Heterogeneous information quality; strategic complementarities and optimal policy design

Journal of Economic Behavior & Organization 83 (2012) 342–352

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Heterogeneous information quality; strategic complementarities and optimal policy design Jonathan G. James, Phillip Lawler ∗ Department of Economics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom

a r t i c l e

i n f o

Article history: Received 7 March 2012 Accepted 5 July 2012 Available online 14 July 2012 JEL classification: C72 D62 D82 E58

a b s t r a c t The beauty-contest framework of Morris and Shin (2002) is extended to allow sub-groups within the population of agents to differ in the quality (i.e. precision) of their private information. We discuss the inefficiency of the resulting model’s equilibrium, and assess the relative effectiveness in remedying this inefficiency of: (i) a Pigouvian tax scheme; (ii) direct policy intervention by means of an instrument which can modify the state of the world. The disclosure-policy implications of each of these two policy approaches are also analyzed. © 2012 Elsevier B.V. All rights reserved.

Keywords: Strategic complementarity Public disclosure Policy intervention

1. Introduction A number of recent studies have highlighted the inefficiencies which potentially arise in economies characterized by heterogeneous information, and where agents’ actions exhibit strategic complementarities: see for example Angeletos and Pavan (2007a), Hellwig (2005), James and Lawler (2008), and Roca (2010). The potential for departures from efficiency in this context is associated with the motive which agents have to align their actions with those of others, but where the incentives facing individual agents may alternatively over-value or understate the social benefit of alignment. A particularly significant implication which follows from this eventuality is that improvements in information quality can be damaging to welfare. This possibility has been central to a recent strand of literature relating to the desirability, or otherwise, of central bank transparency.1 A particularly prominent position within this literature is occupied by the so-called ‘beauty contest’ framework of Morris and Shin (2002), a key feature of which is that individual payoff functions generate the incentive for agents to attempt to coordinate their actions with those of other agents, despite there being no social benefit from such coordination. This feature leads to a relative overweighting of public (i.e. common) information at the expense of private (i.e. individualspecific) information by agents when formulating their actions. The consequence is the possibility, emphasized by Morris and Shin, that more accurate public information might be detrimental to welfare. In the realm of monetary policy, and with the central bank recognized as a potential source of public information, this then implies that transparency might

∗ Corresponding author. Tel.: +44 1792 295 168; fax: +44 1792 295 872. E-mail address: [email protected] (P. Lawler). 1 For wide-ranging overviews of the transparency literature see, for example, Geraats (2002), Cruijsen and Eijffinger (2007) and Blinder et al. (2008). 0167-2681/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jebo.2012.07.003

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be undesirable. Although the policy conclusion drawn by Morris and Shin has been disputed on a number of different grounds, the inefficient use of information by private sector agents which underpins their anti-transparency result appears to be a general property to which the combination of heterogeneous information and strategic complementarities gives rise. Reflecting this, recent work by Angeletos and Pavan (2007b, 2009) and James and Lawler (2011, 2012) has considered possible policy responses directed at correcting this departure from efficiency. The concern of Angeletos and Pavan is to investigate whether a corrective tax scheme can modify the incentives facing individual agents in a way which promotes attainment of the first-best outcome. They demonstrate that this is, indeed, possible with the appropriate policy requiring that the tax liability of each agent be made contingent on the realized values of the economic fundamental, the individual’s choice of action and, crucially, the aggregate action. As Angeletos and Pavan recognize, an evident consequence of the Pigouvian tax’s ability to bring about the first best is that disclosure of the policy authority’s private information must always be beneficial. The approach of James and Lawler to the policy issue is somewhat different and considers the possibility that the policy authority has available to it an instrument which is capable of directly modifying the welfare impact of the state variable. In this respect, it can be interpreted as indicative of a potential (macroeconomic) stabilization role for policy. The principal result to emerge from their analysis is that, when policy is set in accordance with an optimally designed rule, the first-best outcome will ensue. However, and importantly in light of the transparency issue, the maximization of social welfare requires that the private sector does not have the policy authority’s private information disclosed to it. In examining policy, the studies of Angeletos and Pavan and of James and Lawler follow the prevailing approach within the related literature in assuming that agents’ private information, while diverse, is nonetheless of homogeneous quality. The present contribution relaxes this assumption to consider the implications of agent-groups which differ in the quality of their private information. In the interests of analytical tractability, here we differentiate between two agent-types distinguished by private information quality and consider whether non-uniformity of this kind affects the policy-design conclusions arrived at by Angeletos and Pavan (2007b, 2009) and James and Lawler (2011, 2012). The notion that information quality might differ across agents is inherent to both the ‘sticky information’ concept, associated principally with Mankiw and Reis (2002), and the epidemiological model of expectations formation developed by Carroll (2003). Although these two approaches are distinct, they both share the feature that not all agents are equally wellinformed, reflecting the assumption that the updating of information sets is generally not instantaneous across the whole population. One possible explanation for this, emphasized by Mankiw and Reis, lies in terms of the costs of acquiring and processing new information. The approach taken in what follows, which essentially distinguishes between ‘better-informed’ and ‘less well-informed’ individuals, can thus be rationalized in terms of differential information costs between groups of agents. The binary distinction we draw represents the idea in particularly stark terms, but is commonly applied in both informal discussions and formal analyses of financial market behavior. The vehicle chosen for the analysis of this study is the beauty contest model of Morris and Shin (2002). The framework has, as noted, been highly influential in the transparency debate and has been developed and applied in a number of subsequent contributions.2 One such development is that of Cornand and Heinemann (2008), which allows the policy authority to limit the disclosure of its private information to a subset of the private sector, and models transparency in terms of the proportion of agents to whom the policy authority’s private signal is revealed – the degree of publicity, in Cornand and Heinemann’s terminology. In what follows, we apply Cornand and Heinemann’s approach in modeling public disclosure.3 Thus agents differ not only in terms of their private signal quality, but also in terms of whether they observe any (partially) public signal originating from the policy authority. In considering the welfare consequences of transparency, we determine whether it is optimal for the policy authority’s private information to be disclosed to none, all, or an intermediate proportion of the private sector. The principal results found in this context can be summarized as follows: (i) the effectiveness of the Pigouvian tax scheme devised by Angeletos and Pavan is unimpaired; (ii) optimal policy intervention, as identified in James and Lawler (2011), combined with optimal disclosure policy, no longer achieves the first-best; however, we note that it might still be preferable to the Angeletos and Pavan tax scheme if the associated implementation costs of the latter exceed those of the former by a sufficiently large margin; (iii) given optimal policy intervention, the optimal degree of publicity will, as in James and Lawler (2011, 2012), still be zero providing the difference in private-information quality across agents is not too large relative to the strength of the strategic complementarity which characterizes agents’ actions. The remainder of the paper is structured as follows. Section 2 outlines the model, while Section 3 derives and discusses the efficient and equilibrium actions, and the associated welfare outcomes, in the absence of policy. Section 4 proceeds to identify the optimal Pigouvian tax scheme, and compares its welfare properties and information-dissemination implications with those relating to a regime featuring a policy instrument which can directly influence the impact of the state variable on payoffs. Finally, Section 5 offers some concluding remarks.

