Outubro de 2016
Working
Paper
431
Communication Policy with Public
Uncertainty
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Communication Policy with Public Uncertainty
João Lídio Bezerra Bisneto São Paulo School of Economics, FGV
May 24th 2016
Abstract
This paper analyses how to design a communication policy in a setting with both incomplete information and uncertainty about the future. In the model developed, both agents and the information authority have only noisy knowledge of the state of the world and the degree of uncertainty of the authority is unknown to the agents. The information authority chooses how much information to provide to private agents and how precise, or disperse, such information will be. Agents faces costs on information acquisition and choose how much of it to obtain. In the model developed in this paper, the presence of incomplete information does not alter the equilibrium choices, as a separating equilibrium can be achieved. In all cases analysed, the information authority would prefer to offer signals with the lowest dispersion possible.
Key Words: communication policy, incomplete information, uncertainty, information acquisition
1
Introduction
The issue of public communication has never been more discussed in the realm of monetary policy. The last decades have shown an increase in agreement between economists that public communication about policy is important and should be studied further. Indeed, with interest rates extremely low and further traditional monetary stimulus unlikely, the importance of optimally communicating decisions and future policy has grown greatly.
Before the 1990s, the art of monetary policy was a well-kept secret in most of the world. Decisions were shielded from the public. Such secrecy is quite peculiar for democratic institutions like the Fed and other central banks. Since 1994, however, the Fed began publishing statements about monetary policy decisions. From that moment on, monetary policy entered a process of increasing transparency. (Poole, 2006)
Communication policy has been, then, slowly brought to the front of the academic debate. Since the Great Recession, some central banks were caught in what is called the zero lower bound of interest rates. With less room for traditional monetary accommodation, a good communication policy became more important and this importance induced more academic discussion. Questions focus on the different aspects of a communication policy that you can change to increase public welfare. (Reis, 2011) (Chahrour, 2014) (Colombo, et al., 2014)
A problem Central banks and Governments regularly face is of how much public information to communicate. The issue is further complicated not only if communication is noisy, which is possibly the case with complex policies, but also if there is uncertainty about the true extent and/or effects of the policy. As the public entity is uncertain, it is difficult for agents to assess both the policy and the public uncertainty.
This paper attempts to answer the question of how communication policy should be structured in a setting of uncertainty. We build an incomplete information model in which an information authority sends public information to the agents. The agents face costs in acquiring public information and choose how much of it to obtain. The choice of optimal communication policy consists in the amount and dispersion of public information which maximizes the Government preferences. We find that, with little restriction, the Government can formulate a strategy which leads to the social optimal outcome, even when agents do not know how
This work is organized in the following manner. In Section II, related literature is reviewed. In Section III, we lay the structure of the model, the main assumptions and their consequences. In Section IV, we derive the results and discuss them. Finally, Section V concludes the paper.
2
Literature Review
The literature on the role of public information is a growing literature. Woodford (2005) explains some of the origins of the literature and shows that discussions were just beginning to surge at the time. The main point of debate was if public communication was valuable or not. Morris & Shin (2002), for example, argues that in some aspects, public information can be bad for social welfare. Woodford (2005) contended that public information is most likely beneficial. In a literature survey, Blinder et al (2008) shows some advances had been made and most of the research pointed to an important and beneficial role of public communication. Nonetheless, Blinder et al (2008) also emphasized the lack of consensus about the best communication strategies and the need for further research to address numerous deficiencies in such economic literature.
Recently, the literature began to be enriched. Research took form in different but related approaches. The work of Morris & Shin (2002) gave rise to other studies on the social value of information. Notably, Svensson (2006), reviewing that same paper, concludes that actually public information is valuable. Building on them, Angeletos & Pavan (2007) expands this analysis to a broader class of games and shows that the issue is much more ambiguous and scenario dependent. This research induced a greater exploration of the role of externalities and coordination settings.
Hellwig & Veldkamp (2009) shows that strategic complementarities can induce multiple equilibria in information dependent models. Colombo et al. (2014) takes another take on strategic complementarities and show that the social value of public information is dependent on how efficient acquisition of information is. One interesting conclusion of this last study is that efficiency in the acquisition of information can be independent of the efficiency of using the information.
Empirical research has also been undertaken on the field. For example, Ehrmann et al (2012) shows that transparency tend to have beneficial effects on the dispersion of private forecasts. However, decreasing marginal returns were found pertinent. Hansen et al (2014) uses a linguistics method to try to verify any empirical effect of transparency on internal deliberation in monetary policy formulation. The study found both positive and negative effects, but, overall, more transparency seems to improve the debate in monetary policy committees.
Some literature tries to answer the question of how to formulate communication policy. There are many questions to be asked about what makes a communication policy optimal. How much information should the information authority provide to private agents? How precise should the announcements be? When should you make an announcement? How frequent should such announcements be? Formulating an information policy has the potential to be an extremely complex process.Gaballo (2016) warns of the dangers of an ambiguous information policy.
Reis (2011) focuses on the analysis of when should public announcements should be made and finds that announcements should have their timing balanced to avoid being sent too early or too late, as both cases generate suboptimal acquisition of information or private actions.
Chahrour (2014) tries to answer the specific questions of how much information to provide and how precise should they be. The study found that the optimal amount of information made available is dependent on various characteristics, but information should always be as precise as possible. This conclusion on the precision of information is also shared by Roca (2010), which study the effect of the precision of public information in the volatility of aggregate variables, such as the price level.
This paper focus on the issue of optimal communication policy and builds on the works of Chahrour (2014) and Angeletos & Pavan (2007). Incomplete information is introduced and a new dimension for communication policy is introduced, as it may be informative per se.
3
Model
The model developed in this paper can be better understood as two different games played in sequence. The last game to be played is a simple static game in which agents try to
interpretations for such setting. Agents may be trying to set contracts in accordance with future inflation, but future inflation is unknown when the contracts are signed. Similarly, agents may be deciding the right amount of investment or guessing the future path of taxes and/or interest rates.
The first game is a signalling game with information acquisition. The role of this game is to determine the amount of information that will be available to agents on the following static game. The information authority, which will be referred from now on as Government for simplicity, is the Stackelberg leader in the game. The Government chooses the communication policy: how much public information about the future state of the world to provide and how precise or, equivalently, disperse such information is. A key assumption is that the information provided cannot be infinitely precise, as the process of providing information can be imperfect and the agents can interpret a speech or a report differently than intended. Moreover, the Government does not know with certainty the future. One way to interpret this assumption is think of the Government as a monetary authority trying to inform private agents of the future path of interest rates or inflation. Such uncertainty is plausible in both the case of Delphic and Odyssean forward guidance, as it is almost impossible to know with certainty any future shocks to the economy or to the central bank reaction function.
After the Government has chosen the communication policy, agents observe it and choose how much of information to obtain. Information acquisition is not costless and acquiring all information provided by the Government may not be optimal. Also, agents do not know how uncertain the Government is about the future. This degree of incomplete information gives an extra meaning to the communication policy chosen by the Government, as it will also influence agents’ beliefs about public uncertainty. Thus, the game is a version of a traditional signalling game.
The whole model has similarities with the work in Chahrour (2014), Angeleto & Pavan (2007) and Morris & Shin (2002). Indeed, a version of this model with complete information is treated in its entirety by Chahrour (2014). This paper, however, refrains from treating the coordination issue developed by the previous research to focus completely on the impact of incomplete information.
In the following subsections, the detailed structure of preferences, choices and the full timeline of the model will be explained. First and foremost, we define the state of the world as 𝜃 and assume it is normally distributed with variance normalized to one: 𝜃 ~ 𝑁(0,1).
