Bayes correlated equilibrium and inflationary bias
This is particularly applicable in monetary economics, where CB’s can operational ize this theory by setting clear inflation rates, thus creating a BCE scenario. When these signals are perceived as credible, they are decoded and acted upon by EAs, generating a coordinated response that effectively addresses the issue of time inconsistency in monetary policy. Additionally, the application of BCE to CB communications, especially through a transparent inflation targeting scheme, greatly impacts public inflation expectations. This communication strategy not only addresses the credibility problems 12 monetary authorities face, but also enhances the cooperation between the CB and EAs. The employment of BCE means CBs are able to align their forecasts with public expectations more effectively, creating a dynamic and responsive interaction. The alignment disrupts the ‘ entrenched cycle of inflation(s) ’ we proved in Section 2, facilitating a more stable economic environment. 13
Bayes correlated equilibrium
Let 𝐴 = { 𝐴 i } i ∈ N denote the set of actions available to each player in the game 𝐺 , where 𝑁 is the set of players. 14 The probability distribution 𝜇 over 𝐴 represents the correlated strategy (which the CB’s signals aim to implement, as it will be formulated later on). The utility functions of the CB and EA are represented as 𝑢 CB ( 𝑎 CB , 𝑎 EA ) and 𝑢 EA ( 𝑎 EA , 𝑎 CB ) respectively, where 𝑎 CB ∈ 𝐴 CB and 𝑎 EA ∈ 𝐴 EA . The expectation of utility under the correlated strategy 𝜇 for player 𝑖 is denoted as 𝐸 𝜇 [ 𝑢 i ( 𝑎 i , 𝑎 - i )], considering the actions of other players 𝑎 - i . We define a correlation device 𝜎 : Ω → 𝐴 that maps states of the world 𝜔 ∈ Ω to action profiles, implementing the CE. This device can be interpreted as the mechanism through which the CB’s signals are perceived and acted upon by the EAs. For all 𝑖 ∈ 𝑁 and any unilateral deviation 𝑎 i ’ from the suggested action 𝑎 i , the condition for a CE is
where the expectation is taken over the probability distribution introduced by the correlation device, given the current information set.
Extending the game formulation
A game 𝐺 involves two sets of players: the CB and EA, with 𝑁 = { 𝐶𝐵 , 𝐸𝐴 }. For each player 𝑖 ∈ 𝑁 , let 𝐴 i denote their action set, where 𝐴 CB = { 𝑎 CB , 𝑎 CB , … } corresponds to monetary policy actions (e.g. interest rate decisions), and 𝐴 EA = { 𝑎 EA , 𝑎 EA , … } corresponds to economic responses (e.g. spending levels). A correlation device Γ assigns probabilities to action profiles based on signals sent by the CB. This can be represented as mapping Γ:Ω → Δ ( 𝐴 CB × 𝐴 EA ), where Ω represents the set of possible states of the world and Δ( 𝐴 ) represents the set of probability distributions over 𝐴 . Signals 𝜎 ∈ Σ are issued by the CB to inform economic agents about the state of the world, affecting their expectations and actions. These signals adjust the probabilities of choosing certain actions, influencing the game’s outcome towards a 12 Cukierman & Meltzer 1986 discuss how credibility issues arise when monetary policy is perceived as discretionary and subject to change, and illustrates how transparent communication and consistent policy frameworks can mitigate these credibility problems by aligning CB actions with public expectations. 13 Morris & Shin 2002 explore how public signals, when coupled with private information, can realign agent behaviours towards a new equilibrium that differs from those anticipated by models based solely on private information. 14 The following notation borrows from Aumann 1987. 1 1 2 2
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