2 In addition to the papers referred to subsequently, see also Svensson (2006), Morris and Shin (2007), Colombo and Femminis (2008), Wong (2008) and Myatt and Wallace (2012), for example. 3 James and Lawler (2012) also uses Cornand and Heinemann’s concept of the ‘degree of publicity’, but in the context of a payoff function with different properties than that of the Morris and Shin model.

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2. The model The model builds upon, and subsumes as special cases, both the original beauty-contest framework devised by Morris and Shin (2002), and the development of the latter by Cornand and Heinemann (2008). The payoff function for the individual private-sector agent is identical in form to that assumed in these two papers. However, we depart from the assumptions of the previous literature by assuming that each agent belongs to one of two groups, denoted A and B according to whether the quality of private signals of the state of the world variable  is relatively good (type A) or relatively poor (type B). Agents comprise a continuum indexed by i and uniformly distributed on the unit interval. For k = A, B, agent i’s payoff is: 2 uik = −(1 − r)(aik − ) − r(Lik − L¯ )

1

(1)

1

where Lik ≡ 0 (aj − aik )2 dj, L¯ ≡ 0 Lj dj. In Eq. (1), aik denotes agent i’s action, while parameter r ∈ (0, 1) measures the strength of the beauty-contest motive, as well as the resulting degree of strategic complementarity, characterizing agents’ actions. It is assumed that each of the two A, B information-types comprises one-half of the population of agents.4 For notational convenience, but without compromising generality, the type-A agents are assumed to comprise the lower half of the unit interval, while its upper half consists solely of type-Bs. Since the average beauty-contest loss across the population is therefore  1/2 1 LiA di + LiB di, it follows that (normalized) welfare is: given by L¯ = i=0

1 W= (1 − r)



i = 1/2



1/2

uiA di + i=0



1

uiB di i = 1/2





1/2

≡−

2

(aiA − ) di + i=0



1

2

(aiB − ) di

(2)

i = 1/2

Following Morris and Shin and much of the related literature, the state variable is assumed to be uniformly distributed on the real line, i.e.  ∼ U(− ∞ , ∞), while, as in Cornand and Heinemann (2008), a noisy signal of  received by the policy authority (whom we refer to below as ‘the policymaker’) is disclosed in an unmodified form to proportion Q of the population of agents. This (partially) public signal is common knowledge among the proportion of agents to whom it is communicated, while its realized value is unknown to the remaining proportion 1 − Q. Here we assume that the policy authority cannot distinguish between type-A and type-B agents, and hence cannot practice a discriminatory disclosure policy. As a consequence, release of the signal to proportion Q of agents implies that Q is the degree of publicity among both agent types. Denoting this signal by y, we have: y=+

(3)

where ∼N(0, 2 ≡ 1/˛) is a noise term.5 So far as agents’ private signals of  are concerned, an individual of a given type observes, for k = A, B: xik =  + εik ,

(4)

2 ≡ 1/ˇ ) is an idiosyncratic noise term which is uncorrelated with all other stochastic variables in where εik ∼N(0, εk k the model, including the error terms in the contemporaneous private signals of other agents (i.e. E(εik ) = E(εik ) = 0 and E(εiA εjA ) = E(εiB εjB ) = E(εiA εjB ) = 0 ∀ i, j). With type-A agents assumed to receive private xi signals which are of higher informa2 <  2 , implying 0 < ˇ < ˇ . tive value regarding  than the counterpart signals received by a type-B agent, we have 0 < εA B A εB Clearly, the optimal forecasts of  and the policymaker’s signal y formed by an individual agent will depend both on its private-signal quality-type, and on whether that agent is ‘informed’, i.e. belongs to the proportion Q to whom y is (fully) disclosed. An informed agent will therefore form forecasts: I () = Eik

˛y + ˇk xik ˛ + ˇk

I (y) = y Eik

(5a) (5b)

I (.) ≡ E(.|x , y) denotes a conditional expectation for k = A, B. Using E U (.) ≡ E(.|x ), the counterpart expressions for where Eik ik ik ik uninformed agents are as follows: U U Eik () = Eik (y) = xik

(5c)

4 If R ∈ [0, 1] denotes the proportion of type-A agents, then this assumption amounts to imposing R = 1/2. While this parameterization has the advantage of enhancing the model’s tractability, it is in fact primarily motivated by the recognition that in the two R = 0 and R = 1 extreme cases the quality (i.e. precision) of private signals across the population is uniform. These extremes therefore correspond exactly to the Morris and Shin (2002) scenario itself, in respect of which the policy-design conclusions arrived at by Angeletos and Pavan (2007b) and James and Lawler (2011) are already known to apply. Hence imposing the restriction R = 1/2 in fact amounts to assigning this parameter the value which is most remote from the values under which the latter two papers’ findings are known to arise. 5 ˛ thus represents the precision of the public signal, with an analogous interpretation attached to ˇk (see the discussion following (4)).

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3. Efficiency and equilibrium in the absence of policy 3.1. Efficient actions and the first-best welfare outcome In order to establish a clear benchmark against which to evaluate equilibrium outcomes both in the absence of policy, and in the presence of an optimally designed policy regime, we derive initially the collectively efficient strategy for an agent of each information-quality type. Such a strategy would be that which a benevolent authority (planner) would instruct an agent of a particular type to adhere to, given that the planner’s objective is the maximization of ex-ante expected welfare.6 In identifying efficient outcomes it is appropriate to assume that the signal y, as given by (3), is publicly observable and part of the information set of each particular agent. Substitution of aiA = 1A xiA + 2A y and aiB = 1B xiB + 2B y into (2) enables us to express welfare in terms of the responses of a representative agent of each type to its private signal xiA or xiB and the public signal y. The collectively efficient (or socially optimal) such responses are those which maximize the ex ante expectation of this welfare expression. Using ˜ 1A and ˜ 2A to denote the efficient response of a type-A agent to signals xiA and y respectively (and using ˜ 1B , ˜ 2B for the counterpart responses of a type-B agent to xiB and y), the first-order conditions for this optimization exercise yield, for k = A, B: ˜ 1k =

ˇk , ˛ + ˇk

˜ 2k =

˛ ˛ + ˇk

The collectively efficient actions, and the associated first-best welfare outcome, are therefore given by: ˇk xik + ˛y ˛ + ˇk

a˜ ik =

˜ |) = − E(W

(6)