3.1
The Government
In this model, we assume that the Government has no conflict of interest with the agents. In fact, the Government maximizes the aggregate utility of the economy. Let 𝑈𝑖 be the preferences of agent i, the Government preferences are:
𝑈𝐺(𝜃) = ∫ 𝑈𝑖(𝑘𝑖, 𝑝𝑖, 𝜃) 𝑑𝑖
1 0
(1)
As stated before, previous research includes differences between what the Government and the agents want to achieve. Such feature is not present in this model to keep focus on the issue of incomplete information: different information sets for the Government and the agents. The Government does not know the true value of 𝜃. Instead, the Government makes its own forecast about the future state of the world. We assume such forecast is unbiased and that its variance can take two values. Formally, the forecast is 𝑓𝐺 = 𝜃 + 𝜀, with 𝜀 ~ 𝑁(0, 𝜎𝜀2), 𝜎𝜀2 ∈
{𝜎𝐿2, 𝜎
𝐻2} and 𝜎𝐿2 < 𝜎𝐻2. The Government knows the degree of its uncertainty (the true value of
𝜎𝜀2), but the agents do not. Thus, we assume the prior probability distribution of 𝜎𝜀2 is
𝑃𝑟𝑜𝑏{𝜎𝜀2 = 𝜎𝐿2} = 𝜆 and 𝑃𝑟𝑜𝑏{𝜎𝜀2 = 𝜎𝐻2} = 1 − 𝜆.
Before knowing its own forecast, the Government has to choose its communication policy. The policy consists of sending 𝑛 signals of public information for private agents. As the Government does not know 𝜃, such signals (𝑔𝑙) are centred around its forecast and transmitted with some noise: 𝑔𝑙 = 𝑓𝐺 + 𝜂𝑙, ∀𝑙 = 1,2, … , 𝑛, with 𝜂𝑙~𝑁(0, 𝜎𝜂2).
The Government has freedom to choose the value of 𝑛 and 𝜎𝜂2. However, we assume
that the process of transmitting information is endemically noisy and imperfect and restrict 𝜎𝜂2 ≥ 𝜎𝜂2 > 0. Also, all shocks to the signals (𝜂𝑙) are assumed to be independent.
3.2
The Agents
We define agent i’s actions in the last game as 𝑝𝑖. The agent desire is to get its action as close as possible to the state of the world 𝜃. For example, 𝑝𝑖 may be the agent i’s forecast of
future inflation and 𝜃, inflation in the future. Furthermore, we assume acquiring information is costly and define 𝐶(𝑘) = 𝑐𝑘 as the cost of acquiring 𝑘 public signals. Thus, the agent i’s preferences are:
𝑼𝒊(𝒌𝒊, 𝒑𝒊, 𝜽) = −(𝒑𝒊− 𝜽)𝟐− 𝑪(𝒌
𝒊) = −(𝒑𝒊− 𝜽)𝟐− 𝒄𝒌𝒊 (𝟐)
Agents, evidently, cannot acquire more public information than provided by the Government: 𝒌𝒊
≤ 𝑛. Signals can both be acquired randomly or agents can choose which ones
to obtain. The possibility of direct searching can lead to multiple equilibria in similar models, but this problem is a consequence of agent’s desire to coordinate. In this model, coordination is not an issue and only the amount of information acquired is important. (Hellwig & Veldkamp, 2009) (Chahrour, 2014)
In addition, the linear simplification of the cost function of obtaining signals is not so restrictive as the benefits of more information are concave in this model. The results would only be affected by this assumption if the real cost of acquiring more information is even more concave than the benefits, which is unlikely.
Each agent also makes an unbiased forecast regarding the future state of the world. Formally, agent i receives 𝑓𝑖 = 𝜃 + 𝜉𝑖, where 𝜉𝑖 ~ 𝑁(0, 𝜎𝜉2) and 𝜎𝜉2 > 0. We assume that all shocks to agents’ forecast are not only identically, but also independently distributed. Also, agents are assumed to have difficulties in sharing forecasts. Therefore, each agent’s forecast is his own private information.
Furthermore, agents have beliefs about the type of the Government even before they receive their forecasts and public signals. The beliefs of each agent can be represented as a function of the communication policy: 𝜆𝑖(𝑛, 𝜎𝜂2) = 𝑃𝑟𝑜𝑏{𝜎
𝜀2 = 𝜎𝐿2| 𝑛, 𝜎𝜂2}.
3.3
Timeline:
Lastly, the model’s timeline, defining the order actions are taken by each player, is described below. There are four stages, or periods, in the model, but players take action in only
three of them. We included a third period in which shocks are realized and information is acquired:
1st period: Government knows the degree of his uncertainty (the value of 𝜎
𝜀2).
Also, the Government chooses his communication policy: the amount of information to send to private agents (the value of 𝑛), and its dispersion (the value of 𝜎𝜂2).
2nd period: Agents observe the Government’s communication policy, and
decide how much of information to acquire (chooses 𝒌𝒊) according to his beliefs
on the value of the Government’s uncertainty (𝜆𝑖(𝑛, 𝜎𝜂2) = 𝑃𝑟𝑜𝑏{𝜎 𝜀2 =
𝜎𝐿2| 𝑛, 𝜎 𝜂2}).
3rd period: Shocks are realized, Government compiles his forecast (𝑓
𝐺) and
sends his signals to private agents according to the previously decided communication policy (𝑛, 𝜎𝜂2). Agents also make their forecasts (𝑓𝑖).
4th period: Agents choose their actions (𝑝
𝑖) given their available information
(ℐ𝑖). ℐ𝑖 is defined formally as the set of the agent private forecast and the public signals the agent has acquired.
The state of the economy (𝜃) is not revealed in this whole process. Only after all the actions are taken, it is realized and the final payoffs are determined. Throughout the whole game, there is uncertainty regarding the future.
4
Equilibrium and Results
Before solving the model, the last step is to define both what equilibrium is in this specific setting and what is a social optimum outcome. Given the model is a game of incomplete information with signalling characteristics, equilibrium is defined on the lines of the definitions of Perfect Bayesian Equilibrium. The adaptation for this model of the definition provided by Fudenberg & Tirole (1991) follows below:
EQUILIBRIUM – The set of actions {𝑛∗, 𝜎 𝜂2
∗
, 𝑘∗, 𝑝∗} is a pure strategy symmetric equilibrium
if: 1) 𝑝∗ = 𝑝∗(𝑘∗, 𝑛∗, 𝜎𝜂2 ∗ , ℐ𝑖) = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑝𝑖 𝔼[𝑈𝑖(𝑘∗, 𝑝 𝑖)|𝜆𝑖(𝑛∗, 𝜎𝜂2 ∗ ), ℐ𝑖] (3) 2) 𝑘∗ = 𝑘∗(𝑛∗, 𝜎𝜂2 ∗ ) = 𝑎𝑟𝑔𝑚𝑎𝑥 𝒌𝒊 𝔼 [𝑈𝑖(𝒌𝒊, 𝑝∗(𝑘∗, 𝑛∗, 𝜎 𝜂2 ∗ , ℐ𝑖))| 𝜆𝑖(𝑛∗, 𝜎 𝜂2 ∗ )] 𝑠. 𝑡. 0 ≤ 𝒌𝒊 ≤ 𝑛 (4) 3) (𝑛∗, 𝜎𝜂2 ∗ ) = (𝑛∗(𝜎 𝜀2), 𝜎𝜂2 ∗ (𝜎𝜀2)) = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑛,𝜎𝜂2 𝔼 [𝑈𝐺(𝑘∗(𝑛∗, 𝜎 𝜂2 ∗ ), 𝑝∗(𝑘∗, 𝑛∗, 𝜎 𝜂2 ∗ , ℐ𝑖))| 𝜎𝜀2] (5)
4) Agents’ beliefs in the second period are consistent with the Government strategies
(𝑛∗(𝜎 𝜀2), 𝜎𝜂2
∗
(𝜎𝜀2)). For example, if only the Government with low uncertainty (𝜎 𝐿2)
would choose (𝑛𝐿, 𝜎𝜂2) than 𝜆𝑖(𝑛𝐿, 𝜎𝜂2) = 1.
