E FFICIENCY WITH E NDOGENOUS I NFORMATION C HOICE∗ L UIS G ONZALO L LOSA†

V ENKY V ENKATESWARAN‡

C ENTRUM G RADUATE B USINESS S CHOOL

NYU S TERN

S EPTEMBER 1, 2017

Abstract Do private incentives to acquire information reflect the full social value of such information? We show that the answer to this question is typically negative in a standard business cycle model where firms make production/pricing decisions under imperfect information about aggregate productivity. Importantly, this is true even when the ex post sensitivity of actions to information is socially optimal. The wedge between private and social value of information has 3 components related to market power, coordination incentives and ex-post inefficiencies in the use of such information. The first two reduce the private value of information relative to its social value. When firms choose labor input, only these two forces are present, leading to underacquisition of information in equilibrium. Under nominal price setting, actions are inefficiently too sensitive to private signals. This reduces the social value of information, working against the first two effects and making the overall sign of the inefficiency in information acquisition ambiguous. Finally, we characterize optimal policy: with labor input choice, a constant revenue subsidy that corrects the market power distortion is sufficient to restore efficiency. With price-setting, the monetary authority can achieve first best by targeting price stability only if has perfect information. Otherwise, efficiency requires countercyclical prices and revenue subsidies. JEL Classification: D62, D82, E31, E32, E62 Keywords: Incomplete information, Costly information, Externalities, Business cycles, Optimal policy. ∗

This paper supersedes an earlier working paper ’Efficiency of Information Acquisition in a Price-Setting Model”.

We thank Manuel Amador, Andrew Atkeson, Christian Hellwig, Alex Monge-Naranjo, Alessandro Pavan, Andres Rodgriguez-Clare, Laura Veldkamp, Pierre-Olivier Weill, Jennifer La’O, Mirko Wiederholt and Jose Lopez for helpful discussions and comments. We would also like to thank Juan Martin Morelli for superb research assistance. † Email: [email protected] ‡ Email: [email protected]

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1

Introduction

We introduce costly learning about aggregate shocks in a standard business cycle model with dispersed information. Our main contribution is to highlight a novel source of inefficiency in equilibrium outcomes and characterize optimal policy responses. We show that incentives of firms to acquire private information about shocks are typically not aligned social incentives. In other words, the private value of information, the change in expected profits, is typically different from the social value, i.e. the change in expected social surplus. This leads to a suboptimal level of information in equilibrium, which in turn distorts both the average level of economic activity as well as its sensitivity to shocks. This inefficiency arises from 3 channels. All three are linked to imperfect substitutability, a standard assumption in this class of models. The first channel is a market power distortion - a monopolist does not internalize all the benefits of better aligning her decisions with fundamentals and therefore, underinvests in learning. The second channel arises because demand-driven complementarities in production choices also make coordinated information increases more valuable. Agents do not internalize this effect and therefore, tend to undervalue information. Importantly, these two effects are present even when the incentives to adapt actions to such information are undistorted. Finally, if these incentives are also distorted in equilibrium, i.e. information is incorporated into actions suboptimally, then ex-ante information choices are also distorted. The first of these channels is present even in partial equilibrium, but the other two are general equilibrium forces. We explore the effects of these forces in three variations of our setup. In the first, monopolistically competitive firms make labor input decisions in response to aggregate productivity shocks1 . This is very much in the spirit of the real business cycle literature, with the important difference that decisions are made under endogenous imperfect information. In this case, when competition is imperfect, we have an inefficient reduction in the average level of activity (relative to the welfare maximizing level), but the sensitivity of such activity to information is unaffected. Thus, only the first two of the three channels in the previous paragraph are present. Since they both reduce private incentives to learn (relative to the social optimum), the equilibrium features too little information. As the elasticity of substitution between products increases, both distortions become weaker and in the perfectly competitive limit, the wedge between private and social values disappears entirely. 1

In this framework, prices are fully flexible and monetary neutrality holds.

2

Next, we study a variant where firms set nominal prices under uncertainty about aggregate productivity (and let quantities be determined by realized demand conditions). Aggregate nominal demand is determined by a monetary policy policy (as a function of a noisy signal of aggregate conditions)2 . Now, in addition to distortions related to market power and the value of coordinated choices, equilibrium prices are excessively sensitive to private information, because firms do not internalize their contribution to overall uncertainty in the economy3 . More precise information exacerbates this inefficiency, partly (and in some cases, completely) offsetting the direct benefits of taking actions under better information. Private payoffs do not reflect the inefficiency and therefore, tend to overvalue information. With all three channels present and working in opposite directions, the net effect on information choice is ambiguous. We characterize the conditions under which we have over- or under-acquisition in equilibrium. Intuitively, when the elasticity of substitution between the firms’ products is low, market power and the value of coordination are high and so, the first two forces dominate, leading to under-acquisition. The opposite happens when the third channel is more powerful, e.g. when goods are highly substitutable. These results also extend to the third version of our model, a price-setting environment with aggregate nominal shocks. Finally, we characterize policies which can restore constrained efficiency. Under quantity choice, the standard complete information policy response to non-competitive behavior in a CES environment – a constant revenue subsidy equal to the markup – turns out to be sufficient to implement the constrained efficient outcome. In the price-setting variant, optimal policy depends on the information available to the monetary authority. If it is endowed with perfect information, then it can perfectly stabilize aggregate prices, which along with the constant revenue subsidy, can implement the first best. However, if the monetary authority also makes its decisions under uncertainty, first-best is no longer achievable and implementing the constrained efficient outcome requires a combination of fiscal and monetary policies – specifically, countercyclical revenue subsides as well as a monetary policy rule which targets a countercyclical aggregate price level. 2

Without noise, i.e. if monetary policy could perfectly adapt nominal demand to aggregate conditions, the opti-

mal policy would render the price-setting problem trivial by implementing complete price stability. Our specification prevents this and keeps nominal decisions interesting even when policy is set optimally. See Section 5. 3 What is crucial here is not the distinction of prices vs quantities per se but whether individual choices contribute to economy-wide uncertainty. For example, throughout our analysis, we assume that consumption risk is perfectly insured. This essentially implies that the ex-post dispersion in firm profits does not translate into inefficient dispersion in consumption. If, on the other hand, markets were incomplete, uninsured consumption risk would lead to externalities in information use, even in the quantity choice version.

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Our findings have a number of implications. First, they show that business cycles are typically constrained-inefficient once information choice is modeled explicitly, even if they exhibit efficient responses to fundamentals under exogenous information, as is the case in the quantity choice version. If the information was exogenous, the only inefficiency is a distortion to the average level – the fluctuations around that level are exactly what a constrained planner would choose. With endogenous information acquisition, however, the suboptimality of the ex-ante choice leads to fluctuations that exhibit too low sensitivity to aggregate shocks (relative to the planner’s solution). Second, policies aimed at correcting ex-post distortions have additional - and sometimes surprising - effects when information is endogenously chosen. For example, consider the effects of a revenue subsidy aimed at correcting the monopoly distortion. In the quantity choice model, where agents respond to information efficiently, such a policy always improves welfare. However, this is no longer true when actions respond inefficiently to information, as in the price-setting variant. Here, such a policy can actually reduce welfare. This counter-intuitive finding rests on the fact that information is endogenous and subject to multiple forces exerting opposing influences. The constant revenue subsidy eliminates one of these (market power) and therefore exacerbates the distortion in information choice (stemming from inefficient information use). This can, under some circumstances, more than offset the direct benefits of removing the monopoly distortion. It is important to note that this effect arises only when information choice is modeled explicitly. Optimal policy fixes both sources of inefficiency, which requires the combined use of fiscal and monetary instruments as discussed earlier. In a sense, this is the general message of our paper – in this environment, ex-ante incentives to acquire information are aligned with the social value of information if, and only if, all inefficiencies in ex-post responses are eliminated4 . It is worth emphasizing that we impose minimal structure on the learning technology beyond the assumption of interior solutions. Our general specification of information acquisition costs can accommodate several commonly used formulations (e.g. rational inattention5 , costly signals). Finally, while we focus on private signals for most of our analysis, the sources of inefficiency highlighted are relevant to the acquisition of public information as well. In particular, more public information can lead to a reduction in welfare because it crowds out private information production6 . Intuitively, this can occur when the equilibrium features an inefficiently low private 4

There is an important caveat to this insight – it need not hold when markets are incomplete, i.e. there is uninsur-

able consumption risk. Under these circumstances, ex-post efficiency does not guarantee ex-ante efficiency. See the Appendix for details. 5 Though additional restrictions may be necessary in some of these cases to ensure interior solutions. 6 See Colombo, Femminis and Pavan (2014) for a similar result, using a general quadratic specification for payoffs.

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learning. Related literature: This paper bears a direct connection to a large body of work embedding heterogeneous information in business cycle models7 . One branch of this literature8 takes the information structure as exogenous and derives implications for equilibrium responses. A second strand9 , closer to our work, endows agents with a learning technology and endogenously determines the extent of information, as in this paper. The main difference between our analysis and this latter group is that we are concerned primarily with efficiency. There are a few important exceptions10 . In recent papers contemporaneously developed with this one, Colombo, Femminis and Pavan (2014) and Mackowiak and Wiederholt (2011) also study normative properties of equilibrium information choice11 in settings with quadratic payoffs and Gaussian shocks. Working under the rational inattention paradigm, Mackowiak and Wiederholt (2011) study the optimality of attention allocated to rare events. Colombo, Femminis and Pavan (2014) provide a full characterization of the link between payoff externalities and efficiency for a general Gaussian-quadratic model and identify general sources of inefficiencies in the acquisition of information. The insights from these papers are complementary to ours, but differently from them, we work with fully articulated business cycle models and focus on the efficiency implications of various commonly used specifications – decision variables (nominal price setting versus quantity choice) and types of shocks (technology versus nominal)12 . We are able to derive analytical expressions without resorting to approximations under these different assumptions, allowing us to draw robust conclusions about welfare and set the stage for a quantitative evaluation. Our 7

An inexhaustive reading list will include Amador and Weill (2010), Angeletos and La’O (2009, 2011), Hellwig (2005),

Hellwig and Venkateswaran (2009), Lorenzoni (2009, 2010), Mackowiak and Wiederholt (2009, 2011), Moscarini (2004), Reis (2006), Roca (2010), Venkateswaran (2012) and Woodford (2003), to cite a few. 8 Angeletos and La’O (2009, 2011), Hellwig (2005), Hellwig and Venkateswaran (2009), Lorenzoni (2009, 2010), Roca (2010), Venkateswaran (2012) and Woodford (2003) belong to this group. 9 In Mackowiak and Wiederholt (2009, 2015), agents face a constraint on their ability to process information, while Hellwig and Veldkamp (2009), Gorodnichenko (2008) and Reis (2006) introduce explicit costs of planning or acquiring information. 10 Chahrour (2014) also looks at the welfare implications of costly public signals. Myatt and Wallace (2015) also analyze the social value of information in a differentiated product Cournot model. Moreover, they consider endogenous learning in an environment where information can display various degrees of publicity. 11 In an unpublished working-paper version of Hellwig and Veldkamp (2009), information acquisition is shown to be efficient in a beauty-contest model without externalities. 12 Colombo, Femminis and Pavan (2014) analyze a monetary economy, closely related to the third version of our model, as an illustration of how the insights from the Gaussian-quadratic model may help also in fully microfounded applications.

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results - on the sources and magnitude of the inefficiency - are directly interpretable in terms of model primitives, viz. preference and technology parameters. Paciello and Wiederholt (2014) study optimal monetary policy in a price-setting model where firms also choose how much attention to devote to aggregate conditions. They endow the central bank with perfect information and find that price stability is optimal (along with a constant subsidy to correct the monopoly distortion) in response to productivity shocks, but not necessarily in response to markup shocks. We characterize optimal policy under both price and quantity choice as well as under the assumption that the monetary authority is also subject to informational frictions. The latter feature lies at the heart of our finding that imperfect price stabilization and state-contingent subsidies are optimal even in response to productivity shocks. Our analysis thus provides a more comprehensive picture of social and private incentives to both acquire and use information in this otherwise standard business cycle environment. Finally, Angeletos, Iovino and La’O (2016) and Kohlhas (2017) study optimal policies in a environment with endogenous aggregation of information through market prices and/or reported macroeconomic statistics. Our work also complements earlier work on efficiency under exogenous information. Angeletos and Pavan (2007) show that information is used inefficiently in equilibrium when private and social incentives to coordinate are different. Hellwig (2005) and Roca (2010) analyze these incentives in a general equilibrium monetary model while Angeletos and La’O (2009) study them in a real business cycle context. In an important paper, Angeletos and La’O (2011) characterize optimal policy when both real and nominal decisions are subject to informational constraints. They show that the optimal policy is to replicate ‘flexible-price’ allocations, i.e. the monetary authority effectively mitigates or even eliminates the bite of informational constraints on purely nominal decisions. Optimal policy targets a negative correlation between aggregate prices and economic activity, a feature which emerges only because real decisions are subject to the informational frictions. We analyze a setting in which the bite of the informational frictions on nominal decisions cannot be undone by the monetary authority (because of its own informational constraints) and characterize efficient allocations and optimal policy. We show that a negative aggregate pricequantity correlation once again emerges as optimal under these conditions, even though no real variables are chosen under imperfect information. More importantly, however, the main focus of our paper is on social and private incentives to acquire costly information, a margin that is not directly analyzed in Angeletos and La’O (2011). Amador and Weill (2010) also study the efficiency properties of equilibrium when the extent of information is endogenously determined in equilibrium. They find that inefficiency occurs through learning from endogenous objects and

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not, as in this paper, through the acquisition of costly information. Our findings on the effects of market power on information choice also contribute to a broader agenda studying the efficiency implications of imperfect competition – see, for example, Bilbiie et. al. (2008) and the references therein. The rest of the paper is organized as follows. In section 2, we use a partial equilibrium model to show how market power distorts incentives to learn. Section 3 lays out a general equilibrium business cycle model with endogenous information choice. The next 3 sections consider three commonly used versions of this environment. Section 4 is a real business cycle model, where firms make labor input choices under imperfect information about aggregate productivity shocks. Sections 5 and 6 repeat the analysis under price-setting and nominal shocks respectively. Section 7 contains a brief conclusion. Proofs are collected in the Appendix.

