross sectors. I then do this decomposition for the aggregate inputs used and aggregate TFP series.To construct TFP at the sector level, I use the method provided by Basu et al. (2006) in which theauthors developed a “puried” measure of sectoral total-factor productivity (TFP)—a measure of theSolow residual, constructed to take account of non-constant returns to scale in industry productionfunctions, imperfect competition, and varying utilization of labor and capital inputs.

Figure 6 plots the decomposition of total variance into covariance component and sectoral variancefor TFP (left plot) and inputs (right plot) over time. The yellow region corresponds to covariancecomponent and the rosy-brown corresponds to sectoral level variance. It is easy to see that the

covariance region accounts for most of the total variation for inputs (right side plot) and this is nottrue for TFP (left plot). This implies that there is excess comovement in inputs not justied bycomovement in TFP.

In the data, 86% of the aggregate variance of total inputs used is due to co-variation across sectors,while the proportion of the variation in aggregate variance of TFP due to co-variation is only 15%.In the model, the variance of the common error component has to be sufficiently high to justify thiskind of excess comovement in inputs. Without common error, the co-variation in inputs used shouldbe approximately equal to the covariation in TFP across sectors.

24



Figure 6: Plot of decomposition of total variance

This gure plots the decomposition of total variance of TFP and inputs used into sectoral variance and covariancecomponents as in ( 18). Left plot corresponds to TFP series and the right one corresponds to inputs used.

- . 0

0 0 0 2

0

. 0 0 0 0 2

. 0 0 0 0 4

. 0 0 0 0 6

1950 1960 1970 1980 1990year

Covariance Variance

. 0 0 0 1

. 0 0 0 2

. 0 0 0 3

. 0 0 0 4

. 0 0 0 5

. 0 0 0 6

1950 1960 1970 1980 1990year

Covariance Variance

(a) Decomposition of Total Variance of TFP (b) Decomposition of Total Variance of inputs used

Conclusion 1. Sectoral inputs comove highly with each other, even though the comovement in sectoralTFP is very weak

I next do the time series analysis of comovement in inputs used. I regress the covariation of inputson aggregate and idiosyncratic uncertainty to test my hypothesis. I run the following regression:

Covsect = α + β i σsec

i,t + β a σa,t + t .

Results are presented in table 2. Given that I only have 30 yearly observations, I don’t have powerto estimate the coefficients precisely. But to gain robustness, I use different proxies of idiosyncratic and

aggregate uncertainty. The hypothesis I test is that: Co-variation of inputs across sectors increaseswith aggregate uncertainty and decreases with idiosyncratic uncertainty.


I estimate β i to be negative in all the specications and β a to be positive in all the specications.Even though the estimates are not statistically signicant in some specications, the estimates areconsistently of the same sign across all specications. If the measurement error is correlated acrossproxies, then the result might be driven by common measurement error. Since the proxies I use arefrom very different data sources (one is from real data and other from nancial market data), it ishighly unlikely that measurement error is correlated across variables.

Conclusion 2. Covariance of inputs across sectors increases with aggregate uncertainty and decreaseswith idiosyncratic uncertainty.

To gain robustness, I do the analysis using panel data where the dependent variable is the co-variation of inputs used by each sector with the aggregate inputs used and the independent variablesare sectoral level idiosyncratic uncertainty and aggregate uncertainty. This specication will not onlyhave more power, but also allows me to soak up the sector level variation by including the sector levelxed effects.

25



In theory, the covariance of inputs of sector s with the aggregate inputs is given by

Cov(n s , n ) = s a 1τ a

+ 1τ η

.

In theory, same prediction holds: comovement of inputs of each sector with aggregate inputs increaseswith aggregate uncertainty and decreases with each sector’s idiosyncratic uncertainty.

To test this, I use the idiosyncratic volatility constructed at the sectoral level. Next, I constructthe sector-specic co-variation measure, Covs

t , dened as

Covst =

j = s

Cov [γ s,τ ]t+5t−4 , [ω j,τ γ j,τ ]t+5

t−4

and I run the following regression:

Covst = α s + β t + γ i σs

i,t + γ a σa,t + s,t .

Time xed effects will take of the all the time series variation including aggregate uncertainty.Results are reported in table 3. I estimate a negative γ i and a positive γ a in all the specicationsand are signicant even when the standard errors are corrected for auto-correlation of independentvariable (newey west with 5 lags).


Interpretation of column (2): Fix a sector A. If idiosyncratic uncertainty of sector A increasesfrom time t1 to t2, inputs of sector A co-move less with aggregate inputs at t2 than at t1 and vice-

verse. Interpretation of column (3): Fix a time period t. Consider two sectors A and B. If idiosyncraticuncertainty of sector A is higher than sector B, comovement of sector A’s inputs with aggregate inputswill be lower than comovement of sector B’s inputs with aggregate inputs. The result is robust tovarious denitions of idiosyncratic uncertainty.

4.2 Asset pricing implications

My theory predicts that idiosyncratic uncertainty is pro-cyclical and, hence, has positive price of risk,whereas, aggregate uncertainty is counter-cyclical and, hence, has negative price of risk. EmpiricallySchürhoff and Ziegler (2011) document that common shocks to idiosyncratic volatility are positively

priced and systematic variance risk is negatively priced, consistent with my theory. In this subsection,I document some evidence for the proposed channel.

Given that consumption CAPM holds in my economy, a shock is positively priced only if it ispositively correlated with consumption growth. In table 4, I document that aggregate uncertainty iscounter-cyclical whereas idiosyncratic uncertainty is pro-cyclical, providing evidence for consumptionCAPM channel. Since I am interested in the cyclical properties of output and consumption, it isimportant to detrend the series, since the raw series are non-stationary. I use Hodrick-Prescott lterdescribed in Hodrick and Prescott (1997) to extract the cyclical component. I deate all the series

26



to 2009 dollars using the CPI from the BLS to remove any effects from variation in nominal prices.Standard errors corrected for heteroskedasticity and auto-correlation are reported in the table.


To provide more support for the proposed channel, I examine how capital reallocation changeswith aggregate and idiosyncratic uncertainty. In theory, as idiosyncratic uncertainty increases, rmslearn more about idiosyncratic shocks which leads to more efficient reallocation of resources and, hence,higher productivity. For this channel to be true, capital reallocation should increase with idiosyncraticuncertainty. On the other hand, as aggregate uncertainty increases, rms learn less about idiosyncraticshocks and, hence, there should be less reallocation of resources i.e., reallocation should decrease withaggregate uncertainty. I next test this hypothesis.

I measure the amount of reallocation using annual compustat data as sum of acquisitions and salesof property, plant and equipment and focus on the cyclical properties of this series. This is similar to themeasure used by Eisfeldt and Rampini (2006). I use HP lter to de-trend this series. Regression results

are provided in table 5. I nd that reallocation increases with idiosyncratic uncertainty, consistent withmy channel, whereas, reallocation doesn’t decrease with aggregate uncertainty. Note that recessionsare also accompanied by re sales/forced liquidation which has opposite effect on reallocation.


4.3 Market beta hypothesis:

In this subsection, I test the second prediction of the model. I use returns for 49 industries from KenFrench data library for the period 1926-2014; First I estimate market beta for each industry using

daily returns. I use rolling one month window with estimation window of 12 months. Also, followingDimson (1979), I include both current and lagged market returns in the regressions, estimating betaas the sum of the slopes on all lags. I include four lags of market returns, imposing the constraintthat lags 2 – 4 have the same slope to reduce the number of parameters:

R it = α i + β i, 0RM,t + β i, 1RM,t −1 + β i, 2 (RM,t −2 + RM,t −3 + RM,t −4) + i,t .

The market beta is then estimated as β i = β i, 0 + β i, 1 + β i, 2. Then I estimate dispersion in CAPMbetas across the 49 industry portfolios.

First, I show that CAPM betas are indeed time varying. Figure 7 (a) plots the dispersion of CAPM

betas across industries over time. Figure 7(b) plots aggregate and idiosyncratic uncertainty series overtime. It also indicates the historical events over time. Figure 8 plots the dispersion of betas alongwith aggregate (a) and idiosyncratic uncertainty (b) time series. From the plot, you can see that thecorrelation between aggregate uncertainty and dispersion of betas is signicantly negative (-0.21***),but the correlation between idiosyncratic uncertainty and dispersion in betas is signicantly positive(0.23***). I next add some controls in a regression and show that the results remain signicant.


27



Figure 7: Plot of time series of dispersion in betas and uncertainty proxies

Left plot shows the dispersion of CAPM betas over time and the right plots shows the aggregate and idiosyncraticuncertainty along with historical events.

1940 1960 1980 2000

0 . 2

0 . 3

0 . 4

0 . 5

year

1970 1980 1990 2000 2010

0 . 8

0 . 9

1 . 0

1 . 1

1 . 2

year

Vietnam buildup

Cambodia and Kent State

Franklin National

OPEC II

Afghanistan, Iran hostages

Monetary cycle

Black Monday

Gulf War I

Asian Crisis

Tech bubble

Worldcom and Enron

Credit crunc

0.5

1.0

1.5

2.0

2.5

3.0

3.5Macro UncertainityIndustry Uncertainty

(a) Dispersion of CAPM betas over time (b) Time series of uncertainty along with historical events

Figure 8: Plot of Dispersion of CAPM betas with aggregate and idiosyncratic uncertainty

Left plot shows the dispersion of betas and aggregate uncertainty series over time and the right plots shows thedispersion of betas and idiosyncratic uncertainty series over time.

1960 1970 1980 1990 2000

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 6

year

D i s p e r s

i o n

i n b e

t a s

1.0

Dispersion in betasMacro Uncertainity

1980 1985 1990 1995 2000 2005 2010 2015

0 . 2

0 . 3

0 . 4

0 . 5

year

D i s p e r s

i o n

i n b e

t a s

0.5

1.0

1.5

2.0

2.5 Dispersion in betasIndustry Uncertainity

(a) Dispersion of betas and aggregate uncertainty (b) Dispersion of betas and idiosyncratic uncertainty

In table 6, I present regression results of dispersion of market betas on uncertainty proxies. Aggre-gate uncertainty is both economically and statistically negatively signicant. One standard deviationincrease in aggregate uncertainty leads to decrease in dispersion of market betas by 0.02 to 0.03 acrossvarious specications. Idiosyncratic uncertainty is also both economically and statistically signicant.One standard deviation increase in idiosyncratic uncertainty leads to increase in dispersion of market

28



betas by 0.08 to 0.1 across various specications. Both of these are consistent with my theory. Theresults are robust to alternative proxies of uncertainties and alternative ways of estimating betas. 26

Note that R-squared in the regression is also high which implies that uncertainty has a rst ordereffect on time variation in betas.

Conclusion 3. Dispersion in market betas across industries decreases with aggregate uncertainty andincreases with idiosyncratic uncertainty.

5 Extensions

5.1 Information aggregation in the stock market

One of the assumptions of the baseline model is that rm managers do not learn from nancial marketsi.e., I shut down learning from nancial markets in stage 2 when they make real decisions. In appendixC, I introduce aggregate market where rm managers can trade on their aggregate information and

learn from equilibrium prices. Anything that moves stock prices impacts the expectation held by allmarket participants. In this sense, noise in the nancial market will serve as a common error andimpacts the beliefs of all managers. The implications of endogenous learning will qualitatively remainthe same. Recall that for the dispersion in market beta implications to hold, we need substantialcommon error. Since noise in the nancial market adds to the common noise, the results will beamplied with learning from nancial market. This is because, my results depend on the correlationof beliefs across managers and not on any information asymmetry between rm managers and investors.