5) Let 𝑝𝑑𝑓(ℐ𝑖|𝜎𝜀2) be the probability density function of ℐ𝑖 given 𝜎𝜀2. Agents’ beliefs in the last period must be updated using Bayes’ rule:
𝜆𝑖ℐ𝑖 = 𝑃𝑟𝑜𝑏{𝜎 𝜀2 = 𝜎𝐿2| 𝑛, 𝜎𝜂2, ℐ𝑖} = = 𝜆𝑖(𝑛, 𝜎𝜂 2) 𝜆𝑖(𝑛, 𝜎𝜂2) + (1 − 𝜆𝑖(𝑛, 𝜎𝜂2))𝑝𝑑𝑓(ℐ𝑖|𝜎𝜀 2 = 𝜎 𝐻2) 𝑝𝑑𝑓(ℐ𝑖|𝜎𝜀2 = 𝜎 𝐿2) (6)
In the last period, the agent’s optimal action is easily derived by maximizing the agent’s utility given the available information and the beliefs in the previous periods. Agents will choose 𝑝𝑖∗(𝒌
𝒊, 𝑛, 𝜎𝜂2, ℐ𝑖) = 𝔼[𝜃|𝜆𝑖(𝑛, 𝜎𝜂2), ℐ𝑖]. As all the random variables are independent and
normally distributed, we have:1,2 𝔼[𝜃|𝜆𝑖(𝑛, 𝜎𝜂2), ℐ 𝑖] = 𝜆𝑖 ℐ𝑖𝔼[𝜃|𝜎 𝜀2 = 𝜎𝐿2, ℐ𝑖] + (1 − 𝜆𝑖 ℐ𝑖)𝔼[𝜃|𝜎 𝜀2 = 𝜎𝐻2, ℐ𝑖] (7) 1 In solving for 𝔼[𝜃|𝜎
𝜀2, ℐ𝑖], we assumed that agents receive the k first public signals, instead of choosing
which ones to obtain. This assumption is just a way to simplify notation. As discussed before, without desire for coordination, only the number of public signals acquired matter.
2 Appendix A deals with the derivation of 𝔼[𝜃|𝜎
𝔼[𝜃|𝜎𝜀2, ℐ 𝑖] = 𝛾1(𝜎𝜀2, 𝒌𝒊, 𝜎𝜂2) ∑ 𝑔𝑙 𝒌𝒊 𝑙=1 + 𝛾2(𝜎𝜀2, 𝒌 𝒊, 𝜎𝜂2)𝑓𝑖 (8) Where 𝛾1(𝜎𝜀2, 𝑘, 𝜎𝜂2) = 𝜎𝜉2 (1+𝜎𝜉2)(𝑘𝜎 𝜀2+𝜎𝜂2)+𝑘𝜎𝜉2 and 𝛾2(𝜎𝜀2, 𝑘, 𝜎𝜂2) =𝑘𝜎𝜀 2+𝜎 𝜂2 𝜎𝜉2 𝛾1(𝜎𝜀 2, 𝑘, 𝜎 𝜂2).
As a way to simplify notation, take the function ℎ(𝛼, 𝑥, 𝑦) = 𝛼𝑥 + (1 − 𝛼)𝑦, which creates a convex combination of 𝑥 and 𝑦.And let 𝛾1𝐿 = 𝛾1(𝜎𝐿2, 𝑘, 𝜎𝜂2), 𝛾1𝐻 = 𝛾1(𝜎𝐻2, 𝑘, 𝜎𝜂2), 𝛾2𝐿 = 𝛾2(𝜎𝐿2, 𝑘, 𝜎𝜂2) and 𝛾2𝐻= 𝛾2(𝜎𝐻2, 𝑘, 𝜎𝜂2). 3 Then, the agent’s best action in the last period
is: 𝑝∗(𝑘, 𝑛, 𝜎 𝜂2, ℐ𝑖) = ℎ(𝜆𝑖ℐ𝑖, 𝛾1𝐿, 𝛾1𝐻) ∑ 𝑔𝑙 𝑘 𝑙=1 + ℎ(𝜆𝑖ℐ𝑖, 𝛾 2𝐿, 𝛾2𝐻)𝑓𝑖 (9)
Therefore, the remaining problem of the agent, choosing 𝑘, becomes:
𝑚𝑎𝑥 𝑘 𝔼 [− (ℎ(𝜆𝑖 ℐ𝑖, 𝛾 1𝐿, 𝛾1𝐻) ∑ 𝑔𝑙 𝑘 𝑙=1 + ℎ(𝜆𝑖ℐ𝑖, 𝛾 2𝐿, 𝛾2𝐻)𝑓𝑖− 𝜃) 2 − 𝑐𝑘| 𝜆𝑖(𝑛∗, 𝜎𝜂2∗)] 𝑠. 𝑡. 0 ≤ 𝑘 ≤ 𝑛(10)
This agent’s maximization problem can be approximated by a second-order Taylor expansion, as the derivations in Appendix B shows. The simplification results in:
𝑚𝑎𝑥 𝑘 𝔼[𝑈𝑖(𝑘)|𝜆𝑖(𝑛, 𝜎𝜂 2)] 𝑠. 𝑡. 0 ≤ 𝑘 ≤ 𝑛 = 𝑚𝑎𝑥 𝑘 − (𝑘ℎ(𝜆𝑖 ⦁, 𝛾 1𝐿, 𝛾1𝐻) + ℎ(𝜆𝑖⦁, 𝛾2𝐿, 𝛾2𝐻) − 1)2− 𝑘ℎ(𝜆⦁𝑖, 𝛾1𝐿, 𝛾1𝐻)2𝜎𝜂2− 𝑘2ℎ(𝜆 𝑖 ⦁, 𝛾 1𝐿, 𝛾1𝐻)2𝔼𝜆𝑖[𝜎𝜀2] − ℎ(𝜆𝑖⦁, 𝛾2𝐿, 𝛾2𝐻)2𝜎𝜉 2− 𝑐𝑘 𝑠. 𝑡. 0 ≤ 𝑘 ≤ 𝑛 (11) Where 𝔼𝜆𝑖[𝜎𝜀2] = ℎ(𝜆 𝑖(𝑛, 𝜎𝜂2), 𝜎𝐿2, 𝜎𝐻2) and 𝜆𝑖⦁= 𝜆𝑖ℐ𝑖|𝝎=𝟎= 𝜆𝑖(𝑛,𝜎𝜂2) 𝜆𝑖(𝑛,𝜎𝜂2)+(1−𝜆𝑖(𝑛,𝜎𝜂2))√𝛾1 𝐻 𝛾1𝐿 .
Until now, both variables 𝑘 and 𝑛 has been assumed discrete. Nonetheless, the model has a continuous version as well and such equivalence is proved in the Appendix C. For simplicity, we treat 𝑘 and 𝑛 as continuous variables from now on.
ASSUMPTION 1 – The cost of information is not too large: 𝑐 = 1 𝜎̅𝜂2(1 +𝜎1 𝜉2) 2 (12) For some 𝜎̅𝜂2 > 𝜎𝜂2.
This is a trivial assumption that assures that costs of obtaining public information are not prohibitive. Agents will want to acquire some public signals every time 𝜎𝜂2 < 𝜎̅𝜂2. If the
assumption failed for all 𝜎𝜂2 > 𝜎
𝜂2, (e.g. 𝑐 > 1 𝜎𝜂2(1+1
𝜎𝜉2)
2), agents would never acquire any public
signals and the model would be fruitless.