2

A Simple Example

The purpose of this section is to show the connection between market power and the value of information in a simple partial equilibrium setting. We study the problem of a monopolist who makes production choices under uncertainty. She is endowed with a technology that transforms the numeraire, denoted N , into final goods, denoted Q, according to 1

Q = AN δ ,

δ > 1,

where A is a log-normally distributed technology shock13 , i.e. a ≡ ln A ∼ N (0, σa2 ). The profit of the monopolist is given by Π = P Q − N, where P is the price of the final good in terms of the numeraire. The monopolist faces a representative consumer with a utility function   θ−1 θ C= Q θ − P Q, θ−1

θ > 1.

Optimization by the consumer implies 1

P = Q− θ . The total social surplus is  U= 13

θ θ−1

 Q

θ−1 θ

− N.

Hereafter, variables in small cases denote variables in logs, i.e. x ≡ log (X)

7

Using the consumer’s optimality condition, we can rewrite this as   θ U= P Q − N. θ−1 Thus, in this constant demand elasticity environment, there is a simple relationship between the consumer’s utility and the firm’s revenue. Note that as θ → ∞, the difference vanishes, i.e. profits equal the social surplus. At the time of making her decision, the monopolist faces uncertainty about the realization of the technology shock. In particular, she only observes a noisy signal s = a + e, where e ∼ N (0, σe2 ). Formally, her problem is Π = max E [P Q | s] − N N    θ−1 1 θ | s − N, = max E AN δ N

where the operator E [·|s] represents the expectation conditional on the signal s. The first order condition is θ − 1 h θ−1 i θ−1 −1 E A θ | s N θδ = 1. θδ Standard properties of log-normal random variables then imply the following log-linear policy function n = κ + αs, where   δ(θ − 1) σa2 , α= 1 − θ + θδ σa2 + σe2       2 2  θδ θ−1 1 θδ θ−1 2 σa σe κ= log + . 2 1 − θ + θδ θδ 2 1 − θ + θδ θ σa + σe2 The expression for α has an intuitive interpretation. The first part

δ(θ−1) 1−θ+θδ ,

the full information

elasticity of employment to a technology shock, is downweighted by the signal-to-noise ratio. Before analyzing the value of information, it is instructive to examine the efficiency properties of this policy. Consider the surplus-maximizing response function, i.e. the solution to    θ max E P Q | s − N. N θ−1

8

It is easy to show that the solution takes the same log-linear form as the monopolist’s optimal policy, with n∗ = κ∗ + α∗ s, where α∗ = α, 

∗

κ

= κ+

θδ 1 − θ + θδ



 ln

θ θ−1

 > κ.

In other words, the elasticity of labor input with respect to the signal (and therefore, to the fundamental) is the socially optimal one but the monopolist chooses a suboptimally low average level of labor input. In this sense, the monopolist adapts her actions to the information efficiently even though she finds it optimal to restrict production on average. The private value of information to the monopolist is the sensitivity of the (ex-ante) expected profit to the variance of the noise in the signal. A straightforward application of the envelope theorem yields ∂EΠ α2 = − ∂σe2 2



1 − θ + θδ θδ

 EN < 0.

where E is the unconditional expectation (i.e. over the realizations of the aggregate shocks and the signals). The derivative is negative, i.e. profits decline with poorer information. Analogously, the social value is the change in expected total surplus i.e.

∂EU . ∂σe2

We can show that the expected social

surplus is proportional to - and strictly greater than - expected profits,  EU =  |



θδ θ θ−1 θ−1 − θδ θ−1 − 1

{z

> 1

1

  EΠ, }

which directly implies a wedge between private and social values of information    θ θδ − 1 θ−1 θ−1 ∂EU  ∂EΠ < ∂EΠ . = θδ ∂σe2 ∂σe2 ∂σe2 θ−1 − 1

(1)

Thus, welfare losses from noisier signals are greater than the reduction in profits. The source of the difference, the θ/ (θ − 1) term in the numerator, is the ratio of the consumer’s utility and revenue. Intuitively, the distortion arises because revenues do not fully reflect the utility gained by the consumer. Only in the limiting case of infinite demand elasticity does the private value coincide with the social value.

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Figure 1: Profits and Utility Figure 1 provides a graphical illustration for this intuition. The profit maximizing choice for a given level of the technology shock under perfect information, N , leads to a full information profit of Π. The corresponding social surplus is denoted U . Under imperfect information, the firm chooses a scale of production that is lower on average than under full information14 . The expected profit drops to Πe while the social surplus drops to U e . Since the utility function is steeper than the profit function at N , the private loss from less information (Π − Πe ) underestimates the social loss (U − U e ). As a result, the firm in a laissez-faire equilibrium will acquire less information than the welfare maximizing level. This distortion will play an important role in the richer demand structure in the following sections and will create incentives to underinvest in information. However, general equilibrium linkages will generate additional effects on the private and social values of information. As we will show, when firms choose quantities, these effects further reduce the private value and reinforce the under-acquisition incentives, whereas with price-setting, they lean in the opposite direction, making the net effect on information choice ambiguous. 14

To keep the graph simple, we approximate the firm’s decision under uncertainty with only two levels of labor

input - NL and NH . The exact distribution is log-normal, centered at a point which less than the full information level of employment (denoted by N e on the graph).

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3

A Business Cycle Model

In this section, we lay out a microfounded business cycle model with dispersed and endogenously acquired information. The flexible specification will allow us to examine the efficiency of equilibrium information choice under various assumptions about the nature of shocks (real vs. nominal) and decisions (prices vs. quantity). These cases will be examined in detail in the next 3 sections. Time is discrete. The economy is populated by a continuum of entrepreneurs and a final good producer. The entrepreneurs, or firms as we will sometimes refer to them in our exposition, each have access to a technology, which transforms labor into a differentiated intermediate good. These technologies are located on a continuum of informationally-separate islands, with one firm per island. Firms make two decisions - an ex-ante information choice, modeled as the precision of a private signal about an aggregate shock and an ex-post production/pricing choice. Preferences and Technology: Entrepreneur i enjoys a per-period utility according to15 2 Ci − Ni − υ(σei ),

where Ci is consumption of final goods and Ni the labor input16 . The last term is the cost of acquiring private information17 . The agent is subject to a budget constraint P Ci = Pi Yi . Production of intermediate goods is described by a decreasing returns to scale production function: 1

Yi = ANiδ , where δ > 1 and A is aggregate productivity. 15

The absence of curvature is an important assumption. Note that this does not require the absence of risk aversion

at the aggregate level. For example, if the entrepreneurs were members of a representative household, or had access to complete markets, we would still have linearity at the individual level and our results will go through. The case with incomplete markets and curvature at the individual level has different efficiency properties, and is analyzed in detail in the Appendix. We will return to this point in the following section. 16 We model the entrepreneur as using his own effort in production. This backyard production specification is primarily for simplicity. It is possible to introduce explicit (island-specific) labor markets and our results go through almost exactly. 17 Though we consider only private signals, the analysis can be extended easily to include public signals. In an earlier working paper version, we show how the underlying channels of inefficiency are relevant for the acquisition of public information as well.

11

The final good is a CES composite of the intermediate goods Z Y = 0

1

Yi

θ−1 θ

θ  θ−1 di ,

where the parameter θ is the elasticity of substitution between intermediate goods. Throughout the paper, we will assume that θ > 1. Finally, aggregate variables are linked by the following ad-hoc cash-in-advance constraint on total nominal spending P Y = M, where M is the level money controlled by the monetary authority. In the following three sections, we study in detail 3 versions of this general framework: • Quantity (labor input) choice with aggregate productivity shocks • Price choice with aggregate productivity shocks • Price choice with aggregate nominal shocks

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Model I: Quantity choice with productivity shocks

In this version, the only source of aggregate uncertainty is the level of aggregate technology A. Without loss of generality, we assume that the monetary authority sets a constant money supply, ¯ ∀t18 . Note that under complete information, this is basically the canonical real i.e. M = M business cycle model, with monopolistic competition replacing the standard competitive representative firm assumption19 . Firms observe a private signal about the aggregate productivity shock and choose labor input. Then, production takes place, firms sell their output and buy the final good for consumption. Figure 4 shows the timing of events in each period. We will show that information about the aggregate shock is reflected efficiently in actions, but the incentives to learn are suboptimally low. As a result, the laissez-faire equilibrium with endogenous information exhibits inefficient fluctuations, even though the same economy under the assumption of exogenous information does not. The intuition is similar to the simple example 18 19

This assumption is innocuous to our analysis because money neutrality holds under flexible prices. Angeletos and La’O (2009) study a similar environment with dispersed but exogenous information. The main mod-

eling difference is that they have many firms and labor markets on each island. Additionally, their economy admits a representative consumer. As mentioned earlier, our results are robust to incorporating these features.

12

Period t, Stage I

Period t, Stage II

Period t, Stage III

Period t + 1, Stage I

Agents choose information

Signals realized Labor input chosen

Shocks revealed Production and consumption

...

Figure 2: Timeline of Events in the previous section - imperfect substitutability leads to a wedge between the private value of information and its the social value. As a result, agents in equilibrium expend a suboptimally low level of effort in information acquisition. Only in a limiting case, as goods become perfect substitutes, does the equilibrium achieve efficiency.
Aggregate productivity is log-normally distributed, i.e. log A ≡ a ∼ N 0, σa2 . For simplicity, we focus on the case where this is an i.i.d shock, but it is possible to extend the analysis and the results to an AR(1) specification. For example, if we assume a−1 is common knowledge, our results go through exactly with the aggregate shock now interpreted as the current innovation to the aggregate productivity level. Information structure: Before choosing labor input, each agent observes a private signal si about the current productivity shock si = a + ei ,
2 . In equilibrium, the variance of the noise term, σ 2 , is chosen ex-ante by where ei ∼ N 0, σei ei firm i. Optimality: The competitive firm producing the final good solves Z

1

max P Y −

Pi Yi di , 0

1

Z Y = 0

Yi

θ−1 θ

θ  θ−1 di ,

where Pi is the price of intermediate good i. Optimality yields the usual demand function for good i  Yi =

Pi P

−θ (2)

Y .

Substituting from the budget constraint, we can write the intermediate producer’s objective in Stage II as follows:  Πi = max Ei Ni

1 Pi ANiδ − Ni P

13

 ,

where the operator Ei (·) represents the expectation conditional on firm i’s information Ii , i.e. Ei (·) ≡ E (· |Ii ) .The firm’s information set consists only of the private signal. Substituting from the demand function (2) " Πi = max Ei Ni

Yi Y

− 1

#

1 δ

θ

ANi − Ni

(3)

.

The solution is to choose an input level that equates expected marginal revenue to marginal cost  Ei

θ−1 δθ



1 θ

Y A

θ−1 θ

θ−1−θδ δθ



Ni

(4)

= 1.

Rearranging, 1+θδ−θ δθ

Ni

=

θ − 1 h 1 θ−1 i Ei Y θ A θ . δθ

(5)

Information acquisition: In the first stage of each period, before signals are realized, each agent chooses the extent of information to acquire, taking as given choices of other firms in the economy. The unconditional expectation of profits is denoted
ˆ i σ 2 , σ 2 ≡ EΠi , Π ei e

(6)

where E takes expectations over the realizations of the aggregate shocks and the signals. The problem of the agent in the first stage can then be written as:

ˆ i σ2 , σ2 − υ σ2 , max Π ei e ei

2 ∈ R σei +

where υ (·) is the cost of information, with υ 0 (·) < 0. Our focus in this paper is on differences between the social and private value of information, so we wish to impose as little structure as possible on the cost of information. We will, however, restrict attention to cost functions which lead to interior solutions to the above information choice problem20 , i.e. where optima are characterized by ˆi ∂Π 0 2 2 − υ (σei ) = 0 . ∂σei

4.1

(7)

Equilibrium

A equilibrium is (i) a set of information choices for each firm (ii) island-specific labor inputs as functions of the signal on the island (iii) aggregate consumption and output as functions of the 20

A sufficient condition is that the cost function is sufficiently convex, i.e.

is concave.