I also show that learning from nancial market will make information acquisition choice inefficient.This is because of a free rider problem that rm managers’ face i.e., they can learn from nancialmarkets for free and, hence, each manager imposes positive externality on others and thereby has lessincentive to learn about aggregate shocks. This leads to excess learning about idiosyncratic shocks.

5.2 Variable utilization

In the baseline model, I assumed that input choices are made under imperfect information and are notallowed to change once the true state is realized. In this extension, I relax this assumption and allowrm managers to choose the utilization of the inputs once the state is realized. To do this, assume thatL it = N it h it , where L it denote the total labor input, N it denote the labor chosen (number of workers)under imperfect information in stage 2 and h it denote the level of effort per worker and is chosen oncethe true state is realized. I change the HH utility function to accommodate variable effort:

U =∞

t =0β t E U (C t ) − i N it V (h it )

26 As robustness,1. I estimated market betas controlling for FF3 factors;2. I estimated market betas using dynamic conditional correlation (DCC) method proposed by Engle (2002);3. I used 6 months rolling window instead of 12 months;4. To make sure that my results are not driven by estimation error in estimating betas, I also used average estimation

error as one of the controls in regression and nd that the results remain signicant.

29



where I assumeU (C ) =

C 1−γ

1 − γ and V (h) = 1 +

h1+ ν

1 + ν .

I solve the model in appendix B. The results remain qualitatively the same.

6 Efficiency

In this section, I explore the welfare implications of rational inattention/ information acquisition.Information processing capacity of a rm manager is effectively a factor of production just like laborand capital. The only difference is the way in which it enters the production function. In this section, Iexamine if the new factor of production “capacity”, is allocated efficiently between learning aggregateand idiosyncratic productivity.

The welfare criterion I adopt is the ex-ante expected utility of the representative household. Thereare two related issues regarding welfare: 1. (In)Efficiency in the use of information 2. (In)Efficiency in

acquiring information. First, I check if the efficient use of information coincides with the equilibriumuse of information. Second, I examine whether the equilibrium acquisition of information coincideswith the efficient acquisition of information.

I consider a constrained efficiency concept that permits the planner to choose any resource-feasibleallocation that respects the segregation of information in the economy - by which I mean that theplanner cannot make the production and employment choices of rms and workers based on the privateinformation of other rms.

Planner’s use of information: Choose an employment strategy, N( I 2it ) and an aggregate output function, Y (ωt ), so as to maximize

E it U (Y t ) − i V (N it ) di

subject to Y it = At Z it N αit and Y t = Y θ−1

θit di

θθ−1

.

The problem has a simple interpretation: rst term is the utility of consumption for the repre-sentative household and the second term is the marginal dis-utility of labor for the typical worker ina given rm; and the corresponding integral is the overall dis-utility of labor for the representativehousehold. The solution to planner’s problem is pinned down by the following rst order condition:

E it C 1θ

−γ

t A1

−1θ

t Z 1

−1θ

it αN α 1

−1

it = N ν it .

Comparing this equation to the equilibrium labor allocation (see equation ( 3)), we see that thereis a wedge between the equilibrium and efficient use of information. The wedge arises because of monopoly power: rms internalize that their production decisions affect their prices next period andhence invest/hire less than the optimal level. Alternatively, if rms behave as price takers, there willbe no wedge between the equilibrium and efficient use of information. In the absence of monopolydistortions, the equilibrium use of information is efficient, no matter the information structure. As

30



for the complementarity/substuitability, its origin is in preferences and technologies, not any type of market inefficiency, guaranteeing that private motives in coordinating economic activity are perfectlyaligned with social motives.

Efficient acquisition of information: Next, I solve the planners problem to study how he

allocates the nite capacity to learning about aggregate and idiosyncratic shocks. I distinguish twoscenarios. In the rst one, the planner can control the way the agents use the information they acquire.In the second one, the planner is unable to change the way the agents use their available informationand I ask the question of what allocation of capacity maximizes welfare when information is usedaccording to the equilibrium rule.

Under efficient use, we know that the optimal labor solves the equation:

E it C 1θ −γ

t A1−1

θt Z

1−1θ

it αN α 1−1it = N ν

it .

Planners problem of optimal capacity allocation is given by:

Planner’s problem under efficient use: Choose a strategy, κ1: R+ × R+ → [0, κ] so as tomaximize

E U (Y t ) − i V (N it ) di

subject ton it = log N it = 0 + −1a t−1 + a s iat + z s izt

and Y it = A t Z it N αit and Y t = Y θ−1

θit di

θθ

−1

and

12

log2

τ η τ saτ η + τ sa

τ a+ 1 +

12

log2τ szτ z

+ 1 ≤ κ.

Conjecture that yt = ψ0 + ψaat + ψ−1a t−1 + ψηηt where the coefficients depend on the information

acquired by agents. Maximizing planner’s utility is same as maximizing

E t−Y 1−γ

t

1 − γ −

α1+ δ

1 + ν i

(E it [X it ])1+ δ di .

Proposition 5. If planner can dictate the agents on how to use the information, the information acquired in equilibrium coincides with the efficient information acquisition.

Next, I solve the planners problem under equilibrium use of information:

31



Planners problem under equilibrium use:

Here, the planner is unable to change the way the agents use their available information. The equilib-rium use of information is given by

E it C 1θ −γ

t A1−1

θt Z

1−1θ

it α 1 − 1

θN α 1−1

it = N ν it . (19)

Given this, the planner solves the problem: Choose a strategy, κ1: R+ × R+ → [0, κ] so as to maximize

E U (Y t ) − i V (N it ) di

subject ton it = log N it = 0 + −1a t−1 + a s iat + z s izt

and Y it = A t Z it N αit and Y t =

Y

θ−1θ

it di θθ −1

and

12

log2

τ η τ saτ η + τ sa

τ la+ 1 +

12

log2τ szτ z

+ 1 ≤ κ.

Note that the coefficients will remain exactly the same as in the equilibrium.

Proposition 6. Even if the use of information is not efficient, the acquisition of information is efficient.

The intuition is simple. Equilibrium use of information doesn’t change the relative importance of aggregate and idiosyncratic shocks. It only changes the total marginal value of learning. If informationprocessing capacity is xed, inefficient equilibrium use of information doesn’t impact the efficiency of equilibrium acquisition of information. This leads us to my rst main result on efficiency:

Theorem 5. If managers have an exogenous capacity constraint, the acquisition of information is always efficient even though the use of information could be inefficient.

Next, suppose agents’ incurs a convex cost of capacity and are allowed to choose capacity endoge-nously based on the uncertainty, then the equilibrium acquisition of information will be inefficientand agents’ acquire less information in equilibrium than the socially efficient choice. The relativeallocation of capacity to learning about aggregate and idiosyncratic shocks will remain the same as in

the efficient allocation.

Proposition 7. If managers incur some convex cost to process information, C (κ), then the equilibrium capacity chosen will be lower than the efficient capacity i.e, κe < κ s .

The intuition is simple. Market power introduces a constant distortion to the average level of economic activity. It affects the wedge between private and social value of information, and hence,rms equilibrium acquisition of information will be lower than the socially efficient choice. The mainmessage is that: Monopoly power reduces the value of information to rms.

32



7 Conclusion

In this paper, I show that endogenizing information choice of rm managers is a fruitful approach tounderstand input behavior and asset prices. I developed a tractable general equilibrium model thatuses an observable state variable - the state of the business cycle (i.e., aggregate and idiosyncratic

uncertainty) - to predict information choices and link those information choices to testable patternsin the data. I show that rm managers optimally choose to learn less about aggregate shocks whenidiosyncratic uncertainty is higher and vice-versa. This mechanism has two main implications. First,I show that endogenous learning is helpful to understand the patterns of comovement of inputs and market betas in the cross section of rms . Second, I link rational inattention at the rm level toresource misallocation and, hence, to aggregate productivity and output. Through this channel, Ishow that common idiosyncratic uncertainty is pro-cyclical and, hence, has a positive price of risk,while aggregate uncertainty is counter-cyclical , and, hence, has a negative price of risk.

There are several promising directions for future research. In my modeling approach, I have aimedto strike a balance between realism and transparency of the economic forces at play. In doing so, I havemade a couple of admittedly extreme assumptions. For example, the investment choice is modeled asstatic. Similarly, the learning problem is also static, which perfect revelation at the end of the period,implying that rms are able to quickly correct their past errors. These assumptions limit my abilityto do a full-edged calibration to match moments. Relaxing them is conceptually straightforward butinvolves substantial computational challenges. Also, in this paper, I assumed that uncertainty shocksare exogenous, like rst moment shocks. If uncertainty is endogenous, one could think of a propagationand amplication mechanism. This will be part of future work.

Another important direction for future work is towards a unied theory of information acquisitionof both rm managers and investors. Both of them involve in learning about future shocks and theycan also learn from each other. Anything that moves stocks prices impacts the expectations held byall market participants and rm managers. Also, investors can learn from rm manager’s investmentdecisions. Solving their joint problem will be interesting and complicated task and I leave this tofuture work.

ReferencesAngeletos, G.-M., F. Collard, and H. Dellas (2014). Quantifying condence. Technical report, National Bureau of

Economic Research.

Angeletos, G.-M. and J. La’O (2010). Noisy business cycles. In NBER Macroeconomics Annual 2009, Volume 24 , pp.

319–378. University of Chicago Press.Angeletos, G.-M. and J. La’O (2013). Sentiments. Econometrica 81 (2), 739–779.

Angeletos, G.-M. and A. Pavan (2007). Efficient use of information and social value of information. Econometrica 75 (4),1103–1142.

Baker, M., J. C. Stein, and J. Wurgler (2002). When does the market matter? stock prices and the investment of equity-dependent rms. Technical report, National Bureau of Economic Research.

Bali, T. G., S. Brown, and Y. Tang (2014). Macroeconomic uncertainty and expected stock returns. Georgetown McDonough School of Business Research Paper (2407279).

33



Bali, T. G. and R. F. Engle (2010). The intertemporal capital asset pricing model with dynamic conditional correlations.Journal of Monetary Economics 57 (4), 377–390.

Basu, S., J. G. Fernald, and M. S. Kimball (2006). Are technology improvements contractionary? American Economic Review 96 (5), 1418–1448.

Bloom, N. (2013). Fluctuations in uncertainty. Technical report, National Bureau of Economic Research.

Bloom, N., S. Bond, and J. Van Reenen (2007). Uncertainty and investment dynamics. The review of economic studies 74 (2), 391–415.

Bloom, N., M. Floetotto, N. Jaimovich, I. Saporta-Eksten, and S. J. Terry (2012). Really uncertain business cycles.Technical report, National Bureau of Economic Research.

Bollerslev, T., R. F. Engle, and J. M. Wooldridge (1988). A capital asset pricing model with time-varying covariances.The Journal of Political Economy , 116–131.

Bond, P., A. Edmans, and I. Goldstein (2011). The real effects of nancial markets. Technical report, National Bureauof Economic Research.

Campbell, J. Y. (1992). Intertemporal asset pricing without consumption data. Technical report, National Bureau of Economic Research.

Campbell, J. Y. (1993). Understanding risk and return. Technical report, National Bureau of Economic Research.

Campbell, J. Y., S. Giglio, C. Polk, and R. Turley (2012). An intertemporal capm with stochastic volatility. Technicalreport, National Bureau of Economic Research.

Campbell, J. Y., M. Lettau, B. G. Malkiel, and Y. Xu (2001). Have individual stocks become more volatile? an empiricalexploration of idiosyncratic risk. Journal of nance 56 (1).

Christiano, L. J. and T. J. Fitzgerald (1998). The business cycle: it’s still a puzzle. Economic Perspectives-Federal Reserve Bank Of Chicago 22 , 56–83.