2˛ + ˇA + ˇB 2(˛ + ˇA )(˛ + ˇB )

(7)

We note that (6) indicates that the socially efficient action of each agent places a weight on each of the observed signals which purely reflects its relative accuracy. This is a consequence of the fact that in the beauty contest framework of Morris and Shin, there is no social welfare benefit from coordination and thus the parameter r plays no role in influencing action choices consistent with efficiency. 3.2. Equilibrium in the absence of policy In the absence of either a Pigouvian tax scheme or direct policy intervention, each agent’s individually optimal actionchoice conforms with the following linear equation for k = A, B and h = I, U: h h ¯ ahik = (1 − r)Eik () + rEik (a)

(8)

As established by Morris and Shin (2002) and other key papers in the literature, the structural features of the model imply each agent’s individually optimal strategy can be written as a linear function of the signal it observes. Hence: I I aIik = 1k xik + 2k y

(9a)

U = 1k xik aU ik

(9b)

Since

 1/2 i=0

 1/2 i=0

εiA di =

U /2), aU di = (1A iA

a¯ =

1 2



1







1/2 I 1 I  + I y), I  ε di = 0 implies a di = (1/2)(1A aI di = (1/2)(1B 2A i = 1/2 iB i = 0 iA i = 1/2 iB 1 U /2), it follows that the average action across all agents is: aU di = (1B i = 1/2 iB



I I U U I I Q (1A + 1B ) + (1 − Q )(1A + 1B )  + Q (2A + 2B )y



I y), and + 2B

(10)

Individual agents’ expectations of a¯ are formed conditional on the signal(s) they each receive. For k = A, B an informed agent’s expectation of a¯ will consequently be described by the following: I ¯ = (a) Eik

1 I I U U I I I + 1B ) + (1 − Q )(1A + 1B )] Eik () + Q (2A + 2B )y} {[Q (1A 2

(11a)

I () is given by (5a). By making use of (5c), the counterpart expectation for an uninformed agent may be written as: where Eik U ¯ = (a) Eik

6

1 I I U U I I [ Q (1A + 1B ) + (1 − Q )(1A + 1B ) + Q (2A + 2B ) ]xik 2

In other words, expected welfare unconditional on signal realizations.

(11b)

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Proceeding to combine (5a)–(5c), (11a) and (11b) with (8) for k = A, B and h = I, U, enables us to determine, by means of a comparison of the resulting coefficients on xiA , xiB and y with their counterparts in (9a) and (9b), the following equilibrium values of the response coefficients in private-sector actions: I = 2(1 − rQ )ˇh (˛ + ˇj )−1 1k

(12a)

I = ˛[2(˛ + ˇj ) + rQ (ˇh − ˇj )]−1 2k

(12b)

U 1A

(12c)

=

U 1B

=1

where h = A, j = B if the individual is of type k = A, and h = B, j = A if the individual is of type k = B, and where:  ≡ 2˛2 + 2(1 − rQ)ˇA ˇB + (2 − rQ)˛(ˇA + ˇB ). The equilibrium actions for type-A agents are therefore found to be: aIiA =

2(1 − rQ )ˇA (˛ + ˇB )xiA + ˛[2(˛ + ˇB ) + rQ (ˇA − ˇB )]y 2˛2 + 2(1 − rQ )ˇA ˇB + (2 − rQ )˛(ˇA + ˇB )

(13a)

aU = xiA iA

(13b)

while type-B equilibrium actions are as follows: aIiB =

2(1 − rQ )ˇB (˛ + ˇA )xiB + ˛[2(˛ + ˇA ) + rQ (ˇB − ˇA )]y 2˛2 + 2(1 − rQ )ˇA ˇB + (2 − rQ )˛(ˇA + ˇB )

(13c)

= xiB aU iB

(13d)

Substituting (13a)–(13d) into (2) allows us to derive equilibrium expected welfare: E(W |) = − +

2Q  2

2

1 − Q  ˇ + ˇ  B A 2

2

[˛ + (1 − rQ )2 ˇA ](˛ + ˇB ) + [˛ + (1 − rQ )2 ˇB ](˛ + ˇA ) + ˛rQ

ˇA ˇB

rQ 2

− 1 (ˇA − ˇB )

2



(14)

3.3. Discussion It is immediately apparent from comparison of (13a) and (13c) with (6), and of (14) with (7), that the equilibrium actions of those agents to whom y is disclosed are not efficient; consequently, and as in Morris and Shin (2002) and Cornand and Heinemann (2008), the first-best welfare outcome is not achieved in equilibrium.7 To gain further insight into this finding, it is helpful for the moment to focus more sharply on the implications of non-homogeneity of private signal quality for the Morris and Shin framework, rather than its extension by Cornand and Heinemann. Hence we initially consider (13a), (13c) and (14) evaluated for Q = 1 (in which case (13b) and (13d) cease to be of relevance). So far as the implications of an improvement in public information are concerned, the existence of two agent types, such that ˇB < ˇA , renders the conditions under which such an improvement results in higher welfare somewhat more complex. As in the original Morris and Shin paper, with its implicit focus on the ˇA = ˇB case, r ≤ 1/2 is a sufficient (but not necessary) condition for ∂E(W|)/∂˛ > 0. Of greater interest, however, are strong complementarity situations in which r > 1/2, and in respect of which Morris and Shin identify a necessary and sufficient condition for ∂E(W|)/∂˛ > 0, namely ˇ/˛ < (2r − 1)−1 (1 − r)−1 , where ˇ ≡ ˇA = ˇB in terms of our notation. When r > 1/2, one of the principal implications of allowing for the existence of a second type of private agent is that ˇB /˛ < ˇA /˛ < (2r − 1)−1 (1 − r)−1 then becomes merely a sufficient, rather than strictly necessary, condition for ∂E(W|)/∂˛ > 0. Given the existence of a second, privately less wellinformed, group of agents, the possibility arises that ∂E(W|)/∂˛ > 0 will occur when either ˇB /˛ < (2r − 1)−1 (1 − r)−1 < ˇA /˛ or (2r − 1)−1 (1 − r)−1 < ˇB /˛ < ˇA /˛ hold.8 In other words, while a better-quality public signal may, as in the original Morris and Shin framework, reduce welfare, the set of parameter configurations under which this effect arises is smaller when ˇB < ˇA is the case. Furthermore, note that the validity of Svensson’s (2006) ‘conservative benchmark case’ argument for making public signals as accurate as possible is unaffected by the existence of type-B agents, since the plausible condition ˇA ≤ ˛ is sufficient in itself to ensure that ∂E(W|)/∂˛ > 0. To understand the welfare effects of better quality public information (as embodied in ˛), it is useful to bear in mind that when ˇA = ˇB , as in Morris and Shin (2002), an increase in ˛ may either reduce or improve welfare. Although the improved ability of agents to forecast  is beneficial, this may be insufficient to offset the adverse effect of agents being induced to respond even more strongly to a public signal than collective efficiency requires. Our reported findings regarding ∂E(W|)/∂˛