14

ˆ ∂2Π 2 ∂σ 2 ∂σei ei

2

υ − ∂σ2∂ ∂σ 2 < 0, so that the objective ei

ei

aggregate state such that: (a) the labor input is optimal, given island-specific information and the functions in (iii) above, (b) taking the behavior of aggregates in (iii) as given, the information choice in (i) solves the Stage I problem, (c) markets clear and (d) the functions in (iii) are consistent with individual choices. We focus on symmetric equilibria, where all agents acquire the same amount of information in stage I and follow the same strategies in stage II. The characterization of the equilibrium in stage II essentially follows the same procedure as in Angeletos and La’O (2009). We begin with a conjecture that, in equilibrium, firms follow a symmetric labor input policy of the form (8)

ni = k2 + αsi ,

where k2 and α are coefficients to be determined in equilibrium. The former determines the (unconditional) average level of (the log of) employment, while the latter is the elasticity. The details of this guess-and-verify approach are in the Appendix. The expressions for the response coefficients are given in the following result. Proposition 1 Conditional on a symmetric information choice σe2 , equilibrium labor input is given by (8), with 

δ  >0, δ − 1 σa2 +       1  α 2 θ − 1 1 (θ − 1) 2 2 2 + − 1 α σe + 1+ − α σa2 . = δ log θδ 2 θδ 2 2 δ

α = (δ − 1)k2



σa2  1+θ(δ−1) 2 σ e θ(δ−1)

The expression for α has an intuitive interpretation. The first part

δ δ−1

(9) (10)

is simply the full infor-

mation elasticity of employment to a productivity shock. Under incomplete information, this is downweighted by the second part, an adjusted signal-to-noise ratio. The adjustment essentially increases the weight of the noise (by a factor

1+θ(δ−1) θ(δ−1)

> 1), reflecting the well-known effect of

strategic complementarities. In other words, firms in this economy have an incentive to coordinate their actions (due to the imperfect substitutability of the goods they produce). Since the informational friction dampens the overall response of the economy to the fundamental, agents find it optimal to respond less than one-for-one to their expectations of the fundamental. Finally, we characterize the information acquisition decision in stage I. We begin by noting that the maximized stage II profit function, equation (6), depends on both the information choices of the agent herself as well as everybody else in the economy. The latter enter payoffs through the aggregate response coefficients, α and k2 . Ex-ante expected profits are obtained by taking expectations over the realization of the random variable Ei (a). 15

A symmetric stationary equilibrium can thus be represented as a fixed point problem in σe2  
ˆ σ 2 , α(σ 2 ), k2 (σ 2 ) − υ σ 2 , σe2 = argmaxσ2 Π ei e e ei ei

where α and k2 depend on σe2 as in (9)-(10). We restrict attention to cases where the solution to the information choice problem lies in the interior. Then, the fixed point problem reduces to (details in the Appendix):   1 θ−1 ˆ 2 − Πα = υ 0 (σe2 ), 2 δθ

(11)

ˆ is the unconditional expected profit and α is the equilibrium response coefficient. Since where Π both these objects are themselves functions of σe2 , this is a fixed point relation in σe2 and completes the characterization of equilibrium.

4.2

Efficiency in information use

We now turn to efficiency. We compare the equilibrium response function to that of a utilitymaximizing planner21 . Importantly, the planner is assumed to be information-constrained, i.e. cannot pool information across islands but is free to choose how agents respond to the signals. To facilitate the comparison, we restrict attention to symmetric log-linear policy rules of the form ni = k˜2 + α ˜ si .

(12)

Then, it is straightforward to derive the aggregate labor input, consumption and welfare are   1 2 2 ˜ N = exp k2 + α ˜a + α ˜ σe , 2 "    2 # k˜2 1 θ − 1 α ˜ 2 α ˜ C = Y = exp 1 + a+ + σ , δ δ 2 θ δ2 e and the corresponding ex-ante expectations     1 2 2 ˆ k˜2 , α N ˜ = E(N ) = exp k˜2 + α ˜ (σa + σe2 ) , 2 "  # 2   2 ˜2 1  θ − 1  α 1 α ˜ k ˜ Cˆ k˜2 , α ˜ = E(C) = exp 1+ σa2 + + σ2 , 2 δ δ 2 θ δ2 e ˆ. U = Cˆ − N 21

(13)

This subsection is an application of the welfare results in Angeletos and La’O (2009).

16

The socially optimal response function is characterized by coefficients α∗ and k2∗ that maximize utility, i.e.     ˆ k˜2 , α Cˆ k˜2 , α ˜ −N ˜ .

(α∗ , k2∗ ) = argmaxk˜2 ,α˜ The optimality conditions of this problem are

ˆ, Cˆ = δ N      ∗  ∗
α θ−1 α 2 1 ˆ α∗ σa2 + σe2 . 1+ σa2 + σe = N Cˆ δ δ θδ δ Using the first equation, the second condition can be rewritten as     α∗ θ−1 2 ∗ 2 ∗ 1+ . − α σa = α σe 1 − δ θδ

(14)

(15)

This equation reflects the trade-off faced in the choice of α. The left hand side is the (marginal) benefit from a stronger response to the fundamental component of the signal while the right hand side is (marginal) cost of responding to the noise in the signals. Agents in equilibrium face a similar trade-off. As the Appendix shows, the characterization of the equilibrium α has a similar form,       i θ−1 h α θ−1 θ−1 2 2 1 + − α σa = ασe 1 − . θ δ θ θδ

(16)

Comparing (15) and (16), we see that the private benefits and costs of a stronger response are proportional to those faced by the planner, with a scaling factor (θ − 1) /θ. In other words, the trade-off faced by agents in equilibrium when choosing α is the same as the planner’s trade-off. As a result, market power does not distort the equilibrium α (relative to the planner’s solution). However, it does lead to an inefficient reduction in average level of activity. Formally, as the following result shows, k2 is inefficiently lower than k2∗ even though the equilibrium α coincides with the corresponding socially optimal coefficient, α∗ . Moreover, the difference between k2 and k2∗ is invariant to the information structure and vanishes only in the competitive limit, i.e. as θ → ∞. Proposition 2 Given σe2 , the planner’s optimal response coefficients are: α∗ = α, k2∗

(17)

δ = k2 + log δ−1

where α and k2 are as defined in Proposition 1.

17



θ θ−1

 ,

(18)

Thus, the market power distortion takes the form of a constant (i.e. invariant to information), which scales down the labor input. This result has an important implication - when information is exogenous, the average level of activity in this economy is inefficiently low, but fluctuations (measured by the variation in aggregate variables y, n etc.) are constrained efficient.

4.3

Efficiency of information choice

Next, we show that, despite the optimal response to signals ex-post, the ex-ante information acquisition decision is inefficient. Our benchmark is the level of information that maximizes ex-ante utility in a symmetric equilibrium, i.e.

max U σe2 − υ σe2 , σe2

where U is the expected utility characterized in (13). We restrict attention to the case where the solution to the above problem is interior, i.e. characterized by the first-order condition22 ∂υ ∂U = . 2 ∂σe ∂σe2

(19)

Comparing (19) to (7), it is easy to see that information choice is efficient if, and only if, the 2. ˆ marginal value to the planner, ∂U/∂σ 2 , coincides with the private value to the firm, ∂ Π/∂σ e

ei

The next proposition presents the main result of this section. It shows that, in any symmetric equilibrium, there is a constant wedge between the private value of information and its social value. Proposition 3 In a symmetric equilibrium, the private value of information is always less than its social value, i.e.   ∂U δ = 1+ ∂σe2 (θ − 1) (δ − 1)

ˆ ∂Π 2 ∂σei

! < 0

∀ σe2 ∈ R+ .

(20)

2 σe2 =σei

Therefore, the level of information acquired in equilibrium is inefficiently low. 22

As with the equilibrium information choice, a sufficient condition is that the cost function is sufficiently convex, i.e. ∂2U ∂2υ − <0. ∂σe2 ∂σe2 ∂σe2 ∂σe2

18

To gain intuition for the wedge, we decompose it into two components.    ! θδ θ   ˆ θ−1 θ−1 − 1 ∂U ∂ Π δ  1+ . = 2 θδ ∂σe2 (1 − θ + θδ)2 − δ ∂σei 2 2 θ−1 − 1 σe =σei | {z } {z } | GE

(21)

PE

The first component (PE) is the same as the wedge in (1) from the simple example in Section 2. However, general equilibrium interactions lead to a new source of inefficiency, reflected in the second component (GE). To show this more clearly, we compare the private value in a symmetric equilibrium to the change in expected utility from varying information on a particular island holding information on all other islands fixed. We first derive the objective for a planner who chooses information on island i to maximize total utility, taking as given information and equilibˆ can be rium behavior on all the other islands. The Appendix shows that this objective, denoted U, approximated23 by ˆ ≈ EC − EN + U



θ θ−1



 E

P i Yi P



 − ENi dj .

Note the similarity to the partial equilibrium model of section 2 - the planner’s objective is identical to the firm’s profit in (3), except for the factor θ/ (θ − 1) multiplying revenue. The Appendix follows the same steps as in section 2 to derive    ! θ θδ − 1 ˆ θ−1 θ−1 ∂U  = 2 θδ ∂σei 2 2 θ−1 − 1 σe =σei

ˆ ∂Π 2 ∂σei

! .

(22)

2 σe2 =σei

The intuition for this component is the same as in the simple example. Firm profits understate the total social benefit from production and therefore, marginal revenue is strictly less than marginal utility. Therefore, better information, or equivalently better alignment of actions with fundamentals, causes a smaller improvement in profits relative to total utility. As a result, the monopolist attaches a lower value to learning than the social planner. The second component of the wedge in (21) arises because a coordinated change in the information on all the islands has a bigger effect on utility (compared to the effects of varying information on a single island). This is because imperfect substitutability makes coordinated choices more valuable. Formally,   ∂U δ = 1+ ∂σe2 (1 − θ + θδ)2 − δ

ˆ ∂U 2 ∂σei

! (23) 2 σe2 =σei

It is easy to see that both components of the inefficiency wedge are decreasing in the elasticity of substitution, θ. Intuitively, higher θ reduces market power and thereby, the distinction between 23

The factor dj multiplying the second term reflects the fact that each island is ‘small’ relative to the whole economy.

19

marginal revenue and marginal utility. Greater substitutability also reduces the importance of coordinating choices across islands. In the perfectly competitive limit, as θ → ∞, the gap between the social value and the private value of information vanishes and the laissez-faire equilibrium is constrained-efficient. The implications for efficiency of equilibrium outcomes are immediate. When information is exogenous, the average level of activity is inefficiently low but fluctuations around that level are constrained efficient. However, this is no longer true when information is endogenous. Too little information is acquired in equilibrium and, through its effects on k2 and α, this suboptimality influences both the average level of activity in equilibrium as well as the elasticity with respect to the shock. The sign of the effect on the former is in general ambiguous, but the response coefficient α is lower, i.e. the sensitivity of employment (and therefore, of output) to the technology shock is inefficiently muted. This is a novel source of inefficiency in this class of models - one that is absent both under the canonical full information assumption as well as under exogenous information (e.g. Angeletos and La’O 2009). 4.3.1

Social value of information

A less obvious implication relates to the social value of public information. Suppose entrepreneurs also had access to a costless public signal about aggregate productivity in this environment z =
a + η where η ∼ N 0, ση2 . The object of interest is the effect of ση2 on welfare. If information were exogenous, increasing public information (i.e. reducing ση2 ) is always good for welfare. This is because all information - whether public or private - is used efficiently (in the sense that their effect on actions coincides with the socially optimal one). However, with endogenous information, more public information can reduce welfare. Intuitively, this occurs when it decreases the private value of information in equilibrium, exacerbating the under-acquisition of information. In other words, public information crowds out private information. Since equilibrium information is already inefficiently low, this leads to a first-order welfare loss, which can overwhelm the direct benefit of the public signal24 . Note that this effect is present only when information is endogenously chosen. We sketch the formal argument here. With endogenous information choice, the value of public information is,

  2
∂U ∂U dσe dU 0 2 = + − υ σe . 2 2 2 dση ∂ση ∂σe dση2

The first term captures the direct effects of public information on welfare holding private in24

A similar result also emerges in the general quadratic utility framework of Colombo, Femminis and Pavan (2014).

20

formation choices constant. This term is always negative, i.e. more public information increases welfare. The second term captures the indirect effect of public information through its effects on the acquisition of private information. If private information is at its utility maximizing level, then the indirect effect does not matter, i.e. the expression inside the square bracket is zero. This is just an application of the envelope theorem. When information is under-acquired, as is the case in our equilibrium, this expression is negative, see (7) and (20). The sign of the second term now depends on the sign of dσe2 /dση2 . This is of ambiguous sign, i.e. depending on parameters, more public information can encourage or discourage private learning, so dσe2 /dση2 ≷ 0. Putting these pieces together, we see that the net social value of public information can be positive or negative.   2 
dσe ∂U ∂U dU 0 2 = + − υ σe ≷ 0 2 2 2 dση ∂ση ∂σe dση2 | {z } |{z} | {z } <0

<0

≷0

When the crowding out effect is sufficiently strong, the indirect effect outweighs the direct benefits of more information, leading with lower welfare with better public information. Finally, we turn to the policy implications.

4.4

Policy

In this subsection, we characterize the optimal fiscal policy that restores constrained efficiency. Remarkably, the standard complete information policy response, a constant revenue subsidy equal to the constant markup, not only eliminates underproduction, but also leads to the socially optimal of information acquisition. For an arbitrary revenue subsidy Λ, the problem of the firm becomes25 Πi = max Ei [ΛPi Yi − Ni ] , Ni

It is straightforward to show that the level distortion in activity is removed, i.e. k2 equals k2∗ , if the subsidy satisfies Λ=

θ >1. θ−1

More interestingly, as the next result shows, this subsidy also eliminates (both components of) the wedge between private and social values, leading to both ex-post and ex-ante efficiency. Proposition 4 A symmetric equilibrium with a constant revenue subsidy Λ =

θ θ−1

is constrained efficient,

i.e. it attains the optimal allocation of the information constrained planner. 25

In addition, the lump sum transfer Λ

R

Pi Yi di is subtracted from the right hand side of the budget constraint.