Colombo, L., G. Femminis, and A. Pavan (2014). Information acquisition and welfare*. The Review of Economic Studies ,rdu015.

David, J. M., H. A. Hopenhayn, and V. Venkateswaran (2014). Information, misallocation and aggregate productivity.Technical report, National Bureau of Economic Research.

Driessen, J., P. J. Maenhout, and G. Vilkov (2009). The price of correlation risk: Evidence from equity options. The Journal of Finance 64 (3), 1377–1406.

Eisfeldt, A. L. and A. A. Rampini (2006). Capital reallocation and liquidity. Journal of monetary Economics 53 (3),369–399.

Engle, R. F. (2014). Dynamic conditional beta. Available at SSRN 2404020 .

Fama, E. F. and K. R. French (1997). Industry costs of equity. Journal of nancial economics 43 (2), 153–193.

Foster, L., J. Haltiwanger, and C. Krizan (2000). taggregate productivity growth: Lessons from microeconomic evidenceu.

Gilchrist, S., C. P. Himmelberg, and G. Huberman (2005). Do stock price bubbles inuence corporate investment?Journal of Monetary Economics 52 (4), 805–827.

Hellwig, C., S. Kohls, and L. Veldkamp (2012). Information choice technologies. The American Economic Review 102 (3),35–40.

Hellwig, M. F. (1980). On the aggregation of information in competitive markets. Journal of economic theory 22 (3),477–498.

Herskovic, B., B. T. Kelly, H. Lustig, and S. Van Nieuwerburgh (2014). The common factor in idiosyncratic volatility:Quantitative asset pricing implications. Technical report, National Bureau of Economic Research.

34



Hsieh, C.-T. and P. J. Klenow (2009). Misallocation and manufacturing tfp in china and india. The Quarterly Journal of Economics 124 (4), 1403–1448.

Jagannathan, R. and Z. Wang (1996). The conditional capm and the cross-section of expected returns. journal of nance ,3–53.

Jurado, K., S. C. Ludvigson, and S. Ng (2013). Measuring uncertainty. Technical report, National Bureau of EconomicResearch.

Kacperczyk, M. T., S. Van Nieuwerburgh, and L. Veldkamp (2014). A rational theory of mutual funds’ attentionallocation.

Lewellen, J. and S. Nagel (2006). The conditional capm does not explain asset-pricing anomalies. Journal of nancial economics 82 (2), 289–314.

Lorenzoni, G. (2009). A theory of demand shocks. American Economic Review 99 (5), 2050–84.

Mackowiak, B. and M. Wiederholt (2009). Optimal sticky prices under rational inattention. The American Economic Review 99 (3), 769–803.

Morck, R., A. Shleifer, R. W. Vishny, M. Shapiro, and J. M. Poterba (1990). The stock market and investment: is themarket a sideshow? Brookings papers on economic Activity , 157–215.

Morris, S. and H. S. Shin (2002). Social value of public information. The American Economic Review 92, No. 5 ,1521–1534.

Olley, G. S. and A. Pakes (1992). The dynamics of productivity in the telecommunications equipment industry. Technicalreport, National Bureau of Economic Research.

Polk, C. and P. Sapienza (2009). The stock market and corporate investment: A test of catering theory. Review of Financial Studies 22 (1), 187–217.

Rebelo, S. (2005). Real business cycle models: past, present and future. The Scandinavian Journal of Economics 107 (2),217–238.

Schürhoff, N. and A. Ziegler (2011). Variance risk, nancial intermediation, and the cross-section of expected optionreturns.

Sims, C. A. (2003). Implications of rational inattention. Journal of Monetary Economics 50 (3), 665–690.

Van Nieuwerburgh, S. and L. Veldkamp (2009). Information immobility and the home bias puzzle. The Journal of Finance 64 (3), 1187–1215.

Van Nieuwerburgh, S. and L. Veldkamp (2010). Information acquisition and under-diversication. Review of Economic Studies 77(2) , 779–805.

Veldkamp, L. and J. Wolfers (2007). Aggregate shocks or aggregate information? costly information and business cyclecomovement. Journal of Monetary Economics 54 , 37–55.

35



A Appendix: Proofs

For all the proofs in the appendix, I assume that there are two kinds of uncertainty: learnable and unlearnable uncertainty.The superscripts ’l’ and ’ul’ correspond to learnable and unlearnable uncertainty respectively.Proof of Proposition 1. Conjecture that yt = ψ0 + ψ l

alt + ψul

a ult + ψ−1 a t −1 + ψη ηt where we need to solve for the

coefficients. This implies

x it = 1θ −γ ψ0 + ψ l

alt + ψ ul

a ult + ψ−1 a t −1 + ψη ηt + 1 −

1θ

ρa a t −1 + lt + ul

t + 1 − 1θ

z it

= 1θ −γ (ψ0 ) + ψ−1

1θ −γ + 1 −

1θ

ρa a t −1 + 1θ −γ ψ l

a + 1 − 1θ

lt +

1

θ −γ ψ ul

a + 1

− 1

θ

ult + 1

− 1

θz it + 1

θ −γ ψη ηt

≡ π0 + π−1 a t −1 + π ul ult + π z z it + π l l

t + π η η

Standard Gaussian updating implies

t |I it N τ η τ s

aτ η + τ sa

τ η τ sa

τ η + τ sa+ τ la

r a

s iat , 1

τ η τ sa 1

τ η + τ sa 1

+ τ la+ 1

τ ula

and ηt |I it N τ s

a τ la

τ la + τ s

a

τ η + τ sa τ la

τ la + τ sa

r η

s iat , 1

τ sa τ laτ l

a + τ sa

+ τ η

it

|I it

N τ sz

τ sz + τ lz

r z

s izt , 1

τ sz + τ lz+ 1

τ ulz

π l lt + πη ηt

s iatN 0

0 ,(π l )2

τ la+

π 2η

τ ηπ l

τ la

+ π ητ η

π l

τ la+ π η

τ η1

τ la

+ 1τ η + 1

τ sa 1

The projection theorem implies:

π l lt + π η ηt |I it N Σ 12 Σ −1

22

∆ 1

s iat ,π l 2

τ la+

π 2η

τ η −Σ 12 Σ −122 Σ 12

where Σ 12 = π l

τ la

+ π ητ η

and Σ 22 = 1τ la

+ 1τ η + 1

τ sa 1.

This impliesE it (x it ) = π 0 + π−1 a t −1 + π z ρz z it −1 + π z r z s izt + ∆ 1 s iat

V ar it (x it ) =π ul 2

τ ula

+ π 2z

1τ sz + τ lz

+ 1τ ul

z+

(τ sa + τ η ) π 2 −2π η τ sa π + π 2η (τ a + τ sa )

τ sa τ a + ( τ a + τ sa ) τ ηwhere the coefficients are endogenous and will be solved in equilibrium. I next evaluate the integral in 5. Before

doing this, I highlight a property of log-normal distributions that is utilized repeatedly in this appendix. When a variableX is log-normal with ln X N x, σ 2 , then, for any δ R , we have that

E X δ = exp δ x + 12

δ 2 σ 2 = exp x + 12

σ 2δ

exp 12

(δ −1) δσ 2

36



and thereforeE X δ = ( E [X ])δ exp 1

2 (δ −1) δσ 2

I use this property again and again in the derivations that follow, for various X and δ .

The rm idiosyncratic productivity z it and E it (x it ) are jointly distributed as

z itE it ( x it ) | l

t , η t N 0

π 0 + π − 1 a t − 1 + ∆ 1lt + η t

,

1τ l

z+ 1

τ ulz

11 − ρ 2

z

1τ l

z+ 1

τ ulz

π z ρ 2z

1 − ρ 2z

+ π z r zτ l

z

1τ l

z+ 1

τ ulz

π z ρ 2z

1 − ρ 2z

+ π z r zτ l

z 1τ l

z+ 1

τ ulz

π 2z ρ 2

z1 − ρ 2

z+ (∆ 1 ) 2

τ sa

+ π 2z r 2

z 1τ l

z+ 1

τ sz

(20)We then have that

log i Z1 − 1

θit (E it [X it ]) δ di = log i exp z it 1 −

1

θ+ E it x it +

1

2V ar it ( x it ) δ di

=π 2

z

2 1 − ρ 2z

1

τ lz+

1

τ ulz

+ δ π 0 + π − 1 a t − 1 + ∆ 1lt + η t

+δ

2

π ul 2

τ ula

+ π 2z

1

τ sz + τ lz

+1

τ ulz

+τ s

a + τ η π 2 − 2π η τ sa π + π 2

η τ a + τ sa

τ sa τ a + ( τ a + τ sa ) τ η

+ 12

δ 2 1τ l

z+ 1

τ ulz

π2z ρ

2z

1 − ρ 2z

+ (∆ 1 )2

τ sa

+ π 2z r 2

z 1τ lz

+ 1τ s

z+ π z δπ z r z

τ lz+ δ 1

τ lz+ 1

τ ulz

π2z ρ

2z

1 − ρ 2z

Substituting the above expression in 5 and comparing corresponding coefficients:

ψ 0 + ψ la

lt + ψ ul

a ult + ψ − 1 a t − 1 + ψ η η t 1 −

1

θ= 1 −

1

θρ a a t − 1 + l

t + ult + δ log α 1 + log i Z

1 − 1θ

it (E it [X it ]) δ di

ψ la 1 −

1

θ= 1 −

1

θ+ δ (∆ 1 ) =

π l − 1 + γ 1θ − γ

1 −1

θ= δ

τ sa 1 π η τ a + π l τ η

τ η τ a + τ sa 1 + τ sa 1 τ a

ψ ula 1 −

1

θ= 1 −

1

θ = ψ ul

a = 1 π ul = 1 − γ

ψ − 1 1 −1

θ= 1 −

1

θρ a + π − 1 δ = 1 −

1

θρ a + ψ − 1

1

θ − γ + 1 −1

θρ a δ = ψ − 1 =

1 − 1θ (1 + δ ) ρ a

1 − 1θ + δ γ − 1

θ

π − 1 =1 − 1

θ (1 − γ ) ρ a

1 − 1θ + δ γ − 1

θ

ψ η 1 −1

θ= δ ∆ 1 =

π η 1 − 1θ

1θ − γ

= δτ s

a 1 π η τ a + π l τ η


This implies π η = π l −1 + γ

The endogenous coefficients are given by

π η = (1 −γ ) r a

(1− 1θ )

δ( 1θ −γ ) −r η −r a

and π l = (1 −γ )

(1− 1θ )

δ ( 1θ −γ ) −r η

(1− 1θ )

δ( 1θ −γ ) −r η −r a

ψη = (1 −γ ) r a δ

1 − 1θ −δ 1

θ −γ (r η + r a )and ψ l

a =1 − 1

θ −δ r η1θ −γ − 1 − 1

θ r a

1 − 1θ −δ 1

θ −γ (r η + r a )

Give this, employment chosen by rm i is given by

37



n it = const + E it [x it ] + 1

2 V ar it (x it )ν + 1 −α 1

= const + π 0 + π−1 a t −1 + π z ρz z it −1 + π z r z s izt + ∆ 1 s iat + 1

2 V ar it (x it )ν + 1 −α 1

= const + π −1 a t −1 + π z ρz z it −1 + π z r z s izt + ∆ 1 s iat

ν + 1 −α 1

= const +1 − 1

θ (1 −γ ) ρa

1 − 1θ + δ γ − 1

θ

1ν + 1 −α 1

a t −1 + πz ρz

ν + 1 −α 1z it −1 +

πz r z

ν + 1 −α 1s izt +

τ sa 1 π η τ a + π l τ ητ η (τ a + τ sa 1 ) + τ sa 1 τ a

1ν + 1 −α 1

s iat

= const + (1 −γ ) ρa

ν + 1 −α (1 −γ )a t −1 +

πz ρz

ν + 1 −α 1z it −1 +

1 − 1θ r z

ν + 1 −α 1 − 1θ

s izt + (1 −γ ) r a

ν + 1 −α 1 − 1θ −α 1

θ −γ (r η + r a )s iat

Proof of Proposition 2:

The objective function can be written as

κ 1 = arg maxκ 1

E E it [X it ]1+ δ |a t −1 , z t −1i

= arg maxκ 1

E exp (1 + δ ) E it (x it ) + (1 + δ )2

V ar it (x it ) |a t −1 , z t −1i

= arg maxκ 1

exp(1 + δ )

2π ul 2

τ ula

+ π 2z

1τ sz + τ lz

+ 1τ ul

z+



E exp ((1 + δ ) ( π 0 + π−1 a t −1 + π z ρz z it −1 + π z r z s izt + ∆ 1 s iat 1 )) |a t −1 , z t −1i

= arg maxκ 1

exp(1 + δ )

2π ul 2

τ ula

+ π 2z

1τ sz + τ lz

+ 1τ ul

z+



exp ((1 + δ ) ( π0 + π−1 a t −1 + π z ρz z it −1 ))

exp(1 + δ )2

2(π z r z )2 1

τ lz+ 1

τ sz+ (∆ 1 )2 1

τ la+ 1

τ η+ 1

τ sa

Agent has xed capacity and has to allocate the attention optimally. For this case, we can rewrite the aboveexpression as

κ 1 = arg maxκ 1

(1 + δ)2

(π z r z )2 1τ lz

+ 1τ sz

+ (∆ 1 )2 1τ la

+ 1τ η

+ 1τ sa 1

+ 12

π 2z

τ sz + τ lz+

τ sa 1 + τ η π 2 − 2π η τ sa 1 π + π 2η τ a + τ sa 1

τ sa 1 (τ a ) + τ a + τ sa 1 τ η

= arg maxκ 1

(1 + δ)2

∆ 21

1τ la

+ 1τ η

+ 1τ sa 1

+ 12

τ sa 1 + τ η π 2 − 2π η τ sa 1 π + π 2η τ a + τ sa 1

τ sa 1 (τ a ) + τ a + τ sa 1 τ η+

12

π 2z

τ lz(δr z )

subject to the entropy constraint. Let’s simplify the objective function further:

Obj = (1 + δ) ∆21

1τ la +

1τ η +

1τ sa 1 +

τ sa 1 + τ η π 2

− 2π η τ sa 1 π + π 2

η τ a + τ sa 1

τ sa 1 (τ a ) + τ a + τ sa 1 τ η + π 2

z

τ lz δr z

= (1 + δ) π η τ a + π l τ η2


τ sa 1τ la τ η

+τ sa 1 + τ η π 2 − 2π η τ sa 1 π + π 2

η τ a + τ sa 1

τ sa 1 (τ a ) + τ a + τ sa 1 τ η+

π 2z

τ lzδr z

= δπ η τ a + π l τ η

2


τ sa 1τ la τ η

+ π 2

z

τ lzδr z +

τ η π l 2 + π 2η τ a

τ a τ η

Here, I use the law of total variance for going from 1st equation to the second.

Case 1. Suppose the information processing constraint is as given in 9

38



12

log2 τ η τ sa 1

τ la (τ η + τ sa 1 ) + 1

κ 1

+ 12

log2τ szτ lz

+ 1

κ 2

≤κ

The above problem can be rewritten as

κ 1 t = argminκ 1 tπη τ a + π l τ η 2 2−

2κ 1

τ la τ 2η + π2z

τ lz 2−2( κ −κ 1 )

The solution of this simple minimization problem is given by

κ 1 =

κ if π 2z

(π η τ a + π l τ η )2

τ 2η

τ la

τ lz ≤2−2κ

κ2 − 1

4 log2 π 2

z

(π η τ a + π l τ η )2

τ 2η

τ la

τ lz

if π 2z

(π η τ a + π l τ η )2

τ 2η

τ la

τ lz

2−2κ , 22κ

0 if π 2z

(π η τ a + π l τ η )2

τ 2η

τ la

τ lz

> 22κ

(21)

Case 2. Suppose the information processing constraint is given by 10

The above problem can be rewritten as

κ 1 t = argminκ 1 t

π 2 τ η + π 2η τ a

τ la τ η2−2κ 1 +

π 2z

τ lz2−2( κ −κ 1 )

The solution of this simple minimization problem is given by

κ 1 =

κ if π 2z

(π 2 τ η + π 2η τ a )

τ η

τ laτ lz ≤2−2κ

κ2 − 1

4 log2 π 2

zπ 2 τ η + π 2

η τ aτ η

τ laτ l

z if π 2

z

(π 2 τ η + π 2η τ a )

τ η

τ laτ lz

2−2κ , 22κ

0 if π 2

zπ 2 τ η + π 2η τ a

τ η

τ l

aτ lz > 22κ

(22)

Proof of Lemma 2:

Fixed point is given by solution to

κ 1 − κ2

+ 14

log2π 2

z

(π η ( κ 1 ) τ la + π ( κ 1 ) τ η )2

τ 2η

x1

τ laτ lz

= 0 (23)

If the optimal κ 1 is not in [0, κ ], then choose the boundary values.

Also, given the information processing constraint,

τ sa = 22κ 1 −1τ a + τ η

τ a τ η

Substituting this into the equilibrium coefficients π and π η , we get

πη =δ 1

θ −γ (1 −γ ) r a

1 − 1θ −δ 1

θ −γ (r η + r a )and π = π η + 1 −γ

39



Then,

x1 = π2

z τ 2ηπ η τ la + ( π η + 1 −γ ) τ η

2 = π2

z τ 2ηπη τ la + τ η + (1 −γ ) τ η

2 = π2

z τ 2ηδ( 1

θ −γ ) (1 −γ ) r a (τ la + τ η )

(1− 1θ )−δ ( 1

θ −γ )( r η + r a ) + (1 −γ ) τ η2

This can be simplied to

x 1 =1 − 1θ −δ 1θ −γ 2

2 κ 1

−122 κ 1 τ

η+ τ l

aτ η

2

(1 −γ )2

Substituting this into the xed point problem 23,

κ 1 − κ2

+ 14

log2 1

(1 −γ )2τ laτ lz

+ log 2 1 − 1θ −δ 1

θ −γ 1 −2−2κ 1 τ η + τ laτ η

2

= 0

A sufficient condition for the above equation to have unique solution is to have the derivative of the above equationwrt κ 1 to have the same sign. Differentiating the above equation wrt κ 1 , I get

1 − 1θ −δ 1

θ −γ τ η + τ la

τ η

1 − 1θ −δ

1θ −γ (1 −2−

2κ 1

) τ η + τ l

a

τ η

I want this expression to have the same sign for all values κ 1 (0, κ ). Given that the above expression is monotonicin κ 1 ,I only need to verify the boundaries.

It is easy to verify that the above expression is always positive at both the boundaries since it is true under thelimits κ 1 →0 and κ 1 → ∞ and the expression is monotonic.

Proof of Theorem 2:

Part 1: If τ η = ∞, then comovements in inputs should be of similar magnitude as comovement in TFP. For nitevalue of τ η , comovement in inputs is higher than comovement in productivity.

Part 2:

Comovement = 2a

1τ a

+ 1τ η

= (1 −γ )2 r 2a

ν + 1 −α 1 − 1θ −α 1

θ −γ (r η + r a )2

1τ a

+ 1τ η

Without endogenous learning, the above object doesn’t change with idiosyncratic uncertainty. With endogenouslearning, as industry uncertainty increases, agents pay less attention to aggregate uncertainty, which implies r a and rη

decrease.

∂Comovement∂τ z

∂ ∂τ z

r a

ν + 1 −α 1 − 1θ −α 1

θ −γ (r η + r a )

=ν + 1 −α 1 − 1

θ

ν + 1

−α 1

− 1

θ −α 1

θ −γ (r η + r a )

2∂τ s

a

∂τ z

> 0

The last inequality follows from agents substituting learning away from aggregate conditions as idiosyncratic uncer-tainty increases.

Proof of Proposition 3:

In the static model,

β i = Cov (D i , D )

V ar ( D )

40



where dividends of rm i are given by

D i = Y 1θ A1− 1

θ Z 1− 1

θi N α 1

i − N 1+ ν

i

E i [U (Y )]

= eyθ + π z a ez i π z + α 1 n i −e(1+ ν ) n i + γE i ( y )− γ 2

2 var i ( y )

= eyθ + π z a + δ log α 1 + 1

2 δV ar i ( x i ) ez i π z + δE i ( x i ) −e(1+ δ )log α 1 + 12 (1+ δ ) V ar i ( x i )− γ 2

2 var i ( y ) e(1+ δ ) E i ( x i )+ γE i ( y )

≡ Ge z i π z + δE i ( x i ) −He (1+ δ ) E i ( x i )+ γE i ( y )

In my model,

E i (y) = ψ0 + E i ψ la

l + ψη η = ψ0 +ψ l

a

τ la+

ψ η

τ η 1τ la

+ 1τ η

+ 1τ sa 1

−1

s ia = ψ0 + Λ 1 s ia

D i = eyθ + π z a + δ log α 1 + 1

2 δV ar i ( x i ) ez i π z + δ ( π 0 + π z r z s iz +∆ 1 s ia )−

e(1+ δ ) π 0 + γψ 0 +(1+ δ ) log α 1 + 1+ δ2 V ar i ( x i )− γ 2

2 var i ( y )

χ

e

(1+ δ ) π z r z s iz + [(1 + δ ) ∆ 1 + γ Λ1 ]

χ 1

s ia

The market dividend is given by

D = eδπ 0 + δ log α 1 + δ2 V ar i ( x i ) e

yθ + π z a i ez i π z + δ (∆ 1 s ia + π z r z s iz ) di −χ i eχ 1 s ia +(1+ δ ) π z r z s iz di

= ey t −χe12 χ 2

11

τ sa

+ 12 (1+ δ ) 2 π 2

z r 2z

1τ z

+ 1τ s

z

χ 3

eχ 1 ( l + η )

= eψ 0 eψ la

l + ψ ula ul + ψ η η −χ 3 eχ 1 ( l + η )

Variance of market dividends is given by

V ar ( D ) = V ar eψ 0 eψ la

l + ψ ula ul + ψ η η −χ 3 eχ 1 ( l + η ) = e2ψ 0 V ar eψ l

al + ψ ul

a ul + ψ η η + χ 23 V ar eχ 1 ( l + η ) −

2eψ 0 χ 3 Cov (eψ la

l + ψ ula ul + ψ η η t , e χ 1 ( l + η ) )

= e2ψ 0 +

ψ 2a

τ la

+ 1τ ul

a+

ψ 2η

τ η eψ 2

aτ l

a+

ψ 2η

τ η + 1

τ ula −1 + χ 2

3 eχ 2

1 1τ l

a+ 1

τ η eχ 2

1 1τ l

a+ 1

τ η

−1 −

2eψ 0 χ 3 eχ 2

12 τ la +

ψ 2a

2 τ la +

ψ 2η

2 τ η + 1

2 τ ula +

χ 21

2 τ η eχ 1 ψ a

τ la +χ 1 ψ η

τ η −1

Expected market dividends is given by

E D = eψ 0 eψ 2

a2 τ a

+ψ 2

η2 τ η

+ 12 τ ul

a −χ 3 eχ 2

1 12 τ a

+ 12 τ η

41



Expected rm dividends is given by

E (D i |ν ia ; ν iz ) = E eδπ 0 + δ log α 1 + δ2 V ar i ( x i ) e

yθ + π z a ez i π z + δ (∆ 1 s ia + π z r z s iz ) −χe χ 1 s ia +(1+ δ ) π z r z s iz