Before getting into the results of the model, we need also to define social optimal actions to use as benchmark:
SOCIAL OPTIMUM – The social optimal set of actions is (𝑘, 𝑝𝑖, 𝜎𝜂2) = (𝑘∘, 𝑝∘(ℐ𝑖), 𝜎𝜂2°),
where 𝑘∘ is the number of public signals acquired by agents, 𝑝∘(ℐ𝑖) is a well-defined function of the agent’s information set to the action 𝑝𝑖 and (𝑘∘, 𝑝∘(ℐ𝑖), 𝜎𝜂2°) satisfy:
(𝑘∘, 𝑝∘(ℐ
𝑖), 𝜎𝜂2°) = argmax 𝑘,𝑝(ℐ𝑖),𝜎𝜂2
[𝑈𝐺(𝑘, 𝑝(ℐ𝑖∗), 𝜎
𝜂2)|𝜎𝜀2] (13)
The social optimal pair of actions is the same that a social planner would choose in a world without incomplete information, but with costly acquisition of information and uncertainty regarding the future state of the economy. The number of public signals sent, 𝑛, is excluded from the definition because it can easily be changed by the social planner to allow the agent to choose 𝑘∘. We use this definition as it can, clearly, show if there is any inefficiency arising with the introduction of incomplete information.
4.1
First-Best Scenario
PROPOSITION 1 – In a First-Best scenario where agents know the true value 𝜎𝜀2, the symmetric equilibrium actions are:
1) 𝜆𝑖𝐹𝐵(𝑛, 𝜎𝜂2) = 𝜆𝑖ℐ𝑖 𝐹𝐵 = {1, 𝑖𝑓 𝜎𝜀2= 𝜎𝐿2 0, 𝑖𝑓 𝜎𝜀2= 𝜎𝐻2 (14) 2) 𝑝𝐹𝐵(𝜎𝜀2) = 𝛾1(𝜎𝜀2, 𝑘𝐹𝐵(𝜎𝜀2), 𝜎𝜂2) ∑𝑘𝑙=1𝑔𝑙+ 𝛾2(𝜎𝜀2, 𝑘𝐹𝐵(𝜎𝜀2), 𝜎𝜂2)𝑓𝑖 (15) 3) 𝑘𝐹𝐵(𝜎 𝜀2) = 𝑘̈(𝜎𝜀2, 𝜎𝜂2) = √ 𝜎𝜂2 𝑐 ⁄ − 𝜎𝜂2(1+𝜎𝜉−2 ) 1+𝜎𝜀2(1+𝜎𝜉−2) (16) 4) 𝑛𝐹𝐵∈ [𝑘̈(𝜎𝜀2, 𝜎𝜂2), ∞) (17) 5) 𝜎𝜂2𝐹𝐵 = 𝜎𝜂2 (18) s
Proof. 1. As the agents know the true value of 𝜎𝜀2, their beliefs must be in accordance with the
truth and the result follows directly.
2. The formula for 𝑝𝐹𝐵(𝜎
𝜀2) is easily obtained by substituting (14), 𝑘𝐹𝐵 and 𝜎𝜂2𝐹𝐵 = 𝜎𝜂2
into expression (9).
4. By solving the agent’s maximization problem in (11), we obtain the best-action function:4 𝑘∗(𝜎 𝜀2, 𝜎𝜂2) = { 0, 𝑖𝑓 𝑘̈(𝜎𝜀2, 𝜎 𝜂2) < 0 𝑘̈(𝜎𝜀2, 𝜎𝜂2) = √ 𝜎𝜂2 𝑐 ⁄ − 𝜎𝜂2(1 + 𝜎𝜉−2 ) 1 + 𝜎𝜀2(1 + 𝜎𝜉−2) , 𝑖𝑓 0 ≤ 𝑘̈(𝜎𝜀2, 𝜎𝜂2) ≤ 𝑛 𝑛, 𝑖𝑓 𝑘̈(𝜎𝜀2, 𝜎 𝜂2) > 𝑛 (19)
Concavity in 𝑘 of 𝔼[𝑈𝑖(𝑘, 𝜃)|𝜎𝜀2] is still necessary to ensure 𝑘∗(𝜎𝜀2, 𝜎𝜂2) really solves
the problem. Indeed, 𝔼[𝑈𝑖(𝑘, 𝜃)|𝜎𝜀2] is concave in 𝑘:
𝜕2𝔼[𝑈 𝑖(𝑘, 𝜃)|𝜎𝜀2] 𝜕𝑘2 = −2𝛾1(𝜎𝜀2, 𝑘, 𝜎𝜂2) 2 ( (1 + 𝜎𝜉 2)𝜎 𝜀2+ 𝜎𝜉2 (1 + 𝜎𝜉2)(𝑘𝜎 𝜀2+ 𝜎𝜂2) + 𝑘𝜎𝜉2 ) < 0 (20)
With this in mind, the problem of the Government becomes: max 𝑛,𝜎𝜂2 𝔼[𝑈𝐺(𝑛, 𝜎𝜂2, 𝜃)|𝜎𝜀2] = 𝔼 [∫ 𝑈𝑖(𝑘∗(𝜎𝜀2, 𝜎𝜂2), 𝜃) 1 0 𝑑𝑖 |𝜎𝜀2] (21) With 𝜕𝔼[𝑈𝐺(𝑛, 𝜎𝜂2, 𝜃)|𝜎 𝜀2] 𝜕𝑛 = ∫ 𝜕𝔼[𝑈𝑖(𝑘∗(𝜎 𝜀2, 𝜎𝜂2), 𝜃)|𝜎𝜀2] 𝜕𝑘 𝜕𝑘∗(𝜎 𝜀2, 𝜎𝜂2) 𝜕𝑛 𝑑𝑖 1 0 = {∫ 𝜕𝔼[𝑈𝑖(𝑘∗(𝜎 𝜀2, 𝜎𝜂2), 𝜃)|𝜎𝜀2] 𝜕𝑘 𝑑𝑖 1 0 > 0, 𝑖𝑓 𝑘̈(𝜎𝜀2, 𝜎 𝜂2) > 𝑛 0, 𝑖𝑓 𝑛 ≥ 𝑘̈(𝜎𝜀2, 𝜎 𝜂2) (22)
Clearly, the Government will choose any possible 𝑛 ≥ 𝑘̈(𝜎𝜀2, 𝜎 𝜂2).
5. As the Government does not restrict the acquisition of information by the agents in equilibrium, we can use the Envelope Theorem, for 𝜎𝜂2 ≤ 𝜎̅
𝜂2: 𝜕𝔼[𝑈𝐺(𝑛, 𝜎𝜂2, 𝜃)|𝜎 𝜀2] 𝜕𝜎𝜂2 = ∫ 𝜕𝔼[𝑈𝑖(𝑘∗(𝜎 𝜀2, 𝜎𝜂2), 𝜃)|𝜎𝜀2] 𝜕𝜎𝜂2 𝑑𝑖 1 0 = −𝛾1(𝜎𝜀2, 𝑘𝐹𝐵, 𝜎 𝜂2) 2 𝑘∗(𝜎 𝜀2, 𝜎𝜂2) < 0(23)
In addition, when 𝜎𝜂2 > 𝜎𝜂2, 𝑘∗(𝜎𝜀2, 𝜎𝜂2) = 0 and 𝜕𝑘
∗(𝜎 𝜀2,𝜎𝜂2)
𝜕𝜎𝜂2 = 0. So equation (23) hold
for 𝜎𝜂2 > 𝜎𝜂2 as well. Therefore, the Government will always choose the smallest possible
dispersion of public signals.
3. Combining the results (4) and (5) with the best-action function of the agent, (19), we find the value of 𝑘𝐹𝐵(𝜎
𝜀2).