21

Thus, in this environment, the standard complete information policy response is all it takes to restore efficiency in general equilibrium. As we will see in the following section, this is no longer the case under price-setting - precisely because of inefficiencies in the sensitivity of actions to information. This also turns out not to be the case when entrepreneurs are risk averse and markets are incomplete. We analyze this case in the Appendix and show that risk aversion brings additional externalities from the dispersion of individual actions, that distort the equilibrium α away from the socially optimal one. This inefficiency spills over to the information choice problem and correcting market power is no longer sufficient to implement the information-constrained optimal outcomes. In fact, with risk aversion, we show that even if all ex-post inefficiencies are corrected (except market incompleteness), ex-ante information is inefficient.

5

Model II: Price-setting

In this section, we study a version of the environment in the previous section where firms set nominal prices (instead of choosing labor input) under uncertainty26 . As before, aggregate pro
ductivity is log-normally distributed, i.e. log A ≡ a ∼ N 0, σa2 . The monetary authority receives a noisy signal of aggregate productivity and chooses aggregate money supply (Mt ). Apart from formalizing the realistic notion that policy is often constrained by limited information about the state of the economy, the noise in the signal will also serve to keep the pricing problem of firms interesting27 . Formally, the monetary authority follows a log-linear monetary policy rule 28 : m = αm s m

(24)


2 2 ≥ 0 indexes the where sm = a + ema denotes the signal and ema ∼ N 0, σma . The parameter σma degree of uncertainty faced by the monetary authority. Agents do not see this signal (or equivalently, the chosen value of m) when they make their information acquisition and pricing choices, 26

Whether firms compete by choosing prices or quantities is a matter of some debate. See Aiginger(1999) for a survey.

One of the studies cited in that paper describes a survey of Austrian manufacturing on their main strategic variable. About 38% of the 930 firms surveyed said they produce a specific quantity, thereafter permitting demand to decide price conditions while the remaining said they set prices leaving competitors and the market to determine quantity sold. 27 As we will see, in the absence of noise, i.e. if the monetary authority could choose Mt under perfect information, it can make the pricing problem trivial, eliminating any role for firm-level information. 28 It is straightforward to show that adding a constant term to the policy does not have an effect on allocations.

22

but they know the policy parameter αm . We begin by charactetizing the equilibrium given αm . We then turn to the choice of optimal policy. The intermediate goods producers set a nominal price and commit to producing any amount demanded at that price. The producer’s problem is given by:  max Ei Pi

Pi P

"

1−θ Y −

Pi P

−θ

Y A

#δ .

As before, optimality equates expected marginal revenue to expected marginal cost   h i δ −θ −θδ−1 θ−1 θδ Y (θ − 1) Pi Ei P . Y = θδPi Ei P Aδ

(25)

(26)

A comparison of (26) with (4) reveals an important difference between the price and quantity choice environments. When a firm chooses its labor input under uncertainty, its marginal cost (the term on the right hand side) is unaffected by the actions of other agents. This is no longer the case under price setting - the marginal cost to a firm from changing its own price depends on the aggregate price level P , i.e. the sensitivity of economy-wide prices to the fundamental affects each firm’s uncertainty about its own marginal cost. As we will see, this additional interaction leads to an externality, causing the ex-post response function to exhibit excess sensitivity to signals. Recall that each agent observes a private signal si about the current productivity shock. The solution strategy follows the same guess-and-verify procedure as in the previous section. We guess, and verify, that individual prices are set according to pi = k2 + αsi .

(27)

The expressions for the response coefficients in a symmetric equilibrium are collected in the following result. Proposition 5 In a symmetric equilibrium, firms follow a pricing rule of the form (27), with     2 δ σ   a  , α = αm − 1+θ(δ−1) δ−1 2 σa + σe2 δ−1   i 1 h θδ (δ − 1)k2 = log + σa2 δ 2 (1 + α − αm )2 − (αm − α)2 θ−1 2 
2 2 1 1 2 + (1 − θ) [1 − δ(1 − θ)] + θ2 δ 2 α2 σe2 + δ − 1 αm σma , 2 2
≡ κ α, σe2 , αm where σe2 is the variance of the error in agents’ signals. 23

As with quantity choice, the adjustment factor

1+θ(δ−1) δ−1

> 1 in the denominator of the expres-

sion for α reflects incentives to coordinate, stemming from strategic complementarities, which dampen responsiveness to the signal, in addition to the dampening due to the presence of noise. Note also that monetary policy (summarized by αm ) plays a role as well. For instance, if the monetary sets αm =

δ δ−1 ,

agents have no incentives to respond to their information, effectively

stabilizing prices. However, as we will show later in the section, when the monetary authority’s signal is noisy, this choice is not necessarily optimal.

5.1

Efficiency: Response to Information

In this section, we characterize the efficient allocation29 . We consider a social planner who maximizes social surplus by choosing the agents’ responsiveness to information as well as the level of private information30 . In addition, the planner also chooses monetary policy optimally. For clarity, we separate the social planner maximization problem in the following three stages. In the third and final stage, the planner optimally chooses responsiveness of prices to information, taking as given the policy coefficients and information precision. At the second stage, the planner chooses the welfare-maximizing level of private information σe2 . Then, in the first stage, the planner designs the optimal log-linear monetary policy rule by picking αm . The next result characterizes the optimal coefficients α∗ and k2∗ (the derivations are relegated to the appendix): Proposition 6 The utility maximizing response coefficients are     2 σ δ   a  , α∗ = αm − 1+θ(δ−1) δ−1 2 σa + θ σe2 δ−1  
θ (δ − 1)k2∗ = − log + κ α∗ , σe2 , αm , θ−1
where κ α∗ , σe2 , αm is the function characterized in Proposition 5.

(28)

Thus, the equilibrium features prices that are too responsive31 to signals, i.e. |α∗ | < |α|. In other words, monopolistically competitive firms respond to information suboptimally when set29 30

This subsection is an application of the welfare results in Angeletos and La’O (2009). One interpretation is that this represents the choice of an information-constrained planner, but is subject to all the

other equilibrium constraints. In particular, given a cross-sectional distribution of prices {Pi } , the aggregate price level P and output Y are determined by the zero-profit condition of the final goods producer and the quantity equation, respectively. 31 Hellwig (2005) arrives at a similar result in an environment with monetary shocks.

24

ting prices – recall that their response corresponded to the socially optimal one when the choice variable was labor input. The intuition is related to the marginal cost uncertainty mentioned earlier. Firms do not take into account their contribution to the uncertainty faced by other firms in the economy and as a result, set prices that are too responsive to private signals32 . The level coefficient k2 is also suboptimal – but now it includes both the usual markup distortion and the effects of the inefficient sensitivity to information. Importantly, the latter persist even as the former disappears, e.g. as θ tends to infinity. The next equation characterizes the optimal level of private learning σe2∗ at the efficient use of information: −


1 ∗2
δθ [1 + θ (δ − 1)] ∗ α U ≤ υ 0 σe2∗ 2 δ−1

where σe2∗ is the optimal information choice and U∗ is the maximum level of social surplus given the information constraint. The equation holds with equality at an interior solution, i.e. σe2∗ < ∞. The left hand side of the equation represents the social value of information at the optimal information use. Clearly, this social value is non-negative which means that, at the optimum, more private information cannot decrease aggregate utility. Finally, we characterize the optimal monetary policy rule. The utility maximizing response ∗ is given by : coefficient αm

∗ αm =



δ δ−1



σa2

 

σe2 θ

2 + σ2  2 σma a σe θ

h

h

1+θ(δ−1) δ−1

1+θ(δ−1) δ−1

i

+

i

 

σa2 2  2 σma σa2 +σma

To see the intuition behind this expression, note first that

δ δ−1

≥ 0.

(29)

is simply the full information

elasticity of output to a productivity shock under flexible prices. Under incomplete information, this is downweighted by the signal-to-noise ratio faced by the monetary authority,

σa2 2 +σ 2 . σma a

The

last term (in brackets) captures the effect of dispersed information and strategic complementarities, which dampen the responsiveness of nominal prices. They cause the monetary authority to further dampen the responsiveness of money supply to its own signal (notice that this term is < 1). The above equation also shows the effect of the imperfect nature of information available to the monetary authority. As a benchmark, suppose the monetary authority has complete information, 32

What is crucial for the efficiency results is not that the marginal cost is uncertain but that it depends on the actions

of other firms. Suppose firms committed to an output level (Yi ) in advance. Then, marginal cost is uncertain (because A is not known), but is unaffected by the behavior of other agents. As a result, the results from the labor input choice model go through. Alternatively, in the single firm environment of Section 2, the responsiveness of prices to the signal coincides with the socially optimal one.

25

2 = 0. Then, optimal monetary policy sets α∗ to σma m

δ δ−1 ,

which in turn implies that α∗ collapses to

zero and allocations converge to their first-best levels. In others words, by choosing policy to target full price stability, the monetary authority is able to implement the first best allocation. This also eliminates the need for information acquisition, i.e. the social value of information is zero33 . When the monetary authority’s information is imperfect, however, this is no longer optimal. Targeting price stability in this case leads to inefficient fluctuations in aggregate economic activity (stemming from the noise in the monetary authority’s signal and the quantity equation). The optimal policy trades off this inefficiency against the inefficient dispersion of relative prices (and through them, production). ∗ and α∗ imply that the aggregate price under optimal Finally, note that the expressions for αm

policy is countercyclical. This is consistent with the results in Angeletos and La’O (2011), though the driving forces are somewhat different. In that paper, the countercyclicality stems from the fact that firms make both real and nominal choices under imperfect information. In the quantity choice version of our model, only real decisions are subject to the informational friction, monetary policy is irrelevant. In this version, where only nominal prices are chosen under uncertainty, countercyclical prices emerge as an optimal response to the informational constraints faced by monetary policy.

5.2

Efficiency: Information choice

Next, we examine the efficiency of the equilibrium ex-ante information choice. Perhaps unsurprisingly, we will find that choice of signal precision in a laissez-faire equilibrium is not the socially optimal one. Recall that information choice was suboptimal in the quantity choice model, even with an optimal ex-post effect on actions, so the results in Proposition 6 further diminish the prospects for ex-ante efficiency. As in section 4.3, we compare the equilibrium information choice to the level that maximizes ex-ante utility. Under this metric, equilibrium information choice is efficient if, and only if, the 2 . Imporˆ marginal social value of learning, ∂U/∂σ 2 , coincides with the private value, ∂ Π/∂σ e

ei

tantly, we carry this evaluation at the equilibrium information choice. The next result shows that, generically, this is never the case. Proposition 7 In a symmetric equilibrium, the social and private value of information in equilibrium are 33


Recall that υ 0 σe2 < 0. Thus,

∂U∗ 2 ∂σe


> υ 0 σe2 which implies σe2∗ → ∞.

26

linked by the following relationship  2     ∂U σe dα dΠ δ 1 = 1+ +2 1+ 2 ∂σe2 (δ − 1) (θ − 1) θ (δ − 1) α dσe2 dσei σ 2 =σ 2 ei

e

Notice that, relative to the quantity choice case (proposition 3), the wedge under price-setting has an additional component - viz. the last term inside the square brackets. This is precisely the effect of suboptimal information use. Better information exacerbates this problem and therefore

a negative effect on welfare (since σe2 /α dα/dσe2 < 0). This reduces the social value of information relative to the private value and works against the market power/coordination forces identified in quantity choice case (which increase the social value relative to private value). As a result, the wedge can now be greater or less than 1. Whether we have over- or under- acquisition in equilibrium now depends on the relative strength of the market power/coordination effects described in the previous section and the effects of inefficient information use. Holding other parameters fixed , high θ strengthens the latter (and weakens the former). Intuitively, a higher elasticity of substitution makes realized production levels more sensitive to price differences, making marginal cost uncertainty particularly damaging. In the other direction, as we approach unit elasticity, θ → 1, the inefficiency becomes arbitrarily small34 (i.e. the equilibrium sensitivity converges to the planner’s) so only the first effect is present leading to under-acquisition. In fact, the inefficiency in information use can be so severe that it overwhelms the direct benefits of better information and make noise socially desirable, i.e. ∂U/∂σe2 > 0. Note that information 2 < 0. Obviously, if the social value is negative at the equiˆ is always privately valuable, i.e. ∂ Π/∂σ ei

librium choice of σe2 , the equilibrium information choice is trivially inefficient (increasing the noise in the signals raises utility and saves on information costs). We focus on the more interesting case, when the social value of the information acquired in equilibrium is positive (and, as before, optimal information choices are in the interior). A sufficient condition for a positive social value of information at the equilibrium σe2 is35 : Assumption 1 σa2 +

2−θ θ


 1+θ(δ−1)  δ−1

σe2 ≥ 0 .