= E eδπ 0 + δ log α 1 + δ2 V ar i ( x i ) e

ψ 0θ + δ ∆ 1 ν ia + δπ z r z ν iz + ψ a

θ + π z + δ ∆ 1l + ul + π z

uli +( δπ z r z + π z ) l

i +ψ η

θ + δ ∆ 1 η

−E χe χ 1 ( l + η + ν ia )+(1+ δ ) π z r z ( l

i + ν iz )

= eδπ 0 + δ log α 1 + δ2 V ar i ( x i ) e

1θ ψ 0 + δ ∆ 1 ν ia + δπ z r z ν iz e

( 1θ ψ a + π z + δ ∆ 1 )2

2 τ la

+ ( δπ z r z + π z ) 2

2 τ lz

+

ψ ηθ + δ ∆ 1

2

2 τ η + 1

2 τ ula

+ π 2

z2 τ ul

z

−χe χ 1 ν ia +(1+ δ ) π z r z ν iz e( χ 1 ) 2

2 τ la

+ ( χ 1 ) 22 τ η

+(1+ δ ) 2 π 2

z r 2z

2 τ lz

= eδπ 0 + δ log α 1 + δ

2 V ar i ( x i )+ ψ 2

a2 τ a

+π 2

z ( δr z +1) 2

2 τ z +

ψ 2η

2 τ η + ψ 0

θ eδ ∆ 1 ν ia + δπ z r z ν iz

−χe( χ 1 ) 2

2 τ sa

+(1+ δ ) 2 π 2

z r 2z

2 τ z +

(1+ δ ) 2 π 2z r 2

z2 τ s

z eχ 1 r a ν ia + ( χ 1 ) 2

2 τ a + ( χ 1 ) 2

2 τ η −( χ 1 ) 2

2 τ sa

+(1+ δ ) π z r z ν iz −(1+ δ ) 2 π 2

z r 2z

2 τ sz

= eψ 0 eψ 2

a2 τ a

+ψ 2

η2 τ η

+ 12 τ ul

a eδ ∆ 1 ν ia −

( δ ∆ 1 ) 2

2 τ sa

+ δπ z r z ν iz −( δπ z r z ) 2

2 τ sz

ξ 1

−χ 3 e

( χ 1 ) 22 τ a

+ ( χ 1 ) 22 τ η

e

χ 1 ν ia +(1+ δ ) π z r z ν iz −( χ 1 ) 2

2 τ sa −

(1+ δ ) 2 π 2z r 2

z2 τ s

z

ξ 2

where ξ1 i and ξ2 i are random variables of mean 1 and they depend on the realization of signal errors.

Finally,

E ( D i D |ν ia , ν iz ) = E e

ψ 0 + ψ la

l + ψ ula ul + ψ η η

θ + π z l + ul + δ log α 1 + 12 δVar i ( x i )

e z i π z + δ ( π 0 + π z r z s iz +∆ 1 s ia ) − χe χ 1 s ia +(1+ δ ) π z r z s iz

e ψ 0 e ψ la

l + ψ ula ul + ψ η η − χ 3 e

χ 1l + η

= E e ψ 0 e ψ la

l + ul + ψ η η eδ ∆ 1 ν ia −

( δ ∆ 1 ) 2

2 τ sa

+ δπ z r z ν iz −( δπ z r z ) 2

2 τ sz − χ 3 e

χ 1l + η t

eχ 1 ν ia +(1+ δ ) π z r z ν iz −

( χ 1 ) 2

2 τ sa

−(1+ δ ) 2 π 2

z r 2z

2 τ sz

e ψ 0 e ψ la

l + ul + ψ η η − χ 3 eχ 1

l + η

= e2 ψ 0 + (2 ψ a ) 2

2 τ la

+( 2 ψ η ) 2

2 τ η+ 4

2 τ ula ξ1 + + χ 2

3 e(2 χ 1 ) 2 1

2 τ la

+ 12 τ η

ξ 2 − e ψ 0 χ 3 eψ l

a + χ 12 1

2 τ la

+ 12 τ ul

a+ ( ψ η + χ 1 ) 2 1

2 τ ηξ 1 −

e ψ 0 χ 3 eψ l

a + χ 12 1

2 τ la

+ 12 τ ul

a+ ( ψ η + χ 1 ) 2 1

2 τ ηξ 2

This implies that beta of a rm with signal errors {ν iat ; ν izt } is given by

β (ν iat ; ν izt ) = E (D it D t |ν iat ; ν izt ) −E (D it |ν iat ; ν izt )E (D t )

V ar (D t ) =

ξ1 i f 1 + ξ2 i f 2f 1 + f 2

where f 1 ≡e2ψ 0 +

ψ 2a

τ la

+ψ 2

ητ η

+ 1τ ul

a eψ 2

aτ l

a+

ψ 2η

τ η + 1

τ ula −1 −χ 3 e

ψ o +ψ 2

a + χ 21

2 τ la

+ψ 2

η + χ 21

2 τ η + 1

2 τ ula e

ψ a χ 1τ l

a+

χ 1 ψ ητ η −1 and

f 2 ≡χ 23 e

χ 21

1τ l

a+ 1

τ η eχ 2

1 1τ l

a+ 1

τ η

−1 −χ 3 eψ o +

ψ 2a + χ 2

12 τ l

a+

ψ 2η + χ 2

12 τ η e

ψ a χ 1τ l

a+

χ 1 ψ ητ η −1

where ξ1 i and ξ2 i are random variables with mean 1 and f 1 and f 2 depend on the prior uncertainty and equilibriumsignal precisions.

42



Proof of Proposition 4: Dispersion of betas across rms is given by

Disp (β ) = i β 2 (ν iat ; ν izt )di −1 = 1(f 1 + f 2 )2 i ξ2

1 i f 21 + ξ2

2 i f 22 + 2 f 1 f 2 ξ1 i ξ2 i di −1

= 1(f 1 + f 2 )2 i ξ2

1 i −1 f 21 + ξ2

2 i −1 f 22 + 2 f 1 f 2 (ξ1i ξ2i −1) di

This implies dispersion of betas across rms is now given by

Disp = 1(f 1 + f 2 )2 e

( δ ∆ 1 ) 2

τ sa

+ ( δπ z r z ) 2

τ sz −1 f 2

1 + e( χ 1 ) 2

τ sa

+(1+ δ ) 2 π 2

z r 2z

τ sz −1 f 2

2 + 2 f 1 f 2 eδ ∆ 1 χ 1

τ sa

+ δ (1+ δ )( π z r z ) 2

τ sz −1

≈ 1

(f 1 + f 2 )2(δ ∆ 1 )2

τ sa+ (δπ z r z )2

τ szf 2

1 + (χ 1 )2

τ sa+ (1 + δ )2 π 2

z r 2z

τ szf 2

2 + 2 f 1 f 2δ ∆ 1 χ 1

τ sa+

δ (1 + δ ) ( π z r z )2

τ sz

= 1τ sa

(δ ∆ 1 f 1 + χ 1 f 2 )2

(f 1 + f 2 )2 + 1τ sz

(δπ z r z f 1 + (1 + δ ) π z r z )2

(f 1 + f 2 )2

≈ f ∆ 2

1

τ sa+ (π z r z )2

τ sz

Proof of Theorem 4: Dispersion measure can be written as

D = 2

a

τ sa+

2z

τ sz ≈1 −2−2κ 1 2

τ sa+

1 −2−2κ 2 2

τ sz

=τ η 2−2κ 1 −τ a 1 −2−2κ 1 1 −2−2κ 1

τ a τ η+

2−2κ 2 1 −2−2κ 2

τ z

=τ η 2−2κ 1 −τ a 1 −2−2κ 1 1 −2−2κ 1

τ a τ η+

2−2κ 2 1 −2−2κ 2

τ z

=(τ η + τ a ) 2−κ τ a

τ z −τ a 1 −2−κ τ aτ z

τ a τ η+

2−κ τ zτ a 1 −2−κ τ z

τ a

τ z

=(τ η + τ a ) 2−κ

τ aτ z −τ a 1 −2−κ

τ aτ z

τ a τ η+

2−κ

τ zτ a 1 −2−κ

τ zτ a

τ z

= 1τ η

τ η2−κ

√ τ a τ z+ 2 1−κ τ a

τ z −1 −(τ η + τ a ) 2−2κ

τ z+ 2−κ 1

τ z τ a −2−2κ 1τ a

Differentiating with respect to τ z , I get

∂D∂τ z

= 2−κ

τ z √ τ a τ z(2−2κ 1 −1) ≤0

This implies, as idiosyncratic uncertainty increases, dispersion of CAPM betas increases.Differentiating D with respect to τ a , I get

∂D∂τ a

= 2−κ

τ η 1τ a τ z −

2−κ

τ z+ 2−κ

τ a2−κ

τ a − 1τ z τ a

= 2−κ

√ τ a τ z1 −2−2κ 1

τ η

> 0

+ 2−2κ 2 −1τ a

< 0

> 0 τ η is sufficiently small

From the above expression, note that, with out common noise ( τ η → ∞), dispersion of CAPM betas increases withaggregate uncertainty. But, with sufficient common noise term ( τ η is small), dispersion of CAPM betas decreases withaggregate uncertainty.Proof of Theorem 3: We can write SDF as

log St +1

St= log β −γ ∆log C t +1

43



Note that the consumption in the economy is given by

ct = log C t = ψ0 + ψaat + ψη ηt

This implies∆log C t +1 = ∆ ( ψ0 + ψa

at + ψη ηt )

Suppose the steady state values of the state variable ( at , η t , τ at , τ zt ) = (0 , 0, ¯τ a , ¯τ z ). Loglinearizing around the steady

state gives ∆log C t +1 = b4 ∆ τ a,t +1 −b3 ∆ τ z,t +1 + ψa ∆ at +1 + ψη ∆ ηt +1

Effect of Idiosyncratic uncertainty: Higher idiosyncratic uncertainty generates more learning about idiosyncraticshocks, which increases the reallocation of resources and decreases misallocation of resources in the economy. Thisincreases aggregate productivity of the economy and, hence, output increases.

To see this more clearly, lets solve a slightly different economy with no aggregate shocks and lets keep total laborxed in the economy. This is similar to the setup in David et al. (2014). In this case with α = 1 , I can solve for aggregateproductivity and show that

Agg TF P σ 2z −V z

where σz is the prior uncertainty and V z is the posterior uncertianty. If rms do not learn about idiosyncratic shocksat all, posterior uncertainty is same as prior uncertainty and aggregate productivity is zero. This is independent of theuncertainty they face. As rms learn more about their idiosyncratic shocks, the posterior variance decrease and aggregateproductivity increases.

In an economy with endogenous learning, as idiosyncratic uncertainty, rms learn more about idiosyncratic shockswhich increases the aggregate productivity.