COROLLARY 1 – In the first-best scenario, agents acquire less public information when the
Government is more uncertain:
𝑘𝐹𝐵(𝜎
𝐻2) < 𝑘𝐹𝐵(𝜎𝐿2) (24)
This results follows directly from the formula for 𝑘̈(𝜎𝜀2, 𝜎
𝜂2). Indeed, the agent’s actions
in the model is in accordance with intuition. If the Government is less uncertain, public signals help more to predict the future state of the economy and agents acquire more of them. Moreover, when agent’s forecasts become better (𝜎𝜉2 decreases), public signals add less to the agent’s
information set and, then, are acquired in smaller numbers. The dispersion of public signals, however, have an ambiguous effect on 𝑘̈(𝜎𝜀2, 𝜎𝜂2). If costs are small enough, an increase in 𝜎𝜂2
may cause agents to acquire more public signals. Higher 𝜎𝜂2 makes more public signals necessary to better predict the state of the world and this may trump the costs of more information. Nevertheless, if 𝜎𝜂2 and 𝑐 are big enough, any increase in 𝜎𝜂2 will tend to make public information marginally costlier than beneficial.
The Government behaviour can be easily explained as well. In the first-best scenario, Government preferences are the same as the agents’. Agents actions will also be optimal in the view of the Government as they know the true value of 𝜎𝜀2. Therefore, the Government chooses
a communication policy that not only allow agents to freely acquire of public information, but also increases the value of such information (smallest 𝜎𝜂2).
COROLLARY 2 – The first-best outcome is the social optimal.
Proof. The social planner’s preferences are the same as the Government’s, which does not conflict with the agent’s. Thus, if the following three conditions are met, we have a social optimum:
The choice of 𝑝𝑖 is made optimally by the agent taking into account ℐ𝑖 and the true value of 𝜎𝜀2.
The choice of 𝑘 is the same the agent would choose if they have known the true value of 𝜎𝜀2 and 𝑛 → ∞.
□ The social optimal outcome, as defined before, maximizes the Government preferences in regard to the amount of public information acquired by the agents, the agents’ actions on the last period, and the dispersion of public signals given the Government level of uncertainty. In the first-best case, there is conflict of interest between Government and agents and there is no different in the information sets of them in the game. Therefore, it is intuitive that the first-best outcome would be the first-best possible by the social point of view.
An interesting result in the first-best is that the Government would always prefer to choose the smallest dispersion of signals possible. Under no circumstances, a highly dispersed communication policy is desirable. In a practical sense, it would be best for the Government to provide the right amount of information in a compact and concise way, for example, in a highly detailed direct statement.
4.2
Second-Best Scenario
Introducing incomplete information, we achieve the model’s main result:
PROPOSITION 2 – Even when agents do not know the true value of 𝜎𝜀2, the first-best outcome,
which is also social optimum, is an equilibrium. Formally, the set of actions below form an equilibrium 1) 𝜆𝑖𝑆𝐵(𝑛, 𝜎𝜂2) = { 1, 𝑖𝑓 (𝑛, 𝜎𝜂2) = (𝑛𝐿𝑆𝐵, 𝜎𝜂2) 0, 𝑖𝑓 (𝑛, 𝜎𝜂2) = (𝑛𝐻𝑆𝐵, 𝜎𝜂2) 𝑖𝑟𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (25) 2) 𝑝𝑆𝐵(𝜆𝑖𝑆𝐵(𝑛, 𝜎𝜂2), ℐ𝑖) = ℎ(𝜆𝑖ℐ𝑖, 𝛾1𝐿, 𝛾1𝐻) ∑𝑘𝑙=1𝑔𝑙+ ℎ(𝜆𝑖ℐ𝑖, 𝛾2𝐿, 𝛾2𝐻)𝑓𝑖 (26) 3) 𝑘𝐿𝑆𝐵= 𝑘𝐹𝐵(𝜎𝐿2), 𝑘𝐻𝑆𝐵= 𝑘𝐹𝐵(𝜎𝐻2) (27) 4) 𝑛𝐿𝑆𝐵∈ [𝑘𝐿𝑆𝐵, ∞), 𝑛𝐻𝑆𝐵∈ [𝑘𝐻𝑆𝐵, ∞), 𝑏𝑢𝑡 𝑛𝐿𝑆𝐵≠ 𝑛𝐻𝑆𝐵 (28) 5) 𝜎𝜂2𝑆𝐵= 𝜎𝜂2 (29)
Proof. Firstly, the choice of 𝑝 is the optimal choice for the agent regardless of the value of the other variables, as we have derived in the beginning of Section IV. Now, take the Government choices of 𝑛, (28), and 𝜎𝜂2, (29). By choosing different amount of public signals when the public
uncertainty is low compared to when it is high, the agents’ beliefs are restricted to (25) when in equilibrium.
When beliefs are those specified in (25), the agents’ expected utility will be the same as the first-best case when (𝑛, 𝜎𝜂2) = (𝑛𝐿𝑆𝐵, 𝜎𝜂2) or (𝑛, 𝜎𝜂2) = (𝑛𝐿𝑆𝐵, 𝜎𝜂2). Then, it is clear that choosing the same 𝑘 as in the first-best scenario is optimal for the agents. Therefore, given the Government strategies, the agent will optimally choose the first-best actions and will not want to deviate.
The remaining question is if the Government will deviate or not. To answer this question, note that, when the Government chooses the communication policy given the agents’ beliefs, it is implicitly choosing a set of (𝑘, 𝜎𝜂2, 𝜆𝑖) which maximizes its preferences given the optimal action of 𝑝 and the restrictions of 𝑘 maximizing the restricted problem of the agent. Thus, if we can show that the government will choose (𝑘𝐿𝑆𝐵, 𝜎𝜂2, 1), if 𝜎𝜀2 = 𝜎𝐿2, and (𝑘𝐻𝑆𝐵, 𝜎
𝜂2, 0), if 𝜎𝜀2 = 𝜎𝐻2, in the unrestricted model below, there is no incentives for the
Government to deviate: 𝑚𝑎𝑥
(𝑘,𝜎𝜂2,𝜆 𝑖)
𝔼[𝑈𝐺(𝑘, 𝑝𝑆𝐵(𝜆𝑖, ℐ𝑖))|𝜆𝑖, 𝜎𝜀2] 𝑠. 𝑡. 𝑘 ≥ 0 (30)
As a first step, we can conclude that the Government would choose 𝑘 strictly positive in this problem, because:
𝜕𝔼[𝑈𝐺(𝑘, 𝑝𝑆𝐵(𝜆𝑖, ℐ𝑖))|𝜆𝑖, 𝜎𝜀2] 𝜕𝑘 |𝑘=0= 1 𝜎𝜂2(1 + 1 𝜎𝜉2) 2− 𝑐 > 0, ∀𝜎𝜂2 ≤ 𝜎̅𝜂2 (31)
Then, we can assume that in the optimal choices, 𝑘 > 0. Now, we take the derivative of the Government preferences in relation to 𝜆𝑖:
𝜕𝔼[𝑈𝐺(𝑘, 𝑝𝑆𝐵(𝜆𝑖, ℐ𝑖))|𝜆𝑖, 𝜎𝜀2] 𝜕𝜆𝑖 = 2𝜕𝜆𝑖 ℐ𝑖 𝜕𝜆𝑖 ℎ(𝜆𝑖 ℐ𝑖, 𝛾 1𝐿[(𝑘𝜎𝐿2+ 𝜎𝜂2)χ + (𝑘𝜎𝜀2 + 𝜎𝜂2)𝜔], 𝛾1𝐻[(𝑘𝜎𝐻2 + 𝜎𝜂2)χ + (𝑘𝜎𝜀2+ 𝜎𝜂2)𝜔]) (32)
(𝑘𝜎𝜀2 + 𝜎𝜂2)𝜔 ≥ 0 and negative if (𝑘𝜎𝐻2+ 𝜎𝜂2)χ + (𝑘𝜎𝜀2+ 𝜎𝜂2)𝜔 ≤ 0. Analysing the value of
the derivative for 𝜎𝜀2 = 𝜎
𝐿2 and 𝜎𝜀2 = 𝜎𝐻2 and using χ + 𝜔 = 0. We can determine:
𝜕𝔼[𝑈𝐺(𝑘, 𝑝𝑆𝐵(𝜆 𝑖, ℐ𝑖))|𝜆𝑖, 𝜎𝜀2] 𝜕𝜆𝑖 { > 0, 𝑖𝑓 𝜎𝜀2 = 𝜎 𝐿2 < 0, 𝑖𝑓 𝜎𝜀2 = 𝜎𝐻2 (33) With this result, we can conclude that the Government would always prefer the agents’ beliefs to be in accordance to reality. In other words, 𝜆𝑖 = {1, 𝑖𝑓 𝜎𝜀
2 = 𝜎 𝐿2
0, 𝑖𝑓 𝜎𝜀2 = 𝜎𝐻2. That is what happens
when the Government chooses the strategy we set in (28) and (29). With this beliefs, the preferences of the agents and the Government becomes the same and, then, the Government would choose (𝑘𝐿𝑆𝐵, 𝜎
𝜂2, 1), if 𝜎𝜀2 = 𝜎𝐿2, and (𝑘𝐻𝑆𝐵, 𝜎𝜂2, 0), if 𝜎𝜀2 = 𝜎𝐻2, because that is the best
plan of action for the agents and because 𝜕𝔼[𝑈𝐺(𝑛,𝜎𝜂
2,𝜃)|𝜎 𝜀2]
𝜕𝜎𝜂2 < 0. In fact, the Government would
not want to deviate, solidifying the equilibrium.
