Sep 1, 2017 - This is very much in the spirit of the real business cycle literature, with the important difference that
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)
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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. Note that the expression is negative for all γ since (1 − γ) Π
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ˆ 2 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.
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