7 Note that imposing ˇA = ˇB ≡ ˇ in (14) yields the equilibrium expected welfare expression derived by Cornand and Heinemann (2008, p. 725), while imposing Q = 1 as well as ˇA = ˇB ≡ ˇ yields the corresponding expression (i.e. Eq. (17)) in Morris and Shin (2002). 8 A specific parameterization which results in ∂E(W|)/∂˛ > 0 when ˇB /˛ < (2r − 1)−1 (1 − r)−1 < ˇA /˛ is {˛ = 1/80, ˇA = 9/8, ˇB = 1, r = 131/256}, while {˛ = 1/32, ˇA = 3, ˇB = 1, r = 13/24} is an instance in which ∂E(W|)/∂˛ > 0 arises when (2r − 1)−1 (1 − r)−1 < ˇB /˛ < ˇA /˛.

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for ˇB < ˇA reveal that the beneficial forecasting-accuracy effect is dominant for a broader set of parameter configurations than when ˇA = ˇB , a result which accords with intuition, given that in relative terms half of the population is informationally disadvantaged, and therefore has more to gain, as regards forecast accuracy, than the other half. Considering now the repercussions of an improvement in the quality of private information, as measured by the precisions ˇA and ˇB , we find that for Q = 1, equilibrium expected welfare is strictly increasing in the values of both of these parameters. Clearly, this ∂E(W|)/∂ˇk > 0 result for k = A, B parallels one of Morris and Shin’s findings for the case in which agents’ (heterogeneous) private signals are drawn from the same normal distribution: namely, that an improvement in the precision of such signals must be beneficial to welfare. The intuition for this is fairly straightforward. As Angeletos and Pavan (2007a) make clear, Morris and Shin’s formal representation of the beauty-contest scenario is characterized by an equilibrium degree of coordination which exceeds the socially efficient amount of coordination.9 In such contexts, the excessive attention paid to public items of information implies that an improvement in the quality of private information must improve welfare, both because of the consequent improved ability of agents to forecast the fundamental (i.e. ), and because each agent appreciates that other participants will now respond less strongly to the public signal, a consideration which induces each agent to react to the two signals it observes in a more appropriate fashion, given their relative information value. Since the beauty-contest motive to respond less strongly to private information than its information value warrants remains present with two types of agent, it is to be expected that an increase in either of the precision parameters, ˇA and ˇB , will improve welfare. A further aspect of the welfare implications of changes in ˇA and ˇB is also of significance. This is that the existence of type-B agents exacerbates the strength of the beauty contest motive, relative to the scenario in which all agents are of type I A. That type-Bs cause a strengthening of type-As’ over-reaction to the public signal becomes evident when we note that 1A I , as given by (12a) and (12b) for k = h = A, j = B and with Q = 1, are related to their homogeneous-quality (i.e. Morris and and 2A



I  Shin) counterparts as follows: 1A



ˇB <ˇA

I  < 1A



ˇB =ˇA

I  and 2A



ˇB =ˇA

I  < 2A

ˇB <ˇA

. Furthermore, the extent of the departure

of informed type-A response coefficients from their efficient values ˜ 1A and ˜ 2A also worsens when type-B agents are present, as can be seen if we follow Morris and Shin in re-expressing the equilibrium action (13a) as a sum comprised of the efficient action (6) for k = A plus an additional term in y − xi :



aIiA 

Q =1

=

ˇA xiA + ˛y + (y − xiA ) ˛ + ˇA

(15)

where: =

r˛ˇA (2˛ + ˇA + ˇB ) (˛ + ˇA )[2˛2 + 2(1 − r)ˇA ˇB + (2 − r)˛(ˇA + ˇB )]

The positive composite parameter measures the severity of the beauty contest-induced overweighting of y (and underweighting of xiA ) relative to the efficient weights that should be placed upon them by a type-A agent. Differentiating with respect to ˇB , we find that a narrowing of the gap between ˇB and ˇA unambiguously lowers this measure of inefficiency: ∂ 2r(1 − r)˛ˇA (˛ + ˇA ) =− <0 2 ∂ˇB [2˛2 + 2(1 − r)ˇA ˇB + (2 − r)˛(ˇA + ˇB )]

(16)

Hence the less pronounced is the difference between the precisions of the two types’ private signals, the smaller is the extent of the inefficiency in type-A actions occasioned by the beauty contest.10 It is apparent therefore that the existence of a privately less well-informed group of agents does worsen the over-weighting of public information that characterizes this scenario. This is ultimately attributable to the fact that each type-A, desiring in part to align its action with the average action of all agents, recognizes that, even were there no beauty contest, the type-Bs among those agents would necessarily respond more strongly to the public signal than type-As, simply because type-B signals are comparatively poor. In addition, each type-A agent realizes that every other agent is reasoning along identical lines: indeed, agents of both type engage in higher-order belief formation regarding average expectations (and the average thereof, etc.), and it is such beauty-contest second-guessing of course which compounds the tendency for agents to react more strongly to the public signal than its quality justifies. Finally within this section, we describe briefly the implications of there being two types for the relationship between equilibrium welfare and the degree of publicity Q. In analyzing the ˇA = ˇB ≡ ˇ case, Cornand and Heinemann (2008) identify a necessary and sufficient condition for the optimal degree of publicity to be partial, namely that the inequality ˛/ˇ < 3r − 1 should hold. The essential implication of allowing for ˇB < ˇA , however, is that the counterpart to Cornand and Heinemann’s

9 Applying Angeletos and Pavan’s (2007a) concepts, we may identify the equilibrium degree of coordination exhibited by a type-k agent, where k = A, I I I I I I I I ¯ where 1k ¯ by (11a), with Q in all xik + 2k y = (1 − ϕ)Eik () + ϕEik (a), is given by (12a), 2k by (12b), Eik () by (5a) and Eik (a) B, as the ϕ solution to aIik = 1k ˜¯ where a˜ ik is these expressions set to unity. Similarly, the efficient degree of coordination is the ϕ ˜ I () + ϕE ˜ I (a), ˜ solution to a˜ ik = ˜ 1k xik + ˜ 2k y = (1 − ϕ)E ik

ik

I I ˜ I ˜ I ¯ by Eik ¯ = (1/2)[(˜ 1A + ˜ 1B )Eik () by (5a) and Eik (a) (a) () + (˜ 2A + ˜ 2B )y]. Regardless of the agent’s type, given Q = 1 the equilibrium degree of given by (6), Eik ˜ = 0. It is apparent therefore that in the Morris and Shin (2002) framework the existence coordination is ϕ = r, while the efficient degree of coordination is ϕ of two types of agent does not affect the values taken by the coordination measures devised by Angeletos and Pavan. 10 Note that in the limit as ˇB approaches ˇA , takes the value it implicitly has in Morris and Shin’s Eq. (21): lim = r˛ˇA /(˛ + ˇA )[˛ + (1 − r)ˇA ]. ˇ →ˇ B