B Appendix - Variable Utilization

In the baseline model, I assumed that input choice are made under imperfect information and are not allowed to changeonce the true state is realized. In this extension, I assume that rm managers can choose the utilization of the inputsonce the state is realized. I change the HH utility function to account for labor utilization:

U =∞

t =0

β t U (C t ) − i N it V (h it )

where N it and h it denote the labor hiring and labor utilization by rm i at time t. N it is measurable with respect

to I it but h it is measurable with respect to s t . I assume

U (C ) = C 1−γ

1 −γ and V (h ) = 1 +

h1+ ν

1 + ν

I repeat the same steps as in the main paper. Firm revenues/prots are given by

π it = P it Y it −P t W it N dit = P t Y 1− 1

θit Y

1θ

t −P t W it N dit

where Y it = A t Z it L αit and L it = N it h it . So, in stage 2, rm i ’s objective is to choose N dit to maximize

E it U (C t ) Y 1θ

t A1− 1

θt Z

1− 1θ

it (N it h it )α 1 −W it (h it ) N dit

where expectation is taken with respect to rm manager’s information set I it ; Wages are function of workers effortand fundamentals realized and α 1 = α 1

− 1

θ. The rst order condition for rms demand of labor is given by

In stage 3: α 1 C 1θ

t A1− 1

θt Z

1− 1θ

it (h it )α 1 −1 N ditα 1 −1

= ∂W (h it )

∂h

In stage 2: E it α 1 C 1θ −γ

t A1− 1

θt Z

1− 1θ

it (h it )α 1 N ditα 1 −1

= E it C −γ t W it

In stage 3, HHs rst order condition gives 27

27 To get intuition for this, note that for any h, C −γ t W it −1−

h 1+ νit

1+ ν gives marginal benet to household associated witheffort h. This has to be positive for worker to prefer working over leisure. This also has to be non-positive for laborsupply to be nite. Since labor demand is positive for optimal h and zero otherwise, C −γ

t W it = 1 + h 1+ ν

it1+ ν for optimal h.

44



C −γ t W it = 1 +

h1+ νit

1 + ν .

These conditions can be rewritten as

h it ::::::::::::::::::::::::::: C 1θ −γ

t A1− 1

θt Z

1− 1θ

it (h it )α 1 −1 α 1 N ditα 1 −1

= h νit (24)

N it ::::::::::::::::::::: E it α 1 C 1θ −γ t A1−

1θ

t Z 1−1θ

it (h it )α 1 N dit α 1 −1 = E it 1 + h1+ νit

1 + ν (25)

Substituting rst condition in second, I get

E it h 1+ νit = E it 1 +

h1+ νit

1 + ν = E it h 1+ ν

it = 1 + ν ν

Equation ( 24) can be rewritten as

α 1 C 1θ −γ

t A1− 1

θt Z

1− 1θ

it N ditα 1 −1

1+ νν +1 − α 1 = h ν +1

it

N dit( α 1 − 1)(1+ ν )

ν +1 − α 1 E it C 1θ −γ

t A1− 1

θt Z

1− 1θ

it

1+ νν +1 − α 1 = 1 + ν

ν α− 1+ ν

ν +1 − α 11

Let X it = C 1θ −γ

t A1− 1

θt Z

1− 1θ

it ; ι = 1+ νν α− 1+ ν

ν +1 − α 11 and δ = α 1

ν +1 −α 1. Simplifying above expressions, I get

ν + 1 −α 1

(1 −α 1 ) (1 + ν )log E it X 1+ δ

it −log ι = log N dit

α 1 X it (N it )α 1 −1 α

ν +1 − α 1 = hαit

Output of each rm is given by

Y it = A t Z it N αit h αit = ( A t Z it )

ν +1ν +1 − α 1 (Y t )

α ( 1θ

− γ )ν +1 − α 1 α

αν +1 − α 11 (N it )

ναν +1 − α 1

log Y it = να

ν + 1 −α 1 log(N it ) + ν + 1

ν + 1 −α 1 log A t Z it C

α ( 1θ

− γ )ν +1

t

Aggregation: Note that aggregate revenue equals aggregate output, which implies

Y 1− 1

θt = ( Y t )

α 1 ( 1θ

− γ )ν +1 − α 1 α

α 1ν +1 − α 11 (A t )(1− 1

θ )( δ +1) i (Z it )(1− 1θ ) ( δ +1) (N it )νδ di

In logs,

yt 1 − 1θ −

α 11θ −γ

ν + 1 −α 1= 1 −

1θ

(δ + 1) a t + log i Z 1− 1

θit E it X 1+ δ

it

νδ(1 − α 1 )(1+ δ ) di + const.

Conjecture that yt = ψ0 + ψ la

lt + ψ ul

a ult + ψ−1 a t −1 + ψη ηt where we need to solve for the coefficients. This implies

x it =1θ −γ ψ0 + ψ

la

lt + ψ

ula

ult + ψ−1 a t −1 + ψη ηt + 1 −

1θ ρa a t −1 +

lt +

ult + 1 −

1θ z it

= 1θ −γ (ψ0 ) + ψ−1

1θ −γ + 1 −

1θ

ρa a t −1 + 1θ −γ ψ l

a + 1 − 1θ

lt +

1θ −γ ψ ul

a + 1 − 1θ

ult + 1 −

1θ

z it + 1θ −γ ψη ηt

= π0 + π−1 a t −1 + π ul ult + π z z it + π l l

t + π η η

Let W it = C 1θ −γ

t A1− 1

θt Z

1− 1θ

it

1+ δ

w it = (1 + δ ) x it = (1 + δ ) π 0 + π−1 a t −1 + π ul ult + π z z it + π l l

t + π η η

45



Standard gaussian updating implies

t |I it N τ η τ s

aτ η + τ s

aτ η τ s

aτ η + τ sa

+ τ la

r a

s iat , 1

τ η τ sa

τ η + τ sa+ τ la

+ 1τ ul

aand ηt |I it N

τ sa τ laτ l

a + τ sa

τ η + τ sa τ la

τ la + τ s

a

r η

s iat , 1

τ sa τ l

aτ la + τ sa

+ τ η

z it |I it N τ sz

τ sz + τ lz

r z

s izt , 1

τ sz + τ lz+ 1

τ ulz

π l lt + πη ηs iat

N 00 ,

(π l )2

τ la

+ π 2

ητ η

π l

τ la+ π η

τ ηπ l

τ la

+ π ητ η

1τ l

a+ 1

τ η + 1τ s

a

The projection theorem implies:

π l lt + π η η

|I it

N Σ 12 Σ −1

22

∆ 1

s iat ,π l 2

τ la

+ π 2

η

τ η −Σ 12 Σ −1

22 Σ 12

where Σ 12 = π l

τ la

+ π ητ η

and Σ 22 = 1τ la

+ 1τ η

+ 1τ sa

. This implies

E it (w it ) = (1 + δ ) ( π 0 + π−1 a t −1 + π z r z s izt + ∆ 1 s iat )

V ar it (w it ) = (1 + δ )2 π ul 2

τ ula

+ (1 + δ )2 π 2z

1τ sz + τ lz

+ 1τ ul

z+ (1 + δ )2 (τ sa + τ η ) π 2 −2π η τ sa π + π 2

η (τ a + τ sa )τ sa τ a + ( τ a + τ sa ) τ η

= E it (W it ) = exp E it (w it ) + 12

V ar it (w it )

The rm idiosyncratic productivity z it and E it (w it ) are jointly distributed as

zitE it (w it ) | l

t , η t N 0

(1 + δ) π 0 + π−1 a t −1 + ∆ 1lt + ηt

,

1τ l

z+ 1

τ ulz

(1+ δ ) π z r zτ l

z(1+ δ ) π z r z

τ lz

(1+ δ ) 2 (∆ 1 ) 2

τ sa

+ (1 + δ)2 π 2z r 2

z 1τ lz

+ 1τ s

z

(26)We then have that

log i Z 1− 1

θit (E it [W it ])

νδ(1 − α 1 )(1+ δ ) di = log i exp z it 1 −

1θ

+ E it w it + 12

V ar it (w it ) νδ

(1 − α 1 ) (1 + δ)di

= π 2

z

2 1τ lz

+ 1τ ul

z+ π 0 + π−1 a t −1 + ∆ 1

lt + ηt

νδ(1 − α 1 )

+ · · ·Substituting the above expression in expression for output and comparing corresponding coefficients:

yt 1 − 1θ −

α 1 1θ − γ

ν + 1 − α 1= 1 −

1θ

(δ + 1) a t + log i Z 1− 1

θit E it X 1+ δ

it

νδ(1 − α 1 )(1+ δ ) di + const.

lt :: ψ l

a 1 − 1θ − δ

1θ − γ = 1 −

1θ

(δ + 1)+ νδ ∆ 1

1 − α 1=

π l − π z1θ − γ

1 − 1θ − δ

1θ − γ = 1 −

1θ

(δ + 1)+νδ π η r a + π l r η

1 − α 1

ult :: ψ ul

a 1 − 1θ −

α 11θ − γ

ν + 1 − α 1= 1 −

1θ

(δ + 1) = ψula =

ν + 1ν + 1 − α (1 − γ )

a t −1 :: ψ−1 1 − 1θ − δ

1θ − γ = 1 −

1θ

(δ + 1) ρa + π−1νδ

(1 − α 1 ) = ψ−1 =

ρa νδ1 − α (1 − γ )

46



ψη 1 − 1θ − δ

1θ − γ =

νδ(1 − α 1 )

∆ 1 = πη1θ − γ

1 − 1θ − δ

1θ − γ =

νδ π η r a + π l r η

1 − α 1

We can solve these 2 equations in 2 unknowns and there is a unique solution. I don’t report the coefficients here.

So, in stage 1, rm i’s objective is to maximize

E it U (C t ) Y 1θ

t A1

−1θ

t Z 1

−1θ

it (N it h it )α 1

−W it (h it ) N dit = E it Y

1θ

−γ

t A1

−1θ

t Z 1

−1θ

it (N it h it )α 1

−Y −γ

t W it (h it ) N dit

= N νδit α

α 1ν +1 − α 11 E it X 1+ δ

it − 1 + ν

ν N dit

{E it [W it ]}1

(1 − α 1 )(1+ δ )

Agents maximize

κ 1 = argmax κ 1 E {E it [W it ]}1

(1 − α 1 )(1+ δ )

= argmax Eexp 1

(1 − α 1 ) (1 + δ)(1 + δ) E it (x it ) +

(1 + δ)2

2 V ar it (x it )

= argmax exp 1

(1 − α 1 ) (1 + δ)(1 + δ)2

2

π ul 2

τ ula

+ π 2z

1τ sz + τ lz

+ 1τ ul

z+

τ sa 1 + τ η π 2 − 2π η τ sa π + π 2η (τ a + τ sa )

τ sa (τ a ) + ( τ a + τ sa ) τ η

E exp 1(1 − α 1 ) (1 + δ) {(1 + δ) ( π 0 + π−1 a t −1 + π z ρz z it −1 + π z r z s izt + ∆ 1 s iat )}

= argmax exp 1

(1 − α 1 ) (1 + δ)(1 + δ)2

2

π ul 2

τ ula

+ π 2z

1τ sz + τ lz

+ 1τ ul

z+

(τ sa + τ η ) π 2 − 2π η τ sa π + π 2η (τ a + τ sa )

τ sa 1 (τ a ) + ( τ a + τ sa ) τ η

E exp 1

(1 − α 1 ) (1 + δ) {(1 + δ) ( π 0 + π−1 a t −1 + π z ρz z it −1 )}E exp

12 (1 − α 1 )2 (1 + δ)2 (1 + δ)2 (π z r z )2 1

τ lz+

1τ sz

+ (∆ 1 )2 1τ la

+ 1τ η

+ 1τ sa

Agent has xed capacity and has to allocate the attention optimally. For this case, we can rewrite the aboveexpression as

κ 1 = arg maxκ 112 (1 −α 1 )2 (π z r z )2 1τ lz + 1τ sz + (∆ 1 )2 1τ la + 1τ η + 1τ sa

+ (1 + δ )2 (1 −α 1 )

π2z

τ sz + τ lz+


τ sa (τ a ) + ( τ a + τ sa ) τ η

= arg maxκ 1

∆ 21

1τ la

+ 1τ η

+ 1τ sa

+

(1 + δ ) (1 −α 1 )(τ sa 1 + τ η ) π 2 −2π η τ sa π + π 2

η (τ a + τ sa )τ sa (τ a ) + ( τ a + τ sa ) τ η

+ π 2

z r z

τ lzδν (27)

= arg maxκ 1

π η τ a + π l τ η2

τ η (τ a + τ sa ) + τ sa τ a τ saτ la τ η

+ (1 −δν )τ sa (π −π η )2 + τ η π 2 + π 2

η τ aτ sa (τ a ) + ( τ a + τ sa ) τ η

+ π 2

z r z

τ lzδν (28)

= arg maxκ 1

π η τ a + π l τ η2

τ η τ a + τ η τ sa + τ sa τ a τ sa

τ la τ η+

π 2z r z

τ lz(29)

Suppose the constraint is given by

12

log2 τ η τ sa

τ la (τ η + τ sa ) + 1

κ 1

+ 12

log2τ szτ lz

+ 1

κ 2

≤κ

Objective is given by

Min π η τ a + π l τ η2 2−2κ 1

τ la τ 2η+

π 2z

τ lz2−2( κ −κ 1 )

47



κ 1 =κ if x ≤2−2κ

κ2 − 1

4 log2 (x ) if x 2−2κ , 22κ

0 if x > 22κ

(30)

where x = π 2z

(π η τ la + π l τ η )2

τ 2η

τ la

τ lz

and all the coefficients are endogenously determined and they can be solved using xed

point. We can show that this xed point has a unique solution.