Under this assumption36 , to determine whether there is over- or under-acquisition of information, we simply compare the social value of information to the private value. The next proposition shows that information acquired in equilibrium is typically inefficient, though the direction 34

As θ → 1, expenditure shares are close to constant, so the strategic linkage becomes very weak. The Appendix provides a formal statement of the conditions under which this occurs. 36 Note that this always holds for θ ≤ 2. Otherwise, we need σe2 to be sufficiently low. 35

27

is ambiguous. The result essentially divides the information space into two regions, depending on whether the equilibrium exhibits too much or too little information production. Only for a non-generic combination of parameters does the equilibrium choice of σe2 coincide with the utility maximizing level. Proposition 8 Suppose Assumption 1 is met. Then, the equilibrium features too much information relative to the utility maximizing level if the following condition holds at the equilibrium σe2 :   θ (δ − 1) δ 2 σe > σ2 . 1 + θ (δ − 1) 2 (θ − 1) (1 + θ (δ − 1)) − θδ a If the inequality is reversed, there is underacquisition. A full quantitative investigation is beyond the scope of this paper, but a rough calculation indicates that with parameter values commonly used in the literature, the condition in the above result is likely to hold. As an illustrative case, set θ = 4, δ = 1.5. Then, the variance in private signals only needs to be one-twelfth of the variance in aggregate productivity for the above condition to hold. Thus, we need only a very modest departure from full information to see over-acquisition. In other words, in the empirically relevant region of the parameter space, too much information about aggregate productivity is acquired in equilibrium. The implications of this finding for the constrained efficiency of fluctuations as well as the social value of public information are similar to that under the labor input choice model of the previous section, though the presence of both sources of inefficiency makes the overall sign ambiguous in general.

5.3

Optimal Policy

In this subsection, we show how the efficient allocation can be decentralized. Specifically, we will show that this requires state-contingent revenue subsidies (in addition to appropriately chosen monetary policy). We begin by discussing the standard non-state contingent revenue subsidy aimed at correcting the constant monopoly distortion. Recall from section 4.4 that, with labor input choice, this policy was enough to restore constrained efficiency. However, with price-setting, it can actually reduce welfare. To see why, note that subsidizing revenue increases the private value of information and leads to more information acquisition. However, with price setting, this additional investment in information acquisition can be socially suboptimal. To put it differently, in the laissez faire equilibrium, the two sources of inefficiency – market power and inefficient information use – work 28

Figure 3: Effect of subsidy on welfare against each other. If only one of them is removed, the economy bears the full brunt of the other, which could more than overcome the direct benefits of removing one. In other words, incomplete policy responses can do more harm than good. Importantly, this is the case even when monetary policy is set optimally. Figure 3 illustrates such a case. The top panel depicts information choice in equilibrium, where the marginal cost of information (υ 0 ) intersects the marginal benefit (π 0 ). The corresponding level of welfare is shown in the bottom panel. Without the subsidy (the solid lines), the equilibrium features over-acquisition of information (note that the variable on the x-axis is precision, the inverse of σe2 ). In fact, at the equilibrium choice, the social value is negative. The subsidy raises the private value of information and therefore, leads to more learning. The removal of the monopoly distortion to average production raises utility for all levels of information (the direct effect of removing the markup distortion), but the new equilibrium is associated with a lower level of welfare than without the subsidy (the point D in the bottom panel compared to C). We have shown that suboptimal policies lead to an inefficient amount of information acquired in equilibrium which then feedback into inefficient aggregate volatility. Next, we characterize policies that decentralize the efficient allocation. The monetary authority follows the efficient ∗ , characterized in equation (29) . The optimal revenue subsidy takes the form ΛAτ , policy rule αm

29

where the parameters Λ and τ index the level and sensitivity of the subsidy to fundamentals37 . The next result presents the values of Λ and τ , for a given level of noise in signals σe2 . Proposition 9 Given σe2 , equilibrium allocations coincide with the choices of the planner, i.e. (α, k2 ) = (α∗ , k2∗ ) if the revenue subsidy is given by ΛAτ where  ∗  α ∗ τ= − 1 [δ − αm (δ − 1)] < 0 , αeq   2   σa τ (αm ) [2 (αm − α∗ ) + τ (αm )] θ exp − . Λ= θ−1 2

(30)

The optimal revenue subsidy is decreasing in the technology shock A.This countercyclicality dampens the effect of the shock on firm’s profits and through that, fixes the excess sensitivity problem in equilibrium responses. The level coefficient Λ has the usual markup correction, with an adjustment for level effects arising from the responses to signals. More importantly, this combination of monetary and (state-contingent) fiscal policies also implements the socially optimal level of information acquisition, as the following result shows. Formally, Proposition 10 A symmetric equilibrium under the policy described in Proposition 9 and the optimal monetary policy is constrained efficient, i.e. it attains the optimal allocation of the information constrained planner. In other words, the general insight from the quantity choice model goes through here as well - fixing ex-post inefficiencies in equilibrium responses also aligns private and social benefits from learning, leading ex-ante efficiency. Unlike the quantity choice model however, this requires both monetary and fiscal policies to be state-contingent.

6

Model III: Price choice with nominal shocks

In this section, we briefly discuss the results under aggregate nominal shocks. The environment is identical to that of the previous section except that productivity is now constant (i.e. A = A) but aggregate nominal demand is stochastic. In particular, we substitute the monetary authority
2 . policy function, equation (24), with a random money supply, i.e. log M ≡ m ∼ N 0, σm 37

In the quantity choice model, the optimal policy is to set τ = 0 and Λ =

realization of the fundamental.

30

θ , θ−1

i.e. the subsidy is invariant to the

Intermediate goods producers choose nominal prices for their products and commit to producing any amount demanded at that price. Before setting prices, each firm observes a private noisy signal si about the current monetary shock si = m + ei ,
2 , with σ 2 indexing the information choice in stage I by firm i. where ei ∼ N 0, σei ei The appendix shows that the results from the previous section about aggregate productivity shocks extend to learning about nominal shocks as well. First, firms adjust their prices by too much in response to expected changes in money supply. The intuition is very similar to the productivity shocks case - firms do not fully internalize the effect of their pricing decisions on the marginal cost uncertainty faced by other firms. As a result, they react too much to private signals, relative to the planner’s solution. Second, equilibrium can feature both under- and overacquisition. This result mirrors the results under productivity shocks and price setting, see Proposition 8. This ambiguity comes from the two forces identified in the previous section. That is, better information reinforces the inefficiently high level of sensitivity and therefore, inflicts a negative effect on welfare. This reduces the social value of information relative to the private value and works against the market power/coordination forces identified in quantity choice case. Finally, a state-contingent policy is needed to offset the wedge between private and social value of information and ensure that signal precisions in equilibrium are socially optimal.

7

Conclusion

The preceding sections highlight a novel source of inefficiency in a business cycle setting used widely in modern macroeconomics. Ex-post inefficiencies in production/pricing decisions feed back into ex-ante incentives to invest in information, even when these inefficiencies do not distort the responsiveness to such information. This in turn leads to suboptimal levels of learning and therefore, equilibrium outcomes that are constrained inefficient, both in terms of aggregates (average levels and elasticities to fundamentals) and in terms of dispersion of actions across agents. We also characterize policies that restore constrained efficiency, i.e. ex-post and ex-ante efficiency. The nature of such policies depends on whether firm level prices are flexible or not. In the flexible prices case, all that it takes to restore efficiency is a constant revenue subsidy which eliminates the monopoly distortions. In the sticky prices case, efficiency can be implemented through a set of state-contingent subsidies together with an optimal monetary response to fundamentals. 31

Importantly, optimal monetary policy under central bank’s imperfect information does not target full price stability, as it is the case under complete information. Instead, the optimal contingent plan for the monetary authority is to induce countercyclical prices. Finally, we highlight that desirable policies under flexible prices (e.g. revenue subsidy) can do more harm than good under sticky prices. The key behind this result is the endogeneity of information. There are several directions for future work. With a view to maintaining analytical tractability, we have made several simplifying assumptions. For example, we focus exclusively on static decisions, but the channels we highlight also have implications for intertemporal decisions (e.g. through capital accumulation, pricing with nominal frictions etc.). Similarly, for expositional simplicity, we rule out additional shocks (aggregate or idiosyncratic) and other sources of information. Relaxing some of these assumptions might require the use of numerical methods, but will allow a quantitative evaluation of the inefficiency and the policy interventions necessary to correct it. On the theoretical side, exploring the connections between the payoff-linked inefficiencies in this paper with others identified by the literature (e.g. the inefficiency in Amador and Weill (2010)) is another interesting direction for future work.

References Aiginger, Karl. 1999. “The Use of Game Theoretical Models for Empirical Industrial Organization.” In Competition, Efficiency and Welfare - Essays in Honor of Manfred Neumann, 253–277. Kluwer Academic Publishers. Amador, Manuel, and Pierre-Olivier Weill. 2010. “Learning from Prices: Public Communication and Welfare.” Journal of Political Economy 118 (5): pp. 866–907. Angeletos, George-Marios, Luigi Iovino, and Jennifer La’O. 2016. “Efficiency and Policy with Endogenous Learning.” Mimeo, MIT. Angeletos, George-Marios, and Jennifer La’O. 2009. “Noisy Business Cycles.” In NBER Macroeconomics Annual 2009, Volume 24, NBER Chapters, 319–378. National Bureau of Economic Research, Inc. . 2011, November. “Optimal Monetary Policy with Informational Frictions.” Nber working papers 17525, National Bureau of Economic Research, Inc. Angeletos, George-Marios, and Alessandro Pavan. 2007. “Efficient Use of Information and Social Value of Information.” Econometrica 75 (4): 1103–1142. 32

Bilbiie, Florin O., Fabio Ghironi, and Marc J. Melitz. 2008, October. “Monopoly Power and Endogenous Product Variety: Distortions and Remedies.” Working paper 14383, National Bureau of Economic Research. Chahrour, Ryan. 2014. “Public Communication and Information Acquisition.” American Economic Journal: Macroeconomics 6 (3): 73–101 (July). Colombo, Luca, Gianluca Femminis, and Alessandro Pavan. 2014. “Information Acquisition and Welfare.” Review of Economic Studies 81 (4): 1438–1483. Gorodnichenko, Yuriy. 2008. “Endogenous Information, Menu Costs and Inflation Persistence.” Nber working papers 14184, National Bureau of Economic Research, Inc. Hellwig, Christian. 2005. “Heterogeneous Information and the Welfare Effects of Public Information Disclosures.” Mimeo, UCLA. Hellwig, Christian, and Laura Veldkamp. 2009. “Knowing What Others Know: Coordination Motives in Information Acquisition.” Review of Economic Studies 76 (1): 223–251. Hellwig, Christian, and Venky Venkateswaran. 2009. “Setting the right prices for the wrong reasons.” Journal of Monetary Economics 56 (Supplement 1): S57 – S77. Kohlhas, Alexandre N. 2017. “An Informational Rationale for Action over Disclosure.” Lorenzoni, Guido. December 2009. “A Theory of Demand Shocks.” The American Economic Review 99:2050–2084(35). . 2010. “Optimal Monetary Policy with Uncertain Fundamentals and Dispersed Information.” Review of Economic Studies 77 (1): 305–338. Mackowiak, Bartosz, and Mirko Wiederholt. 2009. “Optimal Sticky Prices Under Rational Inattention.” American Economic Review 99 (3): 769–803. . 2011, October. “Inattention to Rare Events.” Cepr discussion papers 8626, C.E.P.R. Discussion Papers. Makowiak, Bartosz, and Mirko Wiederholt. 2015. “Business Cycle Dynamics under Rational Inattention.” The Review of Economic Studies 82 (4): 1502–1532. Moscarini, Giuseppe. 2004. “Limited information capacity as a source of inertia.” Journal of Economic Dynamics and Control 28 (10): 2003 – 2035. Myatt, David P., and Chris Wallace. 2015. “Cournot competition and the social value of information.” Journal of Economic Theory 158 (PB): 466–506. 33

Paciello, Luigi, and Mirko Wiederholt. 2014. “Exogenous Information, Endogenous Information, and Optimal Monetary Policy.” Review of Economic Studies 81 (1): 356–388. Reis, Ricardo. 2006. “Inattentive Producers.” Review of Economic Studies 73 (3): 793–821 (07). Roca, Mauro. 2010, April. “Transparency and Monetary Policy with Imperfect Common Knowledge.” Imf working papers 10/91, International Monetary Fund. Venkateswaran, Venky. 2012. “Heterogeneous Information and Labor Market Fluctuations.” Mimeo, Penn State University. Woodford, Michael D. 2003. “Imperfect Common Knowledge and the Effects of Monetary Policy.” In Knowledge, Information and Expectations in Modern Macroeconomics: In Honor of Edmund S. Phelps, edited by Phillipe Aghion et al.

Appendix A A.1 A.1.1

Proofs of Results

Model I: Quantity choice with productivity shocks Equilibrium

We solve for equilibrium by studying the problem of an individual entrepreneur i, who takes as given the actions of all other entrepreneurs j 6= i in the economy. Note that her objective is a h 1 θ−1 i concave function of Ni and the unique optimum is a function of Ei Y θ A θ . In a symmetric equilibrium, she conjectures (correctly) that all other firms have error variances σe2 and follow a log-linear policy rule nj = k2 + αsj . This implies  nj k2 α α k2 α = a+ =a+ + sj = 1 + a+ + ej , δ δ δ δ δ δ   Z  α k2 1 θ − 1 α 2 2 y = yj dj = 1 + + a+ σ , δ δ 2 θ δ2 e     θ−1 1 θ−1 1 α k2 1 α2 a+ y = + + a+ + (θ − 1) 2 2 σe2 . θ θ θ θ θδ θδ 2 θ δ h 1 θ−1 i Then, in logs, Ei Y θ A θ is an affine function of Ei (a) and therefore, we can express i ’s best yj

response ni = kˆ2 + α ˆ si .