A

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condition, namely ˛/ˇB < 3r − 1, ceases to be necessary to ensure full publicity is welfare-dominated by some particular positive value of Q below one.11 More generally, when type-Bs comprise half of all agents, the set of parameter configurations under which full publicity is optimal is larger compared to when all agents are of type A. Clearly, this result derives from the fact that the effect of higher publicity in improving forecast accuracy within the population of informationally underendowed type-Bs (as well as the proportion Q of type-As to whom y is disclosed), must now, for a more general set of parameter values, be sufficiently powerful to dominate the adverse coordination effect associated with more widespread knowledge of y. 4. Optimal policy design 4.1. The optimal Pigouvian tax We begin our analysis of the effectiveness of alternative policies with two types of agent, distinguished by the precision of their private information, by considering a tax/subsidy scheme similar to that studied by Angeletos and Pavan (2007b, 2009). In particular, we now assume that after every agent has chosen its action, each individual pays a tax which depends ¯ and upon (i) the value of its chosen action ai , and (ii) a marginal tax rate which is a linear function of the average action a, of the realized value of the public signal y.12 The agent also receives ex-post a lump-sum transfer, assumed in this instance to equal the average tax liability of each agent. The individual agent’s ‘net transfer’ is therefore given by: ti = 2(a¯ a¯ + y y)(a¯ − ai )

(17)

where a¯ and  y are the tax-scheme parameters, and 2(a¯ a¯ + y y) is the marginal tax rate on the agent’s action-choice ai . Hence as regards the payoff function for an agent of type k = A, B we now have: 2

uik = −(1 − r)(aik − ) − r(Lik − L¯ ) + 2(a¯ a¯ + y y)(a¯ − aik )

(18)

Focusing appropriately on the Q = 1 case, so that every agent is ‘informed’ about y, the individually optimal action k = A, B is: I I ¯ − y y () + (r − a¯ )Eik (a) aIik = (1 − r)Eik

(19)

I () is given by (5a). By virtue of the fact that the equilibrium action of a type-k agent will be of form aI = I x + where Eik ik 1k ik I I + I ) + (I + I )y], we may 2k y, and that consequently the equilibrium aggregate action will be given by: a¯ = (1/2)[(1A 1B 2B 2A use (5a), (19) and the method of undetermined coefficients to derive the following solutions for the equilibrium response coefficients: I = 2(1 − r)(˛ + ˇj )ˇh (1 + a¯ − r) −1 1k

(20a)

I = [2(˛ + ˇj )[(1 − r)˛ − (˛ + ˇh )y ] + {(1 − r)˛(ˇj − ˇh ) − [2ˇh (˛ + ˇj ) + ˛(ˇj − ˇh )]y }(a¯ − r)] −1 2k

(20b)

where h = A, j = B if the individual is of type k = A, and h = B, j = A if the individual is of type k = B, and where: ≡ (1 − r + a¯ ){2(˛ + ˇA )(˛ + ˇB ) + (a¯ − r)[2ˇA ˇB + ˛(ˇA + ˇB )]}. Combining (20a) and (20b) with the welfare expression, which continues to be given by (2),13 equilibrium welfare may be succinctly written as follows: E(W |) = −

1 2







2(a¯ + y )2 2 1 I 2 1 I 2 1 I 2 I 2 (1A ) + (1B ) + (2A ) + (2B ) +  ˛ ˇA ˇB (1 − r + a¯ )2

 (21)

I , I , I and I are given by (20a) and (20b). The optimal value(s) for the tax-scheme parameters  and  are where 1A y a¯ 1B 2B 2A those which maximize (21) and are found to be given by:

a∗¯ = −y∗ = −

r˛(2˛ + ˇA + ˇB ) ˛(ˇA + ˇB ) + 2ˇA ˇB

(22)

11 A particular parameterization under which Q = 1 is optimal when ˛/ˇA < ˛/ˇB < 3r − 1 is: {˛ = 1, ˇA = 35, ˇB = 3, r = 29/64}. Note that Cornand and Heinemann’s necessary and sufficient condition implies that r ≤ 1/3 (relatively weak complementarity) and 2ˇ ≤ ˛ (relatively good public information) are alternative sufficient (but not necessary) conditions for Q = 1 to be optimal in the ˇA = ˇB ≡ ˇ case. When ˇB < ˇA is the case, r ≤ 1/3 and 2ˇA ≤ ˛ remain sufficient (but non-necessary) conditions for Q = 1 to be optimal. 12 Alternatively, and without affecting the qualitative nature of the results, the tax may be specified to be contingent on the realized value of the state variable  itself, as in Angeletos and Pavan (2007b). 13

 1/2

Note that in aggregating individual-agent payoffs, as given by (18), the aggregate of the transfer term is zero, 2(a¯ a¯ + y y)[

aiB ) di ] = 0, since a¯ =

 1/2 i=0

aiA di +

1

i = 1/2

aiB di.

i=0

(a¯ − aiA ) di +

1

i = 1/2

(a¯ −

J.G. James, P. Lawler / Journal of Economic Behavior & Organization 83 (2012) 342–352

349

Therefore, under the optimal Pigouvian tax scheme, each individual agent for k = A, B will be subject to a net transfer of ∗ = 2 ∗ (a ¯ − y)(a¯ − aik ). The resulting expected equilibrium welfare outcome coincides with the first-best, as described by tik a¯ (7):14



E(W |)

a¯ =  ∗ , y = − ∗ a¯

=−

a¯

(2˛ + ˇA + ˇB ) 2(˛ + ˇA )(˛ + ˇB )

(23)