Total labor chosen by each rm is given by

log Inputs it = 1ν + 1 −α 1

log(X it ) + ν

(1 −α 1 ) (1 + ν ) log E it X 1+ δ

it

= π0 + π−1 a t −1 + π ul ul

t + π z z it + π l lt + π η η

ν + 1 −α 1+

ν (1 −α 1 ) (1 + ν )

((1 + δ ) ( π0 + π−1 a t −1 + π z r z s izt + ∆ 1 s iat ) + con

= const. + π −1 a t −1 + π ul ul

t + π z z it + π l lt + πη η

ν + 1 −α 1+ 1

(1 −α 1 ) ν

ν + 1 −α 1(π−1 a t −1 + π z r z s izt + ∆ 1 s iat )

= const. + πul ul

t

ν + 1 −α 1+

π−1 a t −1

1 −α 1+

π z z it + π l lt + π η η

ν + 1 −α 1+ 1

1 −α 1

ν ν + 1 −α 1

(π z r z s izt + ∆ 1 s iat )

Covariance of inputs across sectors is

Cov t = πul

ν + 1 −α 1

21

τ ul + 1

ν + 1 −α 1

2 π + ν1

−α 1

∆ 12

τ l +π η + ν

1

−α 1

∆ 12

τ η

We can prove that covariance of inputs increases with aggregate uncertainty and decreases with idiosyncratic uncer-tainty.

C Appendix - Learning from nancial marketsOne of the issues with my analysis in the paper is that rm managers are constrained to not learn from asset marketsby shutting down asset markets in stage 2 when they make real decisions. In this appendix, I relax this assumption anddevelop a model of asset markets with dispersed private information in a macroeconomic setting where rm managersalso learn from nancial prices when making their investment decisions. I derive a tractable equilibrium that has afeedback loop between investor trading behavior and rm real investment. 28

One reason rm managers don’t want to learn from asset markets is when it is costly to learn from asset markets (ra-tional inattention setting) and manager is indifferent between learning from asset market or other sources of information(like the way I modeled in the main body of the paper). In a rational inattention setting, cost of processing informationonly depends on the prior and posterior uncertainty and agent is indifferent between different sources of informationif decrease in uncertainty is the same. Moreover, since I assumed that there is only one source of macro information( t + ηt ), asset markets also convey the same information as public source of information and, hence, manager is trulyindifferent in this setting.

If learning from asset markets is free, then the analysis will be more interesting. since the price signal itself isendogenous. Before I go into the details, I will make some assumptions:

Assumption 1: I only model the aggregate stock market and do not model the individual stocks. Since I am modelingrm managers aggregation of information, it is illegal to trade on rm specic information and I assume that managersdon’t indulge in insider trading. So, they only trade based on the aggregate information they have.

Assumption 2: Assume that a random fraction of rm managers are hit by participation shock, which are orthogonalto all cash ow shocks and affect their ability to participate in the stock market. This is similar to the noisy supplyassumption in Grossman and Stiglitz (1980).

The stock market. My specic model structure in this subsection draws heavily from recent work by Albagli,Hellwig, and Tsyvinski (2011a) and Albagli, Hellwig, and Tsyvinski (2011b). For aggregate market, there is a unitmeasure of outstanding stock or equity, representing a claim on the market’s dividends. These claims are traded by allthe agents - imperfectly informed rm managers except for the ones hit by participation shock.

Every period, each manager decides whether or not to purchase up to a single unit of aggregate stock at the currentmarket price q t . This assumption is standard in the literature. Assume that Φ (w t ) of agents are not allowed to tradeeach period, where wt N 0, 1

τ w is i.i.d. and Φ denotes the standard normal CDF. This convenient transformationensures that the total demand of these traders is positive and less than one, the total supply.

28 Even in this economy, q theory doesn’t work because manager still has different beliefs about rms cashows thanthe market.

48



Each rms information set is given by

s iat = t + ηt + ν iat , where ηt N 0, τ −1η (31)

They also see the current stock price q t , or equivalently, place limit orders conditional on q t . My assumption is thatthe random variables ν iat and wt are independent of the fundamental t and the common noise in the rm’s privatesignal ηt . Aggregating demand of both traders and noise traders, the market clearing condition is

d (s iat , q t ) dF (s iat | t + ν t ) = 1 −Φ (w t )

where d(s iat , q t ) [0, 1] is the demand of investor i and F is the conditional distribution of investors’ private signals.The expected payoff to investor i from purchasing the stock is given by

E it [Πt ] = π ( t , η t , P t ) dH ( t , η t |s iat , q t )

The term π (.) denotes the expected current dividends of the stock as a function of fundamentals and stock price. Itis a function of stock price because it enters the rm’s information set and through that, inuences rm decisions. SinceI only model the trading of index, I assume that they trade on the cash-ows of the index. The distribution H is theinvestor i’s posterior over fundamentals. Note that each rm is maximizing their expectation of payoff,

E it U (C t ) D t

since each rm manager is innitesimal, their trading gains wont effect aggregate consumption C t and, hence, rmmanager can be thought of as risk neutral. This implies:

d (s iat , q t ) =1 if E it [Πt ] > q t

[0, 1] if E it [Πt ] = q t0 if E it [Πt ] < q t

An investor purchases the maximum quantity allowed(1 share) when the expected payoff (conditional on her infor-mation) strictly exceeds the price, does not purchase any shares when the expected payoff is strictly less than the price,and is indifferent when the two are equal.

A REE is then a set of functions for prices q t , expected payoffs , investor decision rules d (.) and rms decisions, suchthat, for any history of shocks, all of them are behaving optimally and market clearing sets the prices.

I conjecture that equilibrium outcomes have the following two properties: 1. trading decisions of investors arecharacterized by a threshold rule, i.e., there is a signal s t such that only investors observing signals higher than s t chooseto buy, and 2. the market price is an invertible function of s t .

Aggregating the demand decisions of all investors, market clearing then implies

1 −Φs t − t −ηt

σv= 1 −Φ (wt )

which leads to a simple characterization of the threshold signal

s t = t + ηt + σν w t

This denes a monotonic relationship between q t and s t , implying that observing the stock price is informationallyequivalent to observing s t . The precision of signal „is decreasing in both the variance of the noise in rms’ private signalsand the size of the supply shock. This simple expression for price informativeness is the key payoff of the structure Ihave imposed on stock market trading.

Finally, note that the marginal investor, i.e., the investor whose signal s it = s t ,must be exactly indifferent betweenbuying and not buying. It follows then that the price q t must be equal to her expected payoff from holding the stock:

q t = U (C t ) D t dH ( t , η t |s iat , q t )

= U (C t ) D t dH ( t , η t |s iat , s t )

where with a slight abuse of notation, I replace q t with its informational content s t .

49





ψ − 1 1 −1

θ= 1 −

1

θρ a + π − 1 δ = 1 −

1

θρ a + ψ − 1

1

θ − γ + 1 −1

θρ a δ = ψ − 1 =

1 − 1θ (1 + δ ) ρ a

1 − 1θ + δ γ − 1

θ

π − 1 =1 − 1

θ (1 − γ ) ρ a

1 − 1θ + δ γ − 1

θ

ψp 1 − 1θ

= ∆ p δ =π w 1 − 1

θ1θ −γ

=π η τ a τ sa τ w + π e τ η τ sa τ w + π w √ τ sa (τ sa τ η + τ a (τ sa + τ η ))

τ sa (τ w + 1) τ η + τ a (τ sa (τ w + 1) + τ η )δ

ψη 1 − 1θ

= δ (∆ a + ∆ p ) =π η 1 − 1

θ1θ −γ

= δ (∆ a + ∆ p )

This implies πη = π −1 + γ . We can solve these 3 equations in 3 unknowns and there is a unique solution. Thesolution is

π = −( γ − 1) τ a ( θ − 1) τ η δ τ s

a ( γθ − 1) + θ − 1 + τ sa ( δ ( γθ − 1) + θ − 1) δ τ sa ( γθ − 1) + ( θ − 1) ( τ w + 1) + ( θ − 1) τ s

a τ η δ τ sa ( γθ − 1) + ( θ −

τ a ( θ − 1) τ η δ τ sa ( γθ − 1) + θ − 1 + τ sa ( δ ( γθ − 1) + θ − 1) δ τ sa ( γθ − 1) + ( θ − 1) ( τ w + 1) + τ sa τ η ( δ ( γθ − 1) + θ − 1) δ τ s

a ( γθ − 1) + ( θ

π w =( γ − 1) δ ( θ − 1) τ s

a ( γθ − 1) τ η τ w

τ a ( θ − 1) τ η δ τ sa ( γθ − 1) + θ − 1 + τ sa ( δ ( γθ − 1) + θ − 1) δ τ s

a ( γθ − 1) + ( θ − 1) ( τ w + 1) + τ sa τ η ( δ ( γθ − 1) + θ − 1) δ τ sa ( γθ − 1) + ( θ

π η =( γ − 1) δτ s

a ( γθ − 1) τ η δ

τ s

a ( γθ − 1) + ( θ − 1) ( τ w + 1)

τ a ( θ − 1) τ η δ

τ sa ( γθ − 1) + θ − 1 + τ sa ( δ ( γθ − 1) + θ − 1) δ

τ sa ( γθ − 1) + ( θ − 1) ( τ w + 1) + τ sa τ η ( δ ( γθ − 1) + θ − 1) δ

τ sa ( γθ − 1) + ( θ −

So, in stage 2, rm i’s objective is to maximize

E it U (C t ) C 1θ

t A1− 1

θt Z

1− 1θ

it N α 1it −W it N it = E it (X it ) N α 1

it −E it C −γ t W it N it

= E it (X it ) {E it [X it ] α 1}α 1

ν +1 − α 1 − {E it [X it ] α 1}ν +1

ν +1 − α 1

= E it [X it ]ν +1

ν +1 − α 1 αα 1

ν +1 − α 11 −α

ν +1ν +1 − α 11

Before solving information acquisition, it is useful to rewrite updating formula. Here, I assume that rm i is choosinga signal of precision τ sa and all other rms are choosing signals of precision τ sa 1 .