34

Using this, we write i’s objective function as       2 θ−1 1 ˆ α ˆ θ−1 1+ α + 1 1+ α θδ θδ ˆ θδ exp 1 (θ − 1) α σ 2 Ei A θ−1 ˆ 2 Ei Aαˆ eαˆ θ ( δ ) θ( δ )e Πi = K − K K i 2 2 i 2 θ2 δ 2 e   θ − 1ˆ 1 1 α2 = exp k2 + k2 + (θ − 1) 2 2 σe2 θδ θδ 2 θ δ       α ˆ 1 θ−1 α θ−1 1+ + a+α ˆ 1+ ei exp θ δ θ δ θδ n o − exp kˆ2 + α ˆa + α ˆ ei . The unconditional expectation is n o   ˆ2 + 1 k2 + 1 (θ − 1) α2 22 σ 2 × exp θ−1 k
1 2 2 e θδ θδ 2 θ δ 2 ˆ ˆ o − exp k2 + α n  Πi = ˆ σa + σei .
1

1 2 θ−1 2 2 α ˆ α 2 2 2 σ + σ 1 + + 1 + α ˆ exp 12 θ−1 a ei θ δ θ δ 2 θδ | {z } | {z } ˆ N i

ˆi C

(31) We define:  θ − 1ˆ 1 1 α2 2 k2 + k2 + (θ − 1) 2 2 σe × θδ θδ 2 θ δ (  )       1 θ−1 α ˆ 1 α 2 2 1 2 θ − 1 2 2 exp 1+ + 1+ σa + α ˆ σei , 2 θ δ θ δ 2 θδ  
1 2 2 2 ˆ ˆ σa + σei ≡ exp k2 + α 2

Cˆi ≡ exp

ˆi N



2 as well as Note that the objective of entrepreneur i depends on her own choices kˆ2 , α ˆ and σei   the corresponding equilibrium objects k2 , α and σ 2 . The optimality conditions for kˆ2 , α ˆ are . e





θ−1 ˆ ˆi , Ci = N (32) θδ ( )       
θ−1 α ˆ 1 α 2 θ − 1 θ−1 2 2 ˆ 2 ˆi 1+ + 1+ σa +α ˆ σei Ci = α ˆ σa2 + σei N α ˆ : θ δ θ δ θδ θδ

kˆ2 :

2 = σ2, Π ˆ i = Π, ˆ kˆ2 = k2 , α ˆi = N ˆ . Thus, In a symmetric equilibrium, σei ˆ = α, Cˆi = Cˆ and N e

conditions (32) become: ( )      
1 α2 2 1 2 2 1 α 2 2 1 2 θ − 1 2 2 1 θδ 2 exp k2 + 1+ σa + α σe + (θ − 1) 2 2 σe = exp k2 + α ˆ σa + σe , δ 2 δ 2 θδ 2 θ δ θ−1 2       i θ−1 h α θ−1 θ−1 2 2 1 + − α σa = ασe 1 − . θ δ θ θδ Rearranging, we get the expressions in Proposition 1. Note that this equilibrium is unique within the class of log-linear responses. 35

The expression for the private value of information on the left hand side of (11) is obtained by a direct application of the envelope theorem to (31) along with (32). ˆ ∂Π 2 ∂σei

= = = =

 2 1 θ − 1 2 ˆ 1α Cˆ α ˆ −N ˆ2 2 θδ 2 # "   1 2 θ−1 2 θδ ˆ −1 N α ˆ 2 θ−1 θδ   1 2 1 − θ + θδ ˆ − α ˆ N 2 θδ   1 2 θ−1 ˆ − α ˆ Π, 2 θδ

ˆ i = Cˆi − N ˆi = where the last step makes use of the fact that, in equilibrium, Π A.1.2

(33) 

1−θ+θδ θ−1



ˆi . N

Efficiency in information choice

Expected utility for any given k2 is given by        1 α 2 2 k 2 1 θ − 1 α 2 2 1 2 2 1 2 2 U = exp 1+ + σ − exp k2 + α σa + α σe σa + 2 δ δ 2 θ δ2 e 2 2   ∗ k − k 2 2 ˆ ∗ exp (k2 − k ∗ ) , = Cˆ ∗ exp −N 2 δ ˆ ∗ ) are the corresponding unconditional where k2∗ is the optimal response coefficient and (Cˆ ∗ , N expectations of the optimal consumption and labor input. Using (14), we get     k2 − k2∗ ∗ ˆ∗ U = δ exp − exp (k2 − k2 ) N δ     k2 − k2∗ U∗ ∗ = δ exp − exp (k2 − k2 ) , δ δ−1 ˆ ∗ is the optimal level of social welfare.Then, where U∗ = Cˆ ∗ − N     ∂U k2 − k2∗ 1 ∂U∗ ∗ = δ exp − exp (k − k ) . 2 2 ∂σe2 δ δ − 1 ∂σe2 The term in the square bracket is independent of σe2 , see Proposition 1. The envelope theorem implies ∂U∗ ∂σe2

  1 θ − 1 α2 ˆ ∗ 1 α2 = Cˆ ∗ −N 2 θ δ2 2   1 θ − 1 ˆ∗ = − α2 1 − N . 2 θδ

36

Substituting, ∂U ∂σe2

     ˆ∗ 1 − θ + θδ k2 − k2∗ 1 N δ exp − exp (k2 − k2∗ ) = − α2 2 θδ δ δ−1   1 1 2 1 − θ + θδ = − α U 2 θδ δ−1     1 2 1 − θ + θδ θ−1 = − α U 2 (θ − 1) (δ − 1) θδ     δ θ−1 1 2 U . = − α 1+ 2 (θ − 1) (δ − 1) θδ

(34)

ˆ we have the result in Proposition 3. Since U = Π, A.1.3

A partial equilibrium analysis

Here, we derive the objective of a planner on island i, who takes the choices of all other islands as given but is interested in maximizing economy-wide utility. We start by noting that total consumption (or equivalently, output) is Z Y

=

Yj 

=

Y

θ−1 θ

θ−1 θ

dj + Yi

θ−1 θ



θ θ−1

dj

θ   θ−1 θ−1 θ−1 − 1 + Yi θ dj Y θ

θ   θ−1 θ−1 θ−1 = Y 1 + Yi θ dj Y − θ .

A first-order Taylor expansion around Y of the term in brackets delivers   θ−1 θ − θ−1 θ Y ≈ Y 1+ Y dj Y θ θ−1 i θ−1 θ−1 θ = Y + Yi θ dj Y 1− θ θ−1  θ−1 1 θ θ θ = Y + Yi Y dj θ−1   θ Pi Yi = Y + dj . θ−1 P A similar approximation for the disutility of labor input yields N + Ni dj. Combining, the planner’s objective is approximately ˆ = EY − EN + E U



θ θ−1

37



P i Yi P



 − Ni dj.

Noting that, in equilibrium, 

 P i Yi θδ E = ECi = ENi , P θ−1 we can rewrite the objective as  θ ECi − ENi dj θ−1   θ θδ = EY − EN + − 1 ENi dj θ−1θ−1     θδ θ − 1 θ−1 θ−1 ˆ  Πdj, = EY − EN +  θδ − 1 θ−1    ! θ θδ ˆ θ−1 θ−1 − 1 ∂ Π  =  , 2 θδ ∂σei − 1 2 2 θ−1

ˆ = EY − EN + U

⇒

ˆ ∂U 2 ∂σei

! 2 σe2 =σei



σe =σei

the expression in (22). A.1.4

Policy

To see that the constant revenue subsidy also aligns private and social values of information, note that the only change in the derivation of the private value of information above is in (33). ˆ = (δ − 1) N ˆ instead of Since the level distortion to output is not present under this subsidy, Π   ˆ = 1−θ+θδ N ˆ . Then, it is easy to see that the resulting expression for private value is identical Π θ−1 to the social value in (34).

A.2

Risk aversion

In this subsection, we explore the role of risk aversion using the quantity choice version of our model. In particular, we assume that markets are incomplete so intermediate goods producers have CRRA preferences over consumption: Ui =

Ci1−γ 2 − Ni − υ(σei ) 1−γ

γ ≥ 0.

where γ measures the degree of risk aversion. Clearly, this case nests our baseline model when γ = 0.Then, the labor choice problem is : 1 max Ei Ni 1−γ



Pi Yi P

38

1−γ − Ni .

A.2.1

Equilibrium

The equilibrium characterization follows the same guess and verify procedure used throughout the paper. We start with a log-linear equilibrium policy rule of the form: ni = kˆ2 + α ˆ si . Under this policy rule, the ex-ante profit of a firm becomes,   1 (θ − 1) (1 − γ) ˆ (1 − γ) 1 α2 2 ˆ Πi = exp k2 + k2 + (θ − 1) (1 − γ) 2 2 σe 1−γ θδ θδ 2 θ δ (  )   2     1 (θ − 1) (1 − γ) α ˆ (1 − γ) α 1 2 (θ − 1) (1 − γ) 2 2 2 exp 1+ + 1+ σa + α ˆ σei 2 θ δ θ δ 2 θδ  
1 2 2 2 ˆ ˆ σa + σei − exp k2 + α 2 The associated optimality conditions are Cˆi1−γ (

)         1−γ θ−1 α ˆ 1 α 2 θ − 1 θ − 1 2 2 ˆ 1−γ 1+ + 1+ σa +α ˆ (1 − γ) σe Ci δ θ δ θ δ θ θδ

 =

θ θ−1




ˆi α = N ˆ σa2 + σe2 .

Imposing symmetry and substituting from the first equation into the second equation, we get   
α 2 θ−1 (1 − γ) 1 + σa + (1 − γ) σe2 α = α σa2 + σe2 . δ θδ As in the risk neutral case, the left hand side of the equation captures the benefits of responding to information while the right hand side captures the costs. Note that the benefits are now dampened by the degree of risk aversion. This equation delivers the equilibrium sensitivity to information,

 δ (1 − γ)  α= δ − 1 + γ σa2 +

The first term

δ(1−γ) δ−1+γ



σa2 1+θ(δ−1)+γ(θ−1) 2 σe θ(δ−1+γ)

,

is simply the full information elasticity of employment to a productivity

shock, while the term inside the square brackets is an adjusted signal-to-noise ratio. The factor 1+θ(δ−1)+γ(θ−1) θ(δ−1+γ)

reflects the degree of strategic complementarities in the model. As the expression

shows, risk neutrality dampens incentives to coordinate. This also translates into a dampened value of information38 ,

38

ˆ ∂Π 1 =− 2 2 ∂σei



(θ − 1) (1 − γ) θδ

ˆ > 0. Note that the expression is negative for all γ since (1 − γ) Π

39



ˆ 2<0 Πα

ˆi , δN

A.2.2

Efficiency

We now turn to efficiency. We begin by showing that risk aversion introduces additional externalities that distort the sensitivity of quantity choices to signals. Again, we solve the problem of a constrained social planner whose objective is to maximize ex-ante utility subject to the informationconstraint, i.e. cannot pool information across islands, and the same market incompleteness, i.e. cannot form a risk pool. We posit symmetric log-linear policy rules of the form ni = k˜2 + α ˜ si . Under this rule, the ex-ante utility can be written as, (   2 ) (1 − γ) e 1 (θ − 1) (1 − γ) 2 1 α e 1 exp k2 + α e [1 + (θ − 1) (1 − γ)] σe2 + (1 − γ) 1 + σa2 U = 1−γ δ 2 θ2 δ 2 2 δ  
1 2 2 2 e e σa + σe − exp k2 + α 2 The social planner chooses e k2 and α e to maximize U. As before, combining the two FOCs, we get the following expression characterizing α e,      
α e θ−1 θ−1 γ 2 (1 − γ) 1 + σa2 + (1 − γ) σe2 α e+ σe α e=α e σa2 + σe2 . δ θδ θδ θ Note that the last term on the left hand side was absent in the equilibrium characterization of α. This reflects the fact that the social planner internalizes the effect of the entrepreneur’s responsiveness on consumption risk in the whole economy. Note this term disappears under risk neutrality, i.e. γ = 0 or full information, i.e. σe2 = 0. In the utility maximizing policy rule, the sensitivity to information is given by the following expression,  δ (1 − γ)  α∗ = δ−1+γ



σa2 σa2 +

(1+θ(δ−1)+γ(θ−1)−γ ( θ−1 θ )) θ(δ−1+γ)

Relative to the equilibrium α, this coefficient has an additional γ

 σe2 θ−1 θ




term in the denomina-

tor. Since γ ≥ 0, this implies that the responsiveness to information in equilibrium is suboptimally low, i.e. |α| < |α∗ |. Risk aversion also dampens the gap between the equilibrium level coefficient k2 and the socially optimal one k2∗ : k2∗

δ = k2 + log δ+γ−1 40



θ θ−1



Finally, we compare the equilibrium information choice to the level that maximizes ex-ante utility. The following expression characterizes the wedge between the social value of information, 2, ˆ ∂U/∂σ 2 , and the private value, ∂ Π/∂σ e

ei

     δ−1 θ−1 ∂U δ σe2 ∂α 1−γ = 1+ − 2θγ ∂σe2 (δ − 1) (θ − 1) δ+γ−1 θδ α ∂σe2 The wedge has three parts: the first,

δ (δ−1)(θ−1) ,

ˆ ∂Π 2 ∂σei

! 2 =σ 2 σei e

is the one identified in our baseline setup

without risk aversion - a combination of market power and coordination effects. This is adjusted by a factor

δ−1 δ+γ−1 ,

to reflect the fact that the gap in the level coefficient is decreasing in γ. However,

that is not the only effect of γ. The last term inside the square bracket captures the distortions arising from the consumption risk externality mentioned earlier. It enters the term both directly (through the −γ θ−1 θδ term) as well as because it distorts the response coefficient α. The direct effect is negative (in the sense that it lowers the social value of information relative to the private value) while the indirect effect, through α, pushes in the opposite direction39 . In general, therefore, risk aversion combined with incomplete markets (and the associated consumption risk externality) can lead to a wedge greater or less than 1, making the direction of inefficiency in information choice ambiguous. Depending on parameters, we can now have over- or under-acquisition of information in equilibrium. A.2.3

Policy

Next, we show that even if policy is used to align both the level and response coefficients in the firm’s policy rule with the socially optimal ones, we do not attain constrained efficiency. In other words, aligning social and private incentives to learn requires additional intervention, even if there are no ex-post inefficiencies. We first show that ex-post efficiency can be achieved with a state-contingent revenue subsidy of the form Λ Aτ Under this policy, entrepreneurs solve: max Λ Ni

39

h (θ−1)(1−γ) (θ−1)(1−γ) 1 θ Ni θδ Ei A(1−γ)τ A Y 1−γ

To see the intuition for σ 2 /α




dα/dσ 2




(1−γ) θ

i

− Ni .

< 0, recall that the equilibrium α was too low relative to the utility

∗

maximizing α . Better information alleviates this problem and therefore, a positive effect on welfare.

41

The log-linear structure of policy rules is preserved. It is then straightforward to solve for the policy (τ ∗ , Λ∗ ) that aligns employment to the socially optimal policy, α∗ − 1, αeq       θ α∗ 2 1 ∗ ∗ 2 = exp (1 − γ) τ σa . τ +1+ θ−1 2 δ

τ∗ = Λ∗

where αeq is the laissez-faire equilibrium response coefficient. Note that under risk neutrality,   θ ∗ ∗ i.e. if γ = 0, we have the constant subsidy, that is τ = 0 and Λ = θ−1 . The private value of information at this policy is given by: ˆ 1 ∗2 ∂Π 2 = −2α ∂σei



1 − θ + θδ γ + θδ δ



1 − θ + θδ γ + θδ δ



θ−1 θ



ˆ ∗ < 0. N

Comparing this to the social value, ∂U 1 ∗2 2 = −2α ∂σei

θ−1 θ

2 !

ˆ ∗ < 0, N

we note that they do not coincide unless γ = 0, i.e. under risk neutrality, or in the competitive limit, as θ → ∞. Thus, in general, even if ex-post inefficiencies are corrected, firms will acquire too much information.

A.3 A.3.1

Model II: Price setting with productivity shocks Equilibrium

As with the quantity choice, we begin with the problem of entrepreneur i, who believes (correctly) that everybody else is acting according to pj = k2 + αsj = k2 + αa + αej .

42

The corresponding aggregate relationships are 1 p = k2 + αa + (1 − θ)α2 σe2 , 2 1 y = m − k2 − αa − (1 − θ)α2 σe2 2 1 = −k2 + (αm − α) a + αm ema − (1 − θ)α2 σe2 , 2 yj = −θ (pj − p) + y θ 1 = −αθej + (1 − θ)α2 σe2 − k2 + (αm − α) a + αm ema − (1 − θ)α2 σe2 2 2 1 2 2 2 = αm ema − αθej − k2 + (αm − α) a − α σe (θ − 1) , 2 1 nj = (yj − a) δ = −δk2 + δ (αm − α − 1) a + δαm ema − δαθej − δα2 σe2 (θ − 1)2 , 2 1 2 2 1 2 2 2 2 2 n = −δk2 + δ(αm − α − 1)a + δαm ema − δα σe (θ − 1) + α θ δ σe 2 2  1 2 2  2 = −δk2 + δ(αm − α − 1)a + δαm ema + α σe δ θ δ − (θ − 1)2 . 2 We then solve for i0 s optimal policy pi = kˆ2 + α ˆ si . The objective function can be written as   h i 1 ˆ 1−θ θ−2 2 2 ˆ m m α(1−θ) ˆ − ei eαma Πi = K2 K2 exp (θ − 2)(1 − θ)α σe Ei Aα(θ−2)+α(1−θ)+α 2   h i 1 −θδ δ(θ−1) α ˆ δαm 2 2 2 ˆ K2 K2 exp − δ(1 − θ) α σe Ei Aαδ(θ−1)−δ(1−αm )−θδαˆ e−θδ ema i 2   h i 1 2 2 ˆ = exp (1 − θ)k2 + (θ − 2)k2 exp (θ − 2)(1 − θ)α σe × 2 m exp {[α(θ − 2) + α ˆ (1 − θ) + αm ] a + α ˆ (1 − θ)ei } eαma   h i 1 2 2 2 ˆ − exp −θδ k2 + δ(θ − 1)k2 exp − δ(1 − θ) α σe × 2 m exp {[αδ(θ − 1) − δ (1 − αm ) − θδ α ˆ ] a − θδ α ˆ ei } eδα ma .

The unconditional expectation is h i   exp (1 − θ)kˆ2 + (θ − 2)k2 exp 21 (θ − 2)(1 − θ)α2 σe2 × ˆi = n o Π 2 + 1 α2 σ 2 exp 12 [α(θ − 2) + α ˆ (1 − θ) + αm ]2 σa2 + 12 α ˆ 2 (1 − θ)2 σei 2 m ma {z } | ˆi C

h

i   exp −θδ kˆ2 + δ(θ − 1)k2 exp − 21 δ(1 − θ)2 α2 σe2 ×

− exp |

n

1 2

o 2 + 1 α2 δ 2 σ 2 [αδ(θ − 1) − δ (1 − αm ) − θδ α ˆ ]2 σa2 + 12 θ2 δ 2 α ˆ 2 σei ma 2 m {z } ˆi N

43

We define:   h i 1 2 2 ˆ ˆ Ci ≡ exp (1 − θ)k2 + (θ − 2)k2 exp (θ − 2)(1 − θ)α σe × 2  1 2 1 2 [α(θ − 2) + α ˆ (1 − θ) + αm ]2 σa2 + α ˆ (1 − θ)2 σei + exp 2 2   h i 1 2 2 2 ˆ ˆ Ni ≡ exp −θδ k2 + δ(θ − 1)k2 exp − δ(1 − θ) α σe × 2  1 1 2 exp [αδ(θ − 1) − δ (1 − αm ) − θδ α ˆ ]2 σa2 + θ2 δ 2 α ˆ 2 σei + 2 2

(35) 

1 2 2 α σ , 2 m ma

1 2 2 2 α δ σma 2 m



FOC kˆ2

ˆi , (θ − 1)Cˆi = θδ N  2 α ˆ : Cˆi [α(θ − 2) + α ˆ (1 − θ) + αm ] (1 − θ)σa2 + α ˆ (1 − θ)2 σei  2 ˆi − [αδ(θ − 1) − δ (1 − αm ) − θδ α = N ˆ ] θδσa2 + θ2 δ 2 α ˆ σei . :

2 = σ2, k ˆ2 = k2 , α ˆ N ˆi = N ˆ and Π ˆ i = Π. ˆ The FOC In a symmetric equilibrium, σei ˆ = α, Cˆi = C, e

for k2 becomes

=⇒ (δ − 1) k2

1 1 1 2 2 1 (θ − 2)(1 − θ)α2 σe2 − k2 + (αm − α)2 σa2 + α2 (1 − θ)2 σe2 + αm σma 2  2 2 2  1 1 θδ − δk2 − δ(1 − θ)2 α2 σe2 + [δ(1 + α − αm )]2 σa2 = log θ−1 2 2 1 2 2 2 2 1 2 2 2 + θ δ α σe + αm δ σma , 2  2 h i θδ 1 = log + σa2 δ 2 (1 + α − αm )2 − (αm − α)2 θ−1 2
2 2 1 1 2 + (1 − θ) [1 − δ(1 − θ)] + θ2 δ 2 α2 σe2 + δ − 1 αm σma . 2 2

The FOC for α ˆ simplifies to  h i θδ (αm − α) (1 − θ) σa2 + α (1 − θ)2 σe2 = [α + (1 − αm )] θδ 2 σa2 + θ2 δ 2 ασe2 θ−1 Rearranging yields the expressions in Proposition 5. The private value of information is a direct application of the envelope theorem ˆ ∂Π 2 ∂σei

1 2 1 ˆ α ˆ (1 − θ)2 Cˆ − θ2 δ 2 α ˆ2N 2 2     1 2 θδ 2 2 2 ˆ = − α ˆ −(1 − θ) +θ δ N 2 θ−1 1 2 ˆ = − α ˆ θδ(1 − θ + θδ)N 2 1 2 ˆ. = − α ˆ θδ (θ − 1) Π 2 =

44

A.3.2

Efficiency

Again, we solve the problem of a constrained social planner whose objective is to maximize exante utility. We posit symmetric log-linear policy rules of the form pi = k˜2 + α ˜ si . The problem can be written as: max

˜2 ,αm ,σ 2 } ˜k {α, e


U − υ σe2 ,

s.t. m = αm s m , m = p + y, where: ˆ, U = Cˆ − N   1 1 1 2 2 2 2 2 2 ˜ ˆ C ≡ exp −k2 + (αm − α ˜ ) σa − (1 − θ)˜ α σe + αm σma , 2 2 2     1 2 2 2 1 1 2 2 2 2 2 2 2 ˜ ˆ ≡ exp −δ k2 + δ (αm − α N ˜ − 1) σa + α ˜ σe δ θ δ − (θ − 1) + δ αm σma . 2 2 2 ˆ i , Cˆi and N ˆi evaluated at a symmetric equilibrium (σ 2 = σ 2 , These functions corresponds to Π e ei ˆ ˜ k2 = k2 , α ˆ=α ˜ ). The approach will be to separate the maximization problem assuming the following three stages, that will be compatible with the descentralization of the constrained optimal allocation. At stage three the agent knows the policy rule of the monetary authority and the precision of the private signal σ 2 , and the planner chooses α ˜ and k˜2 . This is when the planner manipulates agents’ e

responsiveness to information. At the second stage, the planner knows the optimality conditions of stage three, and chooses the level of private information σe2 . This is when the planner determines the optimal level of private learning. Given stages three and two optimality conditions, at stage one the planner chooses the optimal log-linear (state contingent) monetary policy rule. At the third stage, the planner’s problem is to choose the optimal response coefficients of the pricing policy rule, max ˜2 α, ˜k

ˆ. Cˆ − N

45

The FOC are: ˆ, k˜2 : Cˆ = δ N h i   α ˜ : −δ (αm − α ˜ ) σa2 + (1 − θ) σe2 = −δ 2 (αm − α ˜ − 1) σa2 + α ˜ σe2 δ θ2 δ − (θ − 1)2 . Solving, we get the coefficients α∗ and k2∗ in Proposition 6. At the second stage, the planner chooses the level of informatioin. The problem of the social planner is:
max U − υ σe2 . σe2

By the envelope theorem, the FOC is40 : −


1 ∗2
δθ [1 + θ (δ − 1)] ∗ α U = υ 0 σe2∗ , 2 δ−1

where σe2∗ is the optimal information choice and U∗ is the optimal level of utility. The left hand side of the equation represents the social value of information at the optimal information use. Clearly, this social value is negative which means that private information is always socially valuable. Finally, the planner’s problem is to choose the optimal monetary policy rule, ˆ. max Cˆ − N αm

By the envelope theorem, the FOC are:     ∗ 2 2 (αm − α∗ ) σa2 + αm σma = δ (αm − α∗ − 1) σa2 + δαm σma . Plugging α∗ into αm yields, h i     δ  σe2 θ 1+θ(δ−1) (δ−1) σa2 ∗ i h ≥ 0. αm = 2 + σ2  σ 2 θ 1+θ(δ−1) + σa2 σ 2  δ − 1 σma a 2 2 e ma (δ−1) σ +σ a

A.3.3

ma

Efficiency in information choice

Expected utility is given by


ˆ α, k ∗ , σ 2 exp (δk ∗ − δk2 ) . U α, k2 , σe2 = Cˆ α, k2∗ , σe2 exp (k2∗ − k2 ) − N 2 e 2



ˆ α, k ∗ , σ 2 and U α, k ∗ , σ 2 = (δ − 1) N ˆ α, k ∗ , σ 2 . Hence, At k2∗ , we know Cˆ α, k2∗ , σe2 = δ N 2 e 2 e 2 e

U α, k2∗ , σe2 2 ∗ ∗ U α, k2 , σe = [δ exp (k2 − k2 ) − exp (δ (k2 − k2 ))] . δ−1 40

As before, restrict attention to cost functions which lead to interior solutions to the above information choice prob-

lem.

46

Note that at equilibrium α, k2∗ − k2 does not depend on σe2 . Now we take the first derivative: " #


∂U α, k2 , σe2 ∂U α, k2∗ , σe2 ∂U α, k2∗ , σe2 ∂α 1 ∗ ∗ . = [δ exp (k2 − k2 ) − exp (δ (k2 − k2 ))] + ∂σe2 δ−1 ∂σe2 ∂α ∂σe2 Clearly, if α = α∗ then by the envelope theorem:
∂U α, k2∗ , σe2 =0 ∂α However, around the equilibrium α :


α2 δ θ2 δ − (θ − 1)2
∂U α, k2∗ , σe2 α2 (θ − 1) ˆ ∗ 2 ˆ α, k ∗ , σ 2 , = C α, k2 , σe − N 2 e 2 ∂σe 2 2
∗ 2 2 α U α, k2 , σe = − δθ [1 + θ (δ − 1)] < 0. 2 δ−1 The latter is the same expression as before. The key is that the derivative was taken at k2∗ . Now we consider: ∂U α, k2∗ , σe2 ∂α


=


  (α − αm ) σa2 + (θ − 1)ασe2 Cˆ α, k2∗ , σe2   
ˆ α, k2∗ , σe2 . −δ δ(1 + α − αm )σa2 + ασe2 θ2 δ − (θ − 1)2 N



ˆ α, k ∗ , σe2 . Hence, At k2∗ , we know Cˆ α, k2∗ , σe2 = δ N 2

  ∂U α, k2∗ , σe2 = (α − αm ) σa2 + (θ − 1)ασe2 − δ(1 + α − αm )σa2 − ασe2 θ2 δ − (θ − 1)2 Cˆ α, k2∗ , σe2 . ∂α Note that, at the equilibrium α : (α − αm ) σa2 + (θ − 1) ασe2 = δ (1 + α − αm ) σa2 + θδασe2 . Thus, ∂U α, k2∗ , σe2 ∂α





= − (θ − 1) [1 + θ (δ − 1)] ασe2 Cˆ α, k2∗ , σe2 ,  
θ−1 = −δ [1 + θ (δ − 1)] ασe2 U α, k2∗ , σe2 . δ−1

Hence: ∂U α, k2 , σe2 ∂σe2




[1 + θ (δ − 1)] = − [δ exp (k2∗ − k2 ) − exp (δ (k2∗ − k2 ))] δ−1
  ∗ 2 2 δU α, k2 , σe α ∂α θ + (θ − 1) ασe2 2 , δ−1 2 ∂σ  2 e 
α [1 + θ (δ − 1)] 2 2 ∂α = − δU α, k2 , σe θ + (θ − 1) ασe 2 . δ−1 2 ∂σe 47


For the rest of the proof, we let U ≡ U α, k2 , σe2 . Therefore, the sign of

∂U ∂σe2

depends on:

α2 ∂α δθ + (θ − 1) ασe2 2 . 2 ∂σe After some algebra, 

α2

∂α θ + (θ − 1) ασe2 2 =  2 ∂σe

σa2 +

2−θ θ

σa2 +




 1+θ(δ−1)  δ−1 1+θ(δ−1) δ−1



σe2

σe2



2

 θα . 2

Notice that if θ ≤ 2, then the social value of information at the equilibrium information choice is always positive, i.e.,

∂U ∂σe2

< 0. If θ > 2, the sign of the social value of information at the equi-

librium information choice is ambiguous and it depends on the level of information chosen in equilibrium. We define the threshold σ ¯e2 : σ ¯e2

 =

δ−1 1 + θ (δ − 1)



θ θ−2



σa2

Then, if θ > 2, the private information space is separated in two zones: ∂U > 0 (equilibrium information is not socially valuable) ∂σe2 ∂U =⇒ < 0 (equilibrium information is socially valuable) ∂σe2

σe2 ≥ σ ¯e2 =⇒ σe2 < σ ¯e2

If the equilibrium information choice falls in the first zone, then trivially a marginal reduction in private information increases social welfare. It follows that there is overacquisition of information in equilibrium. If the equilibrium information choice falls in the second zone, then equilibrium information is socially valuable and the determination of whether there is over or under acquisition will depend on the comparison between the social and the private value of information at the equilibrium information choice. Now we shift the attention to the second zone. First, note that at the symmetric equilibrium ˆ i = U . Thus, the private value of information is: Π ! ˆ ∂Π 1 = − α2 θδ (θ − 1) U < 0. 2 2 ∂σei 2 2 σei =σe

After some manipulations, we have    2  ∂U δ 1 σe ∂α = 1+ +2 1+ ∂σe2 (δ − 1) (θ − 1) θ (δ − 1) α ∂σe2 48

ˆ ∂Π 2 ∂σei

! <0 2 =σ 2 σei e

Finally, whether there is over or under acquisition depends on the term in front. Specifically, if the term is greater (less) than 1, then there is under (over) acquisition of information. Only in a knife edge case, the social and the private value of information will coincide. A few lines of straightforward algebra yield the result in Proposition 8. A.3.4

Policy

Consider a revenue subsidy of the form ΛAτ . The unconditional expectation of profits now becomes: h i   Λ exp (1 − θ)kˆ2 + (θ − 2)k2 exp 21 (θ − 2)(1 − θ)α2 σe2 × ˆi = o n Π 2 + 1 α2 σ 2 ˆ (1 − θ) + αm + τ ]2 σa2 + 12 α ˆ 2 (1 − θ)2 σei exp 12 [α(θ − 2) + α 2 m ma | {z } ˆi C

h i   exp −θδ kˆ2 + δ(θ − 1)k2 exp − 21 δ(1 − θ)2 α2 σe2 ×

− exp |

n

1 2

o 2 + 1 α2 δ 2 σ 2 [αδ(θ − 1) − δ (1 − αm ) − θδ α ˆ ]2 σa2 + 12 θ2 δ 2 α ˆ 2 σei ma 2 m {z } ˆi N

FOC ˆi , kˆ2 : Λ (θ − 1) Cˆi = θδ N  2 α ˆ : Λ [α(θ − 2) + α ˆ (1 − θ) + αm + τ ] (1 − θ)σa2 + α ˆ (1 − θ)2 σei Cˆi =  2 ˆi N − [αδ(θ − 1) − δ (1 − αm ) − θδ α ˆ ] θδσa2 + θ2 δ 2 α ˆ σei Using the FOC for kˆ2 and invoking symmetry:  α = αm −

δ τ − δ−1 δ−1



 



σa2 +

σ2 a  1+θ(δ−1) 2 σe δ−1

Replacing τ (αm ) yields:  α = αm −

δ δ−1



 



σa2 +

σ2 a  σe2 1+θ(δ−1) δ−1

 1+

τ δ − (δ − 1) αm

 .

Invoking symmetry in the FOC for k2 (in logs):   i θδ 1 h (δ − 1)k2 = log − λ + σa2 δ 2 (1 + α − αm )2 − (αm − α + τ )2 θ−1 2 
2 2 1 1 2 + (1 − θ) [1 − δ(1 − θ)] + θ2 δ 2 α2 σe2 + δ − 1 αm σma . 2 2 49

(36)

(37)

Any response function (α, k2 ) can be implemented by setting the policy parameters (τ, λ) to satisfy (36) and (37). In particular, to implement the socially optimal α∗ , we need,   τ αeq (θ − 1) σe2 1 + θ (δ − 1)   = ∗ −1=− < 0, δ − (δ − 1) αm α δ−1 σe2 σa2 + θ 1+θ(δ−1) δ−1 where αeq is the equilibrium sensitity of prices to information from Proposition 5. The level coefficient k2 is also suboptimal - but now it includes both the usual markup distortion and the level effects of the inefficient sensitivity to information. At α = α∗ , k2 reads as follows: 



i 1 h − λ + σa2 δ 2 (1 + α∗ − αm )2 − (αm − α∗ + τ )2 2 
2 2 1 1 2 + (1 − θ) [1 − δ(1 − θ)] + θ2 δ 2 α∗2 σe2 + δ − 1 αm σma . 2 2

(δ − 1)k2 = log

θδ θ−1

Hence, to implement k2∗ we need:   θ 1 λ = log − σa2 [τ (αm ) + 2 (αm − α∗ )] τ θ−1 2 This yields Proposition 9 in the main text. To see that this policy also aligns private and social values, first note that the policy implements the socially optimal response by construction, so we can directly apply the envelope theorem to get ∂U 1 ˆ∗ . = − α∗2 θδ(1 − θ + θδ)N ∂σe2 2 Now, recall that, in equilibrium, private value is ˆ ∂Π 1 2 ˆ. ˆ θδ(1 − θ + θδ)N 2 = −2α ∂σei ˆ =N ˆ ∗ and α Since N ˆ = α∗ with efficient response functions, ! ˆ ∂Π ∂U = 2 ∂σe2 ∂σei 2 2 σei =σe

establishing the result in Proposition 10.

A.4

Model III: Price setting with nominal shocks

In this section, we present the main results from a version where aggregate productivity is constant ¯ but aggregate nominal demand is stochastic. In particular, M is an i.i.d41 , log-normally (i.e. A = A) 41

Again, for simplicity, the iid assumption is primarily for simplicity - it is straightforward to add persistence.

50


2 . Intermediate goods producers choose distributed random variable, i.e. log M ≡ m ∼ N 0, σm nominal prices for their products and commit to producing any amount demanded at that price. Before setting prices, each firm observes a private signal si about the current monetary shock si = m + ei ,
2 . The variance of the noise term, σ 2 is the variance chosen in stage I by the where ei ∼ N 0, σei ei firm. As before, the competitive firm producing the final good operates after the monetary shock is realized. Therefore, the problem of this firm remains the same, i.e. demand for intermediate goods is given by equation (2). The proofs of the results that follow are practically identical to the previous section, so we omit them for brevity. The intermediate producer’s problem is:  max Ei Pi

Pi P

"

1−θ Y −

Pi P

−θ

Y A

#δ .

As before, we guess (and verify) that equilibrium prices follow pi = k2 + αsi .

(38)

The response coefficients in a symmetric equilibrium are collected in the following result. Proposition 11 In a symmetric equilibrium, firms follow a pricing rule of the form (38), with   2 σ  m   , α =  1−θ+θδ 2 σm + σe2 δ−1   i
1 1 2 h 2 θδ (δ − 1) k2 = log − δ (δ − 1) a + α2 σe2 [δθ (1 − θ + θδ) + (δ − 1) (θ − 1)] + σm δ − 1 (1 − α)2 θ−1 2 2 ≡ κ (α) where σe2 is the variance of the error in agents’ signals. A symmetric stationary equilibrium can thus be represented as a fixed point problem in σe2 :  
ˆ σ 2 , α(σ 2 ), k2 (σ 2 ) − υ σ 2 , σe2 = argmaxσ2 Π ei e e ei ei

where, again, α and k2 depend on σe2 according to Proposition 11.

51

A.4.1

Efficiency in information use

The socially efficient response function takes the same form as (38) with the coefficients given in the following result. Proposition 12 For a given σe2 , the socially optimal response coefficients are:   2 σ  m   α∗ =  2 + θ 1−θ+θδ σ 2 σm e δ−1   θ + κ(α∗ ). (δ − 1)k2∗ = − log θ−1 where the dependence of k2 (α∗ ) denotes the equilibrium level coefficient with α∗ replacing α. Again, the equilibrium features prices that are too responsive to signals, i.e. α∗ < α. The intuition is very similar to the productivity shocks case - firms do not fully internalize the effect of their pricing decisions on the marginal cost uncertainty faced by other firms. As a result, they react too much to private signals, relative to the planner’s solution. A.4.2

Efficiency in information choice

Next, we compare the amount of information acquired in equilibrium to the utility-maximizing level. As with productivity shocks case discussed in section 5, the presence of direct and indirect effects means that the marginal social value of information is not always positive. We restrict attention to the region where this value is positive. A sufficient condition is 2 + Assumption 2 σm

2−θ θ


 1+θ(δ−1)  δ−1

σe2 ≥ 0 .

Conditional on being in this region, whether the social planner acquires more or less information than the equilibrium depends only on the marginal value to the planner, ∂U/∂σe2 , versus the 2. ˆ private value to the firm, ∂ Π/∂σ ei

The following result mirrors Proposition 8 and shows that the equilibrium can feature both under- and over-acquisition. Proposition 13 Suppose Assumption 2 is met. Then, there is over-acquisition of information in equilibrium if the following condition holds σe2

  θ (δ − 1) δ > σ2 . 1 + θ (δ − 1) 2 (θ − 1) (1 + θ (δ − 1)) − θδ m 52

A.4.3

Policy

Finally, we characterize optimal policy in this environment. In line with section 5, we consider revenue subsidies of the form Λ M (1−δ)τ . The following proposition characterizes the policy coefficients that correct both sources of inefficiency in the equilibrium response functions. Proposition 14 Given σe2 , equilibrium allocations coincide with the choices of the planner, i.e. (α, k2 ) = (α∗ , k2∗ ) if the subsidy coefficients satisfy α∗ − 1 < 0, αeq   1  2  δ−1 θ σm τ (2 (1 − α∗ ) + (1 − δ) τ ) Λ= exp . θ−1 2 τ=

(39)

Importantly, such a policy also removes the wedge between private and social value of information, ensuring that signal precisions in equilibrium are socially optimal. Formally, Proposition 15 A symmetric equilibrium under the policy described in Proposition 14 (evaluated at the socially optimal σe2 ), is constrained efficient, i.e. it attains the optimal allocation of the information constrained planner.

53