The ability of optimal tax policy to bring about the first-best is, of course, attributable to its nullification of agents’ individual incentives to coordinate their actions. This incentive-neutralization effect is achieved by exploiting information ¯ which only becomes available after actions have been taken. Significantly, the design of the (in this case, regarding a) optimal tax scheme is not affected by the existence of differing types of agents, since the crucial aspect of the scheme is that it specifies each transfer to depend on the action taken by the individual agent concerned: the quality of the private information underlying that action is immaterial to the related transfer. Given these facts, however, it is worth bearing in mind as well that the effectiveness of this tax scheme does depend on the ex-post observability of individual agents’ actions, and that the policy authority’s ability to ascertain the value of individual actions (and hence also the average action) will typically hinge on the existence of an enforcement technology. Clearly, any costs  associated with enforcing and ensuring 15 compliance with the scheme will inevitably reduce the welfare outcome E(W |) ∗ ∗ below its first-best value. a¯ =  , y = − a¯

a¯

4.2. Direct policy intervention: the optimal policy rule Having studied the applicability of Pigouvian taxation with two agent types, we now proceed to investigate whether direct policy intervention can be similarly effective in remedying the welfare losses arising from beauty-contest behavior. Allowing, as in James and Lawler (2011, 2012), for a policy instrument which can modify the state variable’s impact, the individual agent’s payoff now becomes (for k = A, B): 2 uik = −(1 − r)(aik −  − g) − r(Lik − L¯ )

(24)

It is assumed that the policy instrument g is set contingent on the policymaker’s signal y in accordance with the rule g = y, where the policy-rule parameter is common knowledge across all agents. As in Sections 2 and 3, we also assume that the signal y is disclosed to proportion Q of both types of agents. Applying to this variant of the model the solution procedure followed in Section 3.2 allows us to identify the following expressions for the equilibrium response coefficients of private-sector agents, for given values of and Q: I 1k =

I 2k =

2ˇh (˛ + ˇj )[1 − rQ + r(1 − Q ) ]

(25a)

k + 2(1 − rQ )ˇh (˛ + ˇj ) k + [k + 2(1 − r)ˇh (˛ + ˇj )]

(25b)

k + 2(1 − rQ )ˇh (˛ + ˇj )

U U 1A = 1B =1+

(25c)

where: k ≡ ˛[2(˛ + ˇj ) + rQ(ˇh − ˇj )], and h = A, j = B if the individual is of type k = A, and h = B, j = A if the individual is of type k = B. Appropriate aggregation of (24) allows us to express equilibrium welfare as follows: 1 2 2 I 2 −1 I 2 −1 I I E(W |) = − [Q {(1A ) ˇA + (1B ) ˇB + [(2A − ) + (2B − ) ]˛−1 } + (1 − Q )[(1 + )2 (ˇA−1 + ˇB−1 ) + 2 2 ˛−1 ]] (26) 2 where the private-sector response coefficients are given by (25a) and (25b). Solving ∂E(W|)/∂ = 0 to determine the optimal value of for given Q, yields:

∗ = −

1 2 {[2ˇA ˇB + ˛(ˇA + ˇB )]{Q 2 r 2 ˛2 (ˇA − ˇB ) + 4(˛ + ˇA )(˛ + ˇB )[(˛ + ˇA )(˛ + ˇB ) − Q r˛(ˇA + ˇB )]}  2

2

− 8ˇA ˇB (˛ + ˇA ) (˛ + ˇB ) }

(27)

−1

I I I I 14 Note that in the presence of the Pigouvian tax, solving the equation 1A xiA + 2A y = (ˇA xiA + ˛y)(˛ + ˇA ) + (y − xiA ) , where 1A and 2A are given by (20a) and (20b) for k = h = A, j = B, and with the optimal-design requirement y = −a¯ imposed, provides us with a variant of the inefficiency measure stated

after (15): = ˇA {r˛(2˛ + ˇA + ˇB ) + [2ˇA ˇB + ˛(ˇA + ˇB )]a¯ }(˛ + ˇA ) collapses to zero when a¯ is set in accordance with (22). 15

−1

˚−1 where ˚ ≡ 2˛2 + 2(1 − r + a¯ )ˇA ˇB + (2 − r + a¯ )˛(ˇA + ˇB ). Significantly,



Enforcement costs which ensure compliance (truth-telling re ai ) by individuals implies a modification of (23) to E(W |) −1

a¯ =  ∗ , y = − ∗ a¯

= −(2˛ + ˇA +

a¯

ˇB ) [2(˛ + ˇA )(˛ + ˇB )] − C, where C is the (normalized) aggregate welfare loss associated with the enforcement technology. Given that such costs are unlikely to vary with economic fundamentals, their existence implies a lump-sum per-agent tax of (1 − r)C would be appropriate: the fiscal transfer would ∗ = 2a∗¯ (a¯ − y)(a¯ − aik ) − (1 − r)C. thus be modified to tik

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where: 2

2

2

 ≡ [2ˇA ˇB + ˛(ˇA + ˇB )]{Q 2 r 2 ˛2 (ˇA − ˇB ) + 4(˛ + ˇA ) (˛ + ˇB ) − 4Q r(˛ + ˇA )(˛ + ˇB )[(2 − r)ˇA ˇB + ˛(ˇA + ˇB )]} Combining (26) and (27), maximized expected welfare may be expressed as:



E(W |)

= ∗

=

∗ 2Q 2 2 + {˛(˛ + ˇA )(˛ + ˇB )(ˇA − ˇB ) − r[2ˇA ˇB + ˛(ˇA + ˇB )][Q (1 − r)˛(ˇA − ˇB ) ˛ 

+ r(˛ + ˇA )(˛ + ˇB )(ˇA + ˇB )]}

(28)

It is straightforward to demonstrate that, for all admissible values of Q, maximized expected welfare, as given by (28), is less than the first-best level given by (7). An immediate implication is that, provided any implementation or enforcement costs associated with fiscal policy are sufficiently small, direct policy intervention is less effective than a Pigouvian tax in remedying the welfare loss occasioned by the beauty contest. There are two fundamental reasons for this. First, unlike when agents are of a single type, non-homogeneity of private information quality implies the policy rule cannot be tailored to induce agents of both types to react to their private signals in the socially efficient fashion. While it is readily shown that a setting of exists which can elicit efficient responses to xi signals by agents of a particular information-quality type, nevertheless cannot be set in such a way as to induce such behavior by both types simultaneously. Consequently, optimal policy intervention with two agent types, who differ in their private information quality, necessarily involves a compromise between maximizing the average payoff of each type.16 Second, the fiscal scheme exploits an information-set which is far more multifarious than that available to a policymaker whose sole instrument must be set before agents’ actionchoices are known. Given this informational advantage, it is not surprising that the Pigouvian tax outperforms direct policy intervention. An obvious qualification to this conclusion, however, is that the availability of the Pigouvian-policy informationset is contingent on the truthful disclosure by each agent of their individual tax liability under the scheme. As previously noted, enforcement costs are likely to be incurred in ensuring such compliance: the potential for such costs to exceed the implementation costs associated with direct policy intervention implies that, in practice, the relative welfare-performance of the two policies might remain ambiguous. Finally, we turn our attention to the disclosure-policy implications of these results. Identification of the optimal degree  /∂Q = 0. The sole solution of relevance is:17 of publicity involves solving the first-order condition ∂E(W |) ∗