π + π η η + π w w |I it N

τ sa

−π w

τ s

a 1 τ a + τ η + π η τ a + π e τ η

τ a τ sa + τ sa 1 τ w + τ η + τ η τ s

a + τ sa 1 τ w

∆ a ≡ π r a + π η r η + π w r w

s iat +π w

τ s

a 1 τ a τ sa + τ η + τ sa τ η + τ sa 1 τ w π η τ a + π e τ η

τ a τ sa + τ sa 1 τ w + τ η + τ η τ sa + τ s

a 1 τ w

∆ p ≡ π ρ a + π η ρ η + π w ρ w

s pt, Σ

where Σ = −2π η (π w τ a √ τ sa 1 + π e (τ sa + τ sa 1 τ w )) + π 2

η (τ a + τ sa + τ sa 1 τ w )+ π 2

w τ a τ sa 1 + π 2e (τ s

a + τ sa 1 τ w )+ τ η (π e −π w √ τ sa 1 )2

τ a (τ sa + τ sa 1 τ w + τ η )+ τ η (τ sa + τ s

a 1 τ w )

The objective function can be written as

κ 1 = arg maxκ 1

E E it [X it ]1+ δ |a t −1 , z t −1i

= arg maxκ 1

E exp (1 + δ ) E it (x it ) + (1 + δ )2

V ar it (x it ) |a t −1 , z t −1i

= arg maxκ 1 exp(1 + δ )

2 π2z

1τ sz,i + τ z + Σ( i)

E exp ((1 + δ ) ( π 0 + π−1 a t −1 + π z r z,i s izt + ∆ a,i s iat + ∆ p,i s pt )) |a t −1 , z t −1i

= arg maxκ 1

exp(1 + δ )

2π 2

z 1

τ sz,i + τ z+ Σ( i)

exp ((1 + δ ) ( π 0 + π−1 a t −1 ))

exp(1 + δ )2

2(π z r z,i )2 1

τ z+ 1

τ sz,i+ (∆ a,i + ∆ p,i )2

τ a+ (∆ a,i + ∆ p,i )2

τ η+ (∆ a,i )2

τ sa,i+ (∆ p,i )2

τ sa, −i τ w

Agent has xed capacity and has to allocate the attention optimally. For this case, we can rewrite the above

51





Table 1: Correlation matrix

The table presents the correlation matrix of uncertainties proxies I use. (Market refers to proxies constructedusing market data. Real refers to proxies constructed using real data. Refer to variable denitions for moreinformation on each proxy)

Aggregate(Real) Aggregate (Mkt.) Idiosyncratic (Mkt.) Idiosyncratic (Real)

Aggregate (Real) 1

Aggregate (Mkt.) 0.342*** 1

Idiosyncratic (Mkt.) 0.169*** 0.447*** 1

Idiosyncratic (Real) 0.135*** 0.230*** 0.512*** 1

53



Table 2: Sectoral covariance of inputs and outputs-Aggregate

The table presents the results from estimating the regression equation:

Covsect = α + β i σ

seci,t + β a σa,t + t

Panel A: Inputs The sample used is yearly data for 35 sectors to estimate Covariance of inputs. The dependent variable is the covariationof inputs across sectors (Refer to equation 18). The independent variables are aggregate uncertainty and idiosyncraticuncertainty. I use various proxies for each uncertainty (Market refers to proxies constructed using market data. Realrefers to proxies constructed using real data. Refer to variable denitions for more information on each proxy). Differentcolumns correspond to various permutations of proxies.

(1) (2) (3) (4)Aggregate Uncertainty (Market) 0.575*** 0.656***

(0.156) (0.215)Industry Uncertainty (Market) -0.054 -0.135

(0.148) (0.168)Industry Uncertainty (Real) -0.634*** -0.438*

(0.197) (0.243)Aggregate Uncertainty (Real) 0.234 0.043

(0.168) (0.252)Constant -0.129 -0.138 0.000 0.164

(0.152) (0.210) (0.160) (0.265)Observations 32 20 39 20Adjusted R 2 0.273 0.398 0.004 0.070

Panel B: Outputs The dependent variable is the covariation of output across sectors (Refer to equation 18). The independent variables areaggregate uncertainty and idiosyncratic uncertainty. I use various proxies for each uncertainty (Market refers to proxiesconstructed using market data. Real refers to proxies constructed using real data. Refer to variable denitions for moreinformation on each proxy). Different columns correspond to various permutations of proxies.

(1) (2) (3) (4)Aggregate uncertainty (Market) 0.569*** 0.503**

(0.159) (0.230)Idiosyncratic uncertainty (Market) -0.115 -0.180

(0.150) (0.169)Idiosyncratic uncertainty (Real) -0.562** -0.360

(0.211) (0.234)Aggregate uncertainty (Real) 0.178 -0.136

(0.169) (0.243)Constant -0.158 -0.037 -0.000 0.290

(0.155) (0.224) (0.161) (0.255)Observations 32 20 39 20Adjusted R 2 0.260 0.268 -0.006 0.078

54



Table 3: Sectoral covariance of inputs - Panel data


Cov st = α s + β t + γ i σ s

i,t + γ a σa,t + s,t

The sample used is yearly panel data for 35 sectors to estimate covariance of inputs. The dependent variable is thecovariation of inputs of sector s with aggregate inputs. The independent variables are aggregate uncertainty andidiosyncratic uncertainty of sector s . Different columns correspond to various specications of xed effects.

(1) (2) (3)Covariance Covariance Covariance

b/se b/se b/seIdiosyncratic uncertainty -0.00171*** -0.00315*** -0.00137***

(0.00019) (0.00098) (0.00018)Aggregate uncertainty 0.00056*** 0.00066***

(0.00020) (0.00017)Fixed effecs None S TAdjusted R2 0.150 0.145 0.138

55



Table 4: Aggregate Output and consumption

Panel A: Output The table presents the results from estimating the regression equation:

yt = α + β i σ seci,t + β a σa,t + t

The sample used is quarterly data of gdp from FRED (Federal Reserve Economic Data) . The dependent variable iscyclical component of GDP series. I lter the series using Hodrick and Prescott (1997) lter (HP). The independentvariables are aggregate uncertainty and idiosyncratic uncertainty. I use various proxies for each uncertainty (Marketrefers to proxies constructed using market data. Real refers to proxies constructed using real data. Refer to variabledenitions for more information on each proxy). Different columns correspond to various permutations of proxies. Thestandard errors reported are newey-west standard errors with 20 lags.

(1) (2) (3) (4)b/se b/se b/se b/se

Aggregate Uncertainty (Real) -0.183* -0.139

(0.106) (0.102)Idiosyncratic Uncertainty (Market) 0.598** 0.931***(0.242) (0.235)

Idiosyncratic Uncertainty (Real) 0.019 0.210**(0.112) (0.094)

Aggregate Uncertainty (Market) -0.389*** -0.240***(0.092) (0.084)

Constant 0.167* 0.018 0.246*** 0.072(0.097) (0.074) (0.086) (0.074)

Observations 174 152 150 148

Panel B: Consumption The dependent variable is cyclical component of consumption series. I lter the series using Hodrick and Prescott (1997)lter (HP). The independent variables are aggregate uncertainty and idiosyncratic uncertainty. I use various proxiesfor each uncertainty (Market refers to proxies constructed using market data. Real refers to proxies constructed usingreal data. Refer to variable denitions for more information on each proxy). Different columns correspond to variouspermutations of proxies. The standard errors reported are newey-west standard errors with 20 lags.

(1) (2) (3) (4)cons_norm cons_norm cons_norm cons_norm

b/se b/se b/se b/seAggregate Uncertainty (Real) -0.271*** -0.230**

(0.098) (0.102)Idiosyncratic Uncertainty (Market) 0.757*** 0.989***

(0.229) (0.204)Idiosyncratic Uncertainty (Real) 0.020 0.249**(0.133) (0.106)

Aggregate Uncertainty (Market) -0.426*** -0.409***(0.090) (0.096)

Constant 0.178* 0.048 0.235*** 0.093(0.097) (0.089) (0.082) (0.084)

Observations 218 152 248 148

56



Table 5: Aggregate Capital Reallocation


Reallocation t = α + β i σ seci,t + β a σa,t + t

The sample used is annual data from compustat. I measure the amount of reallocation as sum of acquisitions and salesof property, plant and equipment (PPE). The dependent variable is cyclical component of Reallocation series. I lter

the series using Hodrick and Prescott (1997) lter (HP). The rst 4 columns use sum of sales of PPE and acquisitionas dependent variable. The next 4 columns use sales of PPE and the last 4 columns use acquisitions as dependentvariable. The independent variables are aggregate uncertainty and idiosyncratic uncertainty. I use various proxies foreach uncertainty (Market refers to proxies constructed using market data. Real refers to proxies constructed usingreal data. Refer to variable denitions for more information on each proxy). Different columns correspond to variouspermutations of proxies. The standard errors reported are newey-west standard errors with 10 lags

(1) (2) (3) (4) (5) (6)apc apc ppesc ppesc acqc acqcb/se b/se b/se b/se b/se b/se

Aggregate Uncertainty (Real) 0.014 0.004 0.027

(0.037) (0.021) (0.052)Idiosyncratic Uncertainty (Mkt) 0.255*** 0.233* 0.195*** 0.114* 0.265** 0.285*

(0.088) (0.119) (0.050) (0.060) (0.122) (0.167)Aggregate Uncertainty (Mkt) 0.027 0.078** -0.004

(0.067) (0.034) (0.095)Constant 0.059 0.068 0.047* 0.035 0.058 0.081

(0.042) (0.044) (0.024) (0.022) (0.058) (0.062)Observations 44 38 44 38 44 38Adjusted R 2 0.147 0.153 0.250 0.384 0.080 0.071

57



Table 6: Determinants in dispersion of CAPM betas


Disp(CAPM betas) t = α + β 1Aggregate Uncertainity t + β 2Idiosyncratic Uncertainity t + t

Panel A: Using Returns data to estimate Idiosyncratic uncertaintyThe sample used is daily data for 49 industry portfolios to estimate dispersion in betas. I use aggregate uncertainitymeasure from Jurado, Ludvigson, Ng (2014) and Idiosyncratic uncertainity as proxied by industry volatility computedas in Campbell, Lettau, Malkiel and Xu (2001).

(1) (2) (3) (4)std_beta std_beta std_beta std_beta

b/se b/se b/se b/senber_rec 0.008

(0.008)

Aggregate Uncertainty (Real) -0.019*** -0.030***(0.003) (0.003)Idiosyncratic Uncertainty (Mkt) 0.081*** 0.109***

(0.007) (0.009)p/d ratio -0.036*

(0.020)Constant 0.320*** 0.321*** 0.341*** 0.363***

(0.003) (0.003) (0.003) (0.009)Observations 651 651 651 648Adjusted R 2 -0.000 0.059 0.158 0.288

Panel B: Using Census data to estimate idiosyncratic uncertainty as in Bloom et al (2015)Idiosyncratic uncertainty is the standard-deviation of the innovations to establishment TFP in each industry-year in the Annual Survey of Manufacturing. I compute yearly industry uncertainty as mean uncertainty overall industries.

(1) (2) (3) (4)std_beta std_beta std_beta std_beta

b/se b/se b/se b/senber_rec 0.002

(0.009)Aggregate Uncertainty (Real) -0.010*** -0.015***

(0.003) (0.003)Idiosyncratic Uncertainty (Real) 0.021*** 0.023***

(0.003) (0.003)Constant 0.310*** 0.313*** 0.310*** 0.315***

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