Bayes correlated equilibrium and inflationary bias
They use d Bayesian updating for the receiver’s belief upon receiving signals, allowing the receiver to update their probabilities for different states of the world based on new information. Utility functions were formulated for both the sender and the receiver, dependent on the actions taken and the state of the world, and an optimization problem was established, where the sender chooses a signalling mechanism to maximize their expected utility given the receiver’s best responses to signals. This approach brought them to the critical equations of Bayesian persuasion:
T hese equations describe the sender’s optimal signalling mechanism ( 𝜎 ∗ ) that accounts for the receiver’s updated beliefs ( 𝜇 ) and subsequent actions ( 𝑎 ). Solving this optimization problem verifies that the sender’s selected signal structure can indeed manipulate the receiver’s actions in a manner that is aligned with the sender’s preferences, thereby validating their central claim.
Bayes correlated equilibria in a dynamic setting
Consider a dynamic game 𝐺 spanning 𝑇 periods, where the CB and EA iteratively adjust their strategies based on evolving economic conditions and policy signals. Let 𝑡 ∈ {1,2, … , 𝑇 } denote the discrete time periods. Let represent the actions of the CB and EAs at time 𝑡 , respectively. The action space 𝐴 CB and 𝐴 EA include policy tools (interest rate adjustments, quantitative easing) for the CB and economic behaviours (consumption, investment) for EAs.
The state of the economy 𝑠 t ∈ 𝑆 at time 𝑡 is influenced by the actions of the CB and EAs, as well as by exogenous shocks 𝜀 t , leading to the state transition equation
The CB issues signals 𝜎 ∈ Σ at each period, correlating the strategies of the CB and EAs towards a BCE in a dynamic setting. 20 The probability of the CB choosing action 𝑎 CB given the signals 𝜎 t and state 𝑠 t is denoted by 𝜋 CB ( 𝑎 CB | 𝜎 t , 𝑠 t ), and similarly, 𝜋 EA ( 𝑎 EA | 𝜎 t , 𝑠 t ) for EAs.
The CB seeks to minimize the deviation from target inflation 𝜋 ∗ and unemployment rate 𝑢 ∗ over the horizon, leading to the optimization problem
where 𝛽 is the discount factor, and 𝜋 and 𝑢 are the inflation rate and unemployment rate at time 𝑡 , respectively.
EAs derive