=

Q =

2(˛ + ˇA )(˛ + ˇB )[(1 − r)ˇA − (1 + r)ˇB ] r(ˇA − ˇB ){2(2 − r)ˇA ˇB + ˛[(1 − r)ˇA + (3 − r)ˇB ]}

(29)

We consider whether the optimal value of Q is zero, a proper fraction, or unity. Using (29), we can identify two critical values of r, which we denote by r and r

, ordered according to 0 < r < r

< 1, and given by:



1 r ≡ 2(ˇA − ˇB )



2˛ + 3ˇA − ˇB −

˝ 2ˇA ˇB + ˛(ˇA + ˇB )

1/2 

,

r

≡

ˇA − ˇB ˇA + ˇB

where: ˝ ≡ ˛{ˇA3 − ˇB3 + 6ˇB (5ˇA2 − ˇB2 ) + 5ˇA ˇB (ˇA − ˇB ) + 4˛[˛(ˇA + ˇB ) + ˇA2 + ˇB (8ˇA − 3ˇB )]} 2

+ 2ˇA ˇB [(ˇA + ˇB ) + 4(ˇA2 − ˇB2 )] Using these values, the principal conclusions to be drawn regarding the optimal degree of publicity, denoted by Q∗ , can be summarized in the following Proposition: Proposition 1. Under optimal policy intervention, and for given values of ˇA and ˇB such that ˇB < ˇA , Q∗ = 1 if (and only if) r ≤ r , while a partial degree of publicity is optimal and given by Q∗ = Q if (and only if) r is such that r < r < r

. Zero publicity is optimal (Q∗ = 0) if (and only if) r

≤ r. Proof.



(i) Note that E(W |)

= ∗

as given by (28) is continuous in Q for all Q ∈ [0, 1]. (ii) Evaluating the derivative of (28)



with respect to Q for Q = 0, it is found that (∂ [E(W |)

= ∗

 

]/∂Q )

Q =0

> (<)0 ⇔ r < (>)r

. (iii) Analysis of (29) implies 1 ≤ Q

16 In other words, a particular setting of , namely A ≡ −˛/(˛ + ˇA ), combined with Q = 0, induces type-A agents to choose the action aiA = ˜ 1A xiA = ˇA xiA /(˛ + ˇA ), and hence ensures the ex-ante expected payoff for type-As is equal to its first-best value, given by E(u˜ iA |) = −1/(˛ + ˇA ). Similarly, under Q = 0 the alternative rule parameter value B ≡ − ˛/(˛ + ˇB ) induces each type-B agent to choose the action aiB = ˜ 1B xiB = ˇB xiB /(˛ + ˇB ), and thus leads to the first-best ex-ante expected payoff for type-Bs, namely E(u˜ iB |) = −1/(˛ + ˇB ). Consistent with Tinbergen’s Principle, however, a policymaker who adopts both E(u˜ iA |) and E(u˜ iB |) as targets of policy, cannot by means of a single instrument achieve both simultaneously. Such a task amounts to bringing ˜ |) = E[(u˜ iA + u˜ iB )|]/2, as given by (7), and requires two or more independent policy instruments. about the first-best welfare outcome E(W 17 The second solution to this first-order condition is given by: Q

= 2(˛ + ˇA )(˛ + ˇB )[ˇA − ˇB + r(ˇA + ˇB )]r−1 (ˇA − ˇB )−1 {2(2 − r)ˇA ˇB + ˛[(1 − r)ˇB + (3 − r)ˇA ]} −1 .We disregard this solution since, given our assumption that ˇB < ˇA , 1 < Q

is the case.

J.G. James, P. Lawler / Journal of Economic Behavior & Organization 83 (2012) 342–352

351

 iff r ≤ r : (i) and (ii) then imply that for r ≤ r E(W |) is monotonically increasing in Q for all Q ∈ [0, 1], and hence that

= ∗  ∗





Q = 1 ⇔ r ≤ r . (iv) From (29), r < r < r implies 0 < Q < 1: E(W |) then has a unique interior maximum at Q∗ = Q ∈ (0, 1). ∗

=

(v) Also from (29), r

≤ r implies Q ≤ 0: this and (ii) then imply that Q∗ = 0 iff r

≤ r. Before proceeding to discuss these findings, the fact that r

is lower, the smaller the difference between ˇA and ˇB , allows us to formulate Proposition 2 regarding how our optimal-publicity results relate to the non-homogeneity of signal quality: Proposition 2. Given optimal policy intervention, the set of parameter configurations under which zero publicity is socially desirable is larger, the less is the difference in private-signal quality across agent-types. Significantly, Proposition 1 indicates that even when private-signal quality is not homogenous, the anti-transparency conclusions of James and Lawler (2011) continue to apply if the degree of strategic complementarity characterizing agents’ action is sufficiently high.18 Thus in situations in which the degree of complementarity is strong enough to ensure that r

≤ r, the detrimental coordination effect associated with disclosure of the policymaker’s signal will be sufficiently powerful to render zero transparency optimal, despite the limited extent to which optimally formulated policy intervention can compensate for the entailed decrease in agents’ forecasting accuracy regarding fundamentals. Of course, if the incentive to engage in beauty contest behavior is comparatively weak, such that r ≤ r , the disclosure-related trade-off between inducing inefficiently high coordination in respect of y, and enhancing the accuracy of agents’ forecasts of , will be sufficiently favorable to justify the policy authority practicing full publicity. In situations of intermediate complementarity strength, in the sense that r lies between the two identified critical values, disclosure to a subset of both types becomes desirable, with the beneficial welfare impact of enlarging the information-set available to proportion Q of agents balanced by the need to restrict to Q the fraction of agents who engage in inefficiently high coordination on y. Note that in the light of our earlier discussion, the fact that 0 < Q∗ ≤ 1 when r < r