□ When deciding his strategies, the Government knows that the first-best outcome is the optimal and tries to induce the agents to act accordingly. By choosing defined different actions at each level of uncertainty, the Government uses his policy to help agents determine the level of 𝜎𝜀2. In the developed setting, agents are able to fully determine the public uncertainty by the
communication policy and behave just like they would in a first-best scenario, inducing the social optimal outcome.
In the model, we used a simple discrete distribution for 𝜎𝜀2. Nevertheless, the results should not be influenced by this assumption. As the social optimal amount of information obtained by private agents is a continuous and monotonic function of 𝜎𝜀2, given 𝜎
𝜂2, we can see
that choosing 𝑛𝑆𝐵(𝜎𝜀2) = 𝑘𝐹𝐵(𝜎𝜀2) will always induce a separating equilibrium and the social optimum, even with continuous uncertainty.
The key characteristic of the model that induces this result is the lack of conflict of preferences. The Government has neither incentive to restrict the agents’ behaviour nor desire to pose as a Government with different level of uncertainty. Agents know they will not be deceived and correctly infer 𝜎𝜀2 and their expected utility.
One interesting possible extension to this work is to introduce desire for coordination between agents and the Government. If there is any mismatch on the level of coordination desired by the private and the public, the Government may want to restrict agents’ actions or try to pose as if it had more or less uncertainty.
Imagine the Government would like the agents to coordinate more, then it would be better for it to induce the agents to acquire more public signals. Then, faking being a less uncertain Government may be attractive. Such conflicts and frictions may induce pooling equilibria or an interesting role for 𝜎𝜂2, as the Government may use it to indicate his right level
of uncertainty. In the example above, the less uncertain Government would probably want to differentiate itself and could afford to use a higher 𝜎𝜂2 as a hint to the agents.
Even though, the model developed has some drawbacks, it has some interesting implications. The first one was already mentioned: in this setting, lack of conflict of interests leads to the social optimal outcome. The second one is that we introduce another dimension to the communication policy. The Government can inform the agents through not only the public signals but also by his own actions. That is what happens in the second-best scenario. The choice of communication policy alone, without the public signals being sent, informs the true value of public uncertainty and induces an optimal behaviour by the agents. For example, if a central bank’s uncertainty about future policy changes overtime, it may not be desirable to choose a one-fit-all communication policy. The lack of adaptation of the policy may cause agents to be unsure about how the characteristics of the central bank’s signals, which can lead to less efficient use of information.
5
Conclusion
This paper set out to find which kind of public communication policy would be best when the information authority faces uncertainty about the future. We developed a model of incomplete information in which public information is made available for private agents to acquire. Facing costs, agents choose how much information to obtain.
In such a setting, if information is complete, equilibrium will result in the social optimal outcome. Agents optimal choice of how much information to acquire is negatively influenced by the Government’s uncertainty. Moreover, the Government will a communication policy that does not restrict agents’ behavior and maximizes public information value by reducing their dispersion as much as possible.
When agents do not know the magnitude of public uncertainty, the Government will indirectly inform the agent of its uncertainty by choosing different communication policies when its uncertainty changes. Therefore, any mismatch of knowledge is corrected and the social optimum can also be achieved.
The lack of desire for coordination in the whole model may have influenced the results. Further research is needed to introduce this issue on this kind of setting. Any conflict of interests between private and public players may introduce interesting new dimensions for the communication policy, especially for the use of information dispersion as a signaling tool.
Without the presence of externalities or coordination, incomplete information is not enough to introduce inefficiencies in a setting where the Government tries to give information about the future to private agents. There is, however, a higher dimension to the communication policy, as it is also a form of signaling in itself. The results imply that the information authority should distinct policies for different levels of uncertainty and should always send information with the smallest dispersion possible. In other words, central banks should not keep their communication policy constant when confronted with different uncertainty about the future.
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Appendices
Appendix A
CLAIM A1 - 𝔼[𝜃|𝜎𝜀2, ℐ𝑖] = 𝛾1(𝜎𝜀2, 𝑘, 𝜎𝜂2) ∑𝑘𝑙=1𝑔𝑙+ 𝛾2(𝜎𝜀2, 𝑘, 𝜎𝜂2)𝑓𝑖, where 𝛾1(𝜎𝜀2, 𝑘, 𝜎𝜂2) = 𝜎𝜉2 (1+𝜎𝜉2)(𝑘𝜎𝜀2+𝜎 𝜂2)+𝑘𝜎𝜉2 and 𝛾2(𝜎𝜀2, 𝑘, 𝜎𝜂2) =𝑘𝜎𝜀 2+𝜎 𝜂2 𝜎𝜉2 𝛾1(𝜎𝜀2, 𝑘, 𝜎𝜂2).Proof. The state of the world and all shocks are normally distributed and independent; thus, we can say that all the forecasts and public signals received by the agent i are jointly normally distributed as: [ 𝜃 𝑓𝑖 𝑔1 𝑔2 ⋮ 𝑔𝑘](𝑘+2)×1 = 𝑨(𝑘+2)×(𝑘+3) [ 𝜃 𝜉𝑖 𝜀 𝜂1 𝜂2 ⋮ 𝜂𝑘](𝑘+3)×1 ~ 𝑁(𝟎(𝑘+2)×1, 𝑨𝚺𝑨′) (𝐴1) With 𝑨(𝑘+2)×(𝑘+3)= [ 1 1 1 1 ⋮ 1 0 1 0 0 ⋮ 0 0 0 1 1 ⋮ 1 0 0 1 0 ⋮ 0 0 0 0 1 ⋮ 0 … … … ⋱ … 0 0 0 0 ⋮ 1](𝑘+2)×(𝑘+3) (𝐴2) 𝚺(𝑘+3)×(𝑘+3)= [ 1 0 0 0 0 ⋮ 0 0 𝜎𝜉2 0 0 0 ⋮ 0 0 0 𝜎𝜀2 0 0 ⋮ 0 0 0 0 𝜎𝜂2 0 ⋮ 0 0 0 0 0 𝜎𝜂2 ⋮ 0 … … … … ⋱ … 0 0 0 0 0 ⋮ 𝜎𝜂2] (𝑘+3)×(𝑘+3) (𝐴3)
Let 𝜄𝑛′= [1 1 … 1] and Φ = (1 + 𝜎𝜀2)𝜄𝑘𝜄𝑘′ + 𝐼𝑘𝜎𝜂2, then:
𝑨𝚺𝑨′= [ 1
𝜄𝑘+1
𝜄𝑘+1′
𝚺22] (𝐴4)
By the nature of the normal multivariate distribution: 𝔼[𝜃|𝜎𝜀2, ℐ 𝑖] = 𝜄𝑘+1′ 𝚺22−1 [ 𝑓𝑖 𝑔1 𝑔2 ⋮ 𝑔𝑘]ℐ 𝑖 (𝐴5)
Pure algebra gives 𝔼[𝜃|𝜎𝜀2, ℐ𝑖] = 𝛾1(𝜎𝜀2, 𝑘, 𝜎𝜂2) ∑𝑘𝑙=1𝑔𝑙+ 𝛾2(𝜎𝜀2, 𝑘, 𝜎𝜂2)𝑓𝑖, with
𝛾1(𝜎𝜀2, 𝑘, 𝜎𝜂2) = 𝜎𝜉2 (1+𝜎𝜉2)(𝑘𝜎 𝜀2+𝜎𝜂2)+𝑘𝜎𝜉2 and 𝛾2(𝜎𝜀2, 𝑘, 𝜎𝜂2) =𝑘𝜎𝜀 2+𝜎 𝜂2 𝜎𝜉2 𝛾1(𝜎𝜀 2, 𝑘, 𝜎 𝜂2). □
Appendix B
PROPOSITION B1 – Let 𝒙 = [𝑥1 𝑥2 … 𝑥𝑛]′, where 𝑥𝑖 is a random variable, 𝔼[𝒙] = 𝟎
and 𝔼[𝒙𝒙′] = 𝛀. Assume 𝛀 is a diagonal matrix. If 𝑓(𝒙): ℝ𝑛 ⟶ ℝ is a continuous and twice differentiable function, then:
i. 𝔼[𝑓(𝒙)𝑥𝑖2] ≅ 𝑓(𝒙)|𝒙=𝟎∗ 𝔼[𝑥𝑖2], ∀𝑖 = 1,2, … , 𝑛 (𝐵1)
ii. 𝔼[𝑓(𝒙)𝑥𝑖𝑥𝑗] ≅ 0, ∀𝑖, 𝑗 = 1,2, … , 𝑛, 𝑖 ≠ 𝑗 (𝐵2)
Proof. Using a second-order Taylor approximation, we can approximate 𝔼[𝑓(𝒙)𝑥𝑖𝑥𝑗], ∀𝑖, 𝑗 = 1,2, … , 𝑛: 𝔼[𝑓(𝒙)𝑥𝑖𝑥𝑗] ≅ [𝑓(𝒙)𝑥𝑖𝑥𝑗]𝒙=𝟎+ ∑ (𝜕𝑓(𝒙)𝑥𝑖𝑥𝑗 𝜕𝑥𝑙 |𝒙=𝟎∗ 𝔼[𝑥𝑙]) 𝑛 𝑙=1 + 1 2∑ ∑ ( 𝜕2𝑓(𝒙)𝑥 𝑖𝑥𝑗 𝜕𝑥𝑙𝜕𝑥𝑘 |𝒙=𝟎∗ 𝔼[𝑥𝑙𝑥𝑘]) 𝑛 𝑘=1 𝑛 𝑙=1 (𝐵3) As [𝑓(𝒙)𝑥𝑖𝑥𝑗]𝒙=𝟎 = 0, 𝔼[𝑥𝑙] = 𝔼[𝑥𝑙𝑥𝑘] = 0, ∀𝑙, 𝑘 = 1,2, … , 𝑛 ; 𝑙 ≠ 𝑘: 𝔼[𝑓(𝒙)𝑥𝑖𝑥𝑗] ≅ 1 2∑ 𝜕2(𝑓(𝒙)𝑥 𝑖𝑥𝑗) 𝜕𝑥𝑙2 | 𝒙=𝟎 ∗ 𝔼[𝑥𝑙2] 𝑛 𝑙=1 , ∀𝑖, 𝑗 = 1,2, … , 𝑛 (𝐵4)
Note the following: 𝜕2(𝑓(𝒙)𝑥 𝑖𝑥𝑗) 𝜕𝑥𝑙2 | 𝒙=𝟎 = [𝜕 2𝑓(𝒙) 𝜕𝑥𝑙2 𝑥𝑖𝑥𝑗] 𝒙=𝟎 = 0, ∀𝑖, 𝑗, 𝑙 = 1,2, … , 𝑛; 𝑙 ≠ 𝑖 𝑎𝑛𝑑 𝑙 ≠ 𝑗 (𝐵5)
𝜕2(𝑓(𝒙)𝑥 𝑖𝑥𝑗) 𝜕𝑥𝑖2 | 𝒙=𝟎 = [𝜕2𝑓(𝒙) 𝜕𝑥𝑖2 𝑥𝑖𝑥𝑗+ 2 𝜕𝑓(𝒙) 𝜕𝑥𝑖 𝑥𝑗]𝒙=𝟎= 0, ∀𝑖, 𝑗 = 1,2, … , 𝑛, 𝑖 ≠ 𝑗 (𝐵6) And 𝜕2(𝑓(𝒙)𝑥 𝑖2) 𝜕𝑥𝑖2 | 𝒙=𝟎 = [𝜕2𝑓(𝒙) 𝜕𝑥𝑖2 𝑥𝑖2+ 4 𝜕𝑓(𝒙) 𝜕𝑥𝑖 𝑥𝑖+ 2𝑓(𝒙)]𝒙=𝟎 = 2𝑓(𝒙)|𝒙=𝟎, ∀𝑖 = 1,2, … , 𝑛 (𝐵7) Then, substituting (B5), (B6) and (B7) into (B4), when 𝑖 = 𝑗:
𝔼[𝑓(𝒙)𝑥𝑖2] ≅ 𝑓(𝒙)|𝒙=𝟎∗ 𝔼[𝑥𝑖2], ∀𝑖 = 1,2, … , 𝑛 And, when 𝑖 ≠ 𝑗:
𝔼[𝑓(𝒙)𝑥𝑖𝑥𝑗] ≅ 0, ∀𝑖, 𝑗 = 1,2, … , 𝑛, 𝑖 ≠ 𝑗
□
COROLLARY B1 – Let 𝒙 be defined as in Proposition B1. If 𝑓𝑖(𝒙): ℝ𝒏⟶ ℝ, ∀𝑖 = 1,2, … , 𝑛 are
continuous and twice differentiable, it is true that:
𝔼 [(∑ 𝑓𝑖(𝒙)𝑥𝑖 𝑛 𝑖=1 ) 2 ] ≅ ∑(𝑓𝑖(𝒙)|𝒙=𝟎)2∗ 𝔼[𝑥𝑖2] 𝑛 𝑖=1 (𝐵8) Proof. Expanding (∑𝑛𝑖=1 𝑓𝑖(𝒙)𝑥𝑖)2: 𝔼 [(∑ 𝑓𝑖(𝒙)𝑥𝑖 𝑛 𝑖=1 ) 2 ] = ∑ ∑ 𝔼[𝑓𝑖(𝒙)𝑓𝑗(𝒙)𝑥𝑖𝑥𝑖] 𝑛 𝑗=1 𝑛 𝑖=1 (𝐵9) We can rewrite 𝔼[𝑓𝑖(𝒙)𝑓𝑗(𝒙)𝑥𝑖𝑥𝑖] = 𝔼[𝑔𝑖𝑗(𝒙)𝑥𝑖𝑥𝑖], with 𝑔𝑖𝑗(𝒙): ℝ𝑛 ⟶ ℝ continuous and twice differentiable. Straightforward application of Proposition B1 gives the result (B8).
□ COROLLARY B2 – Let 𝝎 = [𝜃 𝜉𝑖 𝜀 𝜂1 𝜂2 … 𝜂𝑘]′.We can approximate the second
maximization problem of the agent as:
𝑚𝑎𝑥 𝑘 𝔼[𝑈𝑖(𝑘, 𝜃)|𝜆𝑖(𝑛, 𝜎𝜂 2)] = 𝑚𝑎𝑥 𝑘 − (𝑘ℎ(𝜆𝑖 ⦁, 𝛾 1𝐿, 𝛾1𝐻) + ℎ(𝜆𝑖⦁, 𝛾2𝐿, 𝛾2𝐻) − 1)2− 𝑘ℎ(𝜆𝑖⦁, 𝛾1𝐿, 𝛾1𝐻)2𝜎𝜂2− (𝐵10)
Where 𝔼𝜆𝑖[𝜎𝜀2] = ℎ(𝜆𝑖(𝑛, 𝜎𝜂2), 𝜎𝐿2, 𝜎𝐻2) and 𝜆𝑖⦁= 𝜆𝑖⦁(𝑘, 𝑛, 𝜎𝜂2) = 𝜆𝑖ℐ𝑖|𝝎=𝟎= 𝜆𝑖(𝑛,𝜎𝜂2) 𝜆𝑖(𝑛,𝜎𝜂2)+(1−𝜆𝑖(𝑛,𝜎𝜂2))√𝛾1 𝐻 𝛾1𝐿 .
Proof. The maximization problem can be written as: 𝑚𝑎𝑥 𝑘 𝔼[𝑈𝑖(𝑘, 𝜃)|𝜆𝑖(𝑛, 𝜎𝜂 2)] = 𝑚𝑎𝑥 𝑘 𝔼 [ − ( (𝑘ℎ(𝜆𝑖ℐ𝑖, 𝛾 1𝐿, 𝛾1𝐻) + ℎ(𝜆𝑖ℐ𝑖, 𝛾2𝐿, 𝛾2𝐻) − 1)𝜃 +ℎ(𝜆𝑖ℐ𝑖, 𝛾 1𝐿, 𝛾1𝐻) ∑ 𝜂𝑙 𝑘 𝑙=1 + 𝑘ℎ(𝜆𝑖ℐ𝑖, 𝛾 1𝐿, 𝛾1𝐻)𝜀 + ℎ(𝜆𝑖ℐ𝑖, 𝛾2𝐿, 𝛾2𝐻)𝜉𝑖 ) 2 − 𝑐𝑘 | | 𝜆𝑖(𝑛, 𝜎𝜂2) ] (𝐵11)
The result follows as a direct application of the Corollary B1. The shocks are all independent and normally distributed with mean zero. Also, all functions of the shocks are continuous and twice differentiable.
□
Appendix C
The transformation for a continuous model is analogous of the one used by Chahrour (2014). The continuous version of the model can be interpreted as the limit of a sequence of models indexed by a parameter 𝑛̇ → ∞. In this sequence, we fix parameters 𝜎̂𝜂2 =𝜎𝜂2
𝑛̇ and 𝑐̂ =
𝑛̇𝑐. Note that the cost of acquiring 𝑛̇ signals is always 𝑐̂. Also, as in Chahrour (2014), the invariance of the transformation is established by verifying that, for 𝑛̇ signals:
𝔼[(𝔼[𝜃|{𝑔𝑙, 𝑙 = 1,2, … , 𝑛̇}, 𝜎𝜀2] − 𝜃)2] = 𝜎𝜀 2+𝜎𝜂2 𝑛̇ 𝜎𝜀2+𝜎𝜂 2 𝑛̇ + 1 = 𝜎𝜀 2+ 𝜎̂ 𝜂2 𝜎𝜀2+ 𝜎̂ 𝜂2+ 1, 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝐶1)
With the invariance settled, we can rewrite the agent’s problem in term of 𝜎̂𝜂2, 𝑐̂, 𝑛 𝑛̇ and 𝑘 𝑛̇. Define 𝑛̂ = lim𝑛̇→∞ 𝑛 𝑛̇ and 𝑘̂ = lim𝑛̇→∞ 𝑘
𝑛̇, and note that 0 ≤ 𝑘̂ ≤ 𝑛̂, because 0 ≤ 𝑘 ≤ 𝑛 holds in all
𝑛̇𝛾1(𝜎𝜀2, 𝑘, 𝜎 𝜂2) = 𝜎𝜉2 (1 + 𝜎𝜉2) (𝑘 𝑛̇ 𝜎𝜀2+ 𝜎𝜂2 𝑛̇ ) +𝑘𝑛̇ 𝜎𝜉2 (𝐶2) 𝛾2(𝜎𝜀2, 𝑘, 𝜎 𝜂2) = 𝑘 𝑛̇ 𝜎𝜀2+ 𝜎𝜂2 𝑛̇ 𝜎𝜉2 𝑛̇𝛾1(𝜎𝜀2, 𝑘, 𝜎𝜂2) (𝐶3)
Taking limits of these expressions, we define: 𝛾̂1(𝜎𝜀2, 𝑘̂, 𝜎̂𝜂2) = lim𝑛̇→∞𝑛̇𝛾1(𝜎𝜀2, 𝑘, 𝜎𝜂2) = 𝜎𝜉2 (1 + 𝜎𝜉2)(𝑘̂𝜎 𝜀2+ 𝜎̂𝜂2) + 𝑘̂𝜎𝜉2 (𝐶4) 𝛾̂2(𝜎𝜀2, 𝑘̂, 𝜎̂ 𝜂2) = lim𝑛̇→∞𝛾2(𝜎𝜀2, 𝑘, 𝜎𝜂2) = 𝑘̂𝜎𝜀2+ 𝜎̂𝜂2 𝜎𝜉2 𝛾̂1(𝜎𝜀2, 𝑘̂, 𝜎̂𝜂2) (𝐶5)
Extending the procedure to the agent’s utility:
𝔼[𝑈𝑖(𝑘̂, 𝑝𝑖, 𝜃)|𝜆𝑖(𝑛̂, 𝜎̂𝜂2)] = − (𝑘 𝑛̇𝑛̇ℎ(𝜆𝑖⦁, 𝛾1𝐿, 𝛾1𝐻) + ℎ(𝜆𝑖⦁, 𝛾2𝐿, 𝛾2𝐻) − 1) 2 −𝑘 𝑛̇𝑛̇2ℎ(𝜆𝑖⦁, 𝛾1𝐿, 𝛾1𝐻)2 𝜎𝜂2 𝑛̇ − (𝑘 𝑛̇) 2 𝑛̇2ℎ(𝜆 𝑖 ⦁, 𝛾 1𝐿, 𝛾1𝐻)2𝔼𝜆𝑖[𝜎𝜀 2] − ℎ(𝜆 𝑖 ⦁, 𝛾 2𝐿, 𝛾2𝐻)2𝜎𝜉2− 𝑐̂𝑘̂ (C6) Observing 𝑛̇ℎ(𝜆𝑖⦁, 𝛾
1𝐿, 𝛾1𝐻) = ℎ(𝜆𝑖⦁, 𝑛̇𝛾1𝐿, 𝑛̇𝛾1𝐻) and taking limits of the expression above:
𝔼[𝑈𝑖(𝑘̂, 𝑝𝑖, 𝜃)|𝜆𝑖(𝑛̂, 𝜎̂𝜂2)] = −(𝑘̂ℎ(𝜆𝑖⦁, 𝛾̂ 1𝐿, 𝛾̂1𝐻) + ℎ(𝜆𝑖⦁, 𝛾̂2𝐿, 𝛾̂2𝐻) − 1) 2 − 𝑘̂ℎ(𝜆⦁𝑖, 𝛾̂ 1𝐿, 𝛾̂1𝐻)2𝜎̂𝜂2− 𝑘̂2ℎ(𝜆 𝑖 ⦁, 𝛾̂ 1𝐿, 𝛾̂1𝐻)2𝔼𝜆𝑖[𝜎𝜀2] − ℎ(𝜆𝑖 ⦁, 𝛾̂ 2𝐿, 𝛾̂2𝐻)2𝜎𝜉2− 𝑐̂𝑘̂ (𝐶7)
An exact analogous of the discrete problem, but now in continuous form. Except for the Appendix, we drop the hat from the variables throughout this paper to simplify notation.