is ultimately attributable to the inability of optimal policy intervention to compensate fully for the information deficiencies experienced by agents who are deprived of the policymaker’s signal. The intuition for Proposition 2 then follows naturally: namely, that when the two types differ relatively little in the quality of their private signals, the compromise nature of the optimal policy rule is then less damaging to the interests of each type, since the rule parameter ∗ then approximates more closely its ideal setting for each, as given by k = − ˛/(˛ + ˇk ) for k = A, B. 5. Concluding remarks The analysis of this paper has extended recent work featuring strategic complementarities and heterogeneously informed agents to capture the idea that all individuals may not make use of information of identical precision. The implications of differences in private information quality for the nature of equilibrium, potential policy responses to correct the identified inefficiency, as well as the desirability of information disclosure by policymakers were each considered. Our results indicate that the inefficiency characterizing equilibrium behavior is exacerbated by non-uniformity of private information quality, though the set of circumstances under which, in the absence of policy, disclosure of public information may be socially beneficial is somewhat broader. In addressing this suboptimality issue, a Pigouvian tax, as originally devised by Angeletos and Pavan (2007b, 2009), was found to provide an effective instrument for inducing socially efficient actions by the private sector and to imply that full transparency by the policymaker is desirable. On the other hand, direct policy intervention of the type considered by James and Lawler (2011, 2012) is unable to attain the first-best welfare outcome, reflecting the fact that the optimal policy response to the policymaker’s signal involves a compromise between what is desirable from the separate perspectives of the two differentially informed groups of private sector agents. Furthermore, the transparency implications of policy intervention of this nature are far from unequivocal: the optimal degree of publicity might be zero, intermediate or full. However, if the strategic complementarity underlying agents’ actions is sufficiently strong relative to the magnitude of the difference in private information quality, the conclusions found in James and Lawler (2011, 2012) regarding the desirability of zero transparency extend to the present setting. Finally, as observed, if the implementation costs of monitoring and enforcing the Pigouvian tax scheme are non-negligible, it is possible that direct policy intervention might be preferred despite its inability to attain the social optimum. Of course, the above conclusions have been arrived at in the context of a specific model, albeit one which has received much attention in the literature. In fact, the beauty contest motive which lies at the center of Morris and Shin’s (2002) contribution might, as the authors recognize and as pointed to by Keynes (1936) in his original formulation of the concept, be viewed as particularly relevant to financial markets. This is also a setting in which the potential presence of so-called ‘noise traders’ gives the idea of differentially informed participants a particular resonance. Nonetheless, it is of interest to briefly consider the likely wider applicability of our findings. With regard to this question, we note that an alternative context in which the consequences of heterogeneous private sector information have been explored is that of price-setting behavior by monopolistically competitive firms, as embedded

18 Precisely what is meant by ‘sufficiently high’ will, of course, depend upon the extent of the disparity in quality of the two types’ private information. It is interesting to note, for example, that when type-A private signals are three times more precise than type-Bs, so that ˇA = 3ˇB and hence r

= 1/2, zero publicity is then optimal if r lies in the upper half of its admissible range of values.

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within the macroeconomic framework of Woodford (2002). An important feature of such models is that, unlike in Morris and Shin, the benefits of coordination are undervalued by private sector agents.19 Despite this difference, and reflecting the undervaluation referred to, the equilibrium is similarly characterized by a departure from the social optimum. It is evident that, in models of this nature, the principle underlying the effectiveness of the Pigouvian tax scheme will remain operative, and allow the correction of the inefficiency which is present and exacerbated by differences in the quality of private information across agents. With regard to direct policy intervention, the presence of two groups differentiated by the quality of their private information will inevitably mean that, just as in the beauty contest setting, such intervention is unable to achieve the social optimum. It is also clear that the potential remains for differences in the optimal degree of transparency to arise for different parameter combinations. However, reflecting the different nature of the individual incentives associated with the payoff functions arising in Woodford-type models, the precise nature of the influences of the strength of the strategic complementarity, and the extent of the divergence in information quality between groups, on optimal transparency may well differ somewhat from those identified in the context of the Morris and Shin framework. Notwithstanding this potential difference, it is clear that the general tenor of our results extend largely unchanged to this alternative setting, thus indicating wider relevance and validity for our findings. References Angeletos, G.-M., Pavan, A., 2007a. Efficient use of information and social value of information. Econometrica 75 (4), 1103–1142. Angeletos, G.-M., Pavan, A., 2007b. Socially optimal coordination: characterization and policy implications. Journal of the European Economic Association 5 (2–3), 585–593. Angeletos, G.-M., Pavan, A., 2009. Policy with dispersed information. Journal of the European Economic Association 7 (1), 11–60. Blinder, A.S., Ehrmann, M., Fratzscher, M., De Haan, J., Jansen, D.-J., 2008. Central bank communication and monetary policy: a survey of theory and evidence. Journal of Economic Literature 46, 910–945. Carroll, C.D., 2003. Macroeconomic expectations of households and professional forecasters. Quarterly Journal of Economics 118 (1), 269–298. Colombo, L., Femminis, G., 2008. The social value of public information with costly information acquisition. Economics Letters 100 (2), 196–199. Cornand, C., Heinemann, F., 2008. Optimal degree of public information dissemination. Economic Journal 118 (528), 718–742. Cruijsen, C.A.B. van der, Eijffinger, S.C.W., 2007. The Economic Impact of Central Bank Transparency: A Survey. CEPR Discussion Paper 6070. Centre for Economic Policy Research, London. Geraats, P.M., 2002. Central bank transparency. Economic Journal 112, F532–F565. Hellwig, C., 2005. Heterogeneous Information and the Welfare Effects of Public Information Disclosures. UCLA Economics Online Papers 283. Department of Economics, University of California, Los Angeles. James, J.G., Lawler, P., 2008. Aggregate demand shocks, private signals and employment variability: can better information be harmful? Economics Letters 100, 101–104. James, J.G., Lawler, P., 2011. Optimal policy intervention and the social value of public information. American Economic Review 101 (4), 1561–1574. James, J.G., Lawler, P., 2012. Strategic complementarity, stabilization policy and the optimal degree of publicity. Journal of Money, Credit and Banking 44 (4), 551–572. Keynes, J.M., 1936. The General Theory of Employment, Interest, and Money. Macmillan, London. Mankiw, N.G., Reis, R., 2002. Sticky information versus sticky prices: a proposal to replace the new Keynesian Phillips curve. Quarterly Journal of Economics 117 (4), 1295–1328. Morris, S., Shin, H.S., 2002. Social value of public information. American Economic Review 92 (5), 1521–1534. Morris, S., Shin, H.S., 2007. Optimal communication. Journal of the European Economic Association 5, 594–602. Myatt, D., Wallace, C.F., 2012. Endogenous Information Acquisition in Coordination Games. Review of Economic Studies 79, 340–374. Roca, M.F., 2010. Transparency and Monetary Policy with Imperfect Common Knowledge. IMF Working Paper WP/10/91. Svensson, L.E.O., 2006. Social value of public information: Morris and Shin (2002) is actually pro transparency, not con. American Economic Review 96, 448–452. Wong, J., 2008. Information acquisition, dissemination, and transparency of monetary policy. Canadian Journal of Economics 41 (1), 46–79. Woodford, M., 2002. Inflation stabilization and welfare. Contributions to Macroeconomics 2, 1–53.

19 In Angeletos and Pavan’s (2007a) terminology, the equilibrium degree of coordination lies below the socially optimal degree of coordination – the opposite configuration to that exhibited by Morris and Shin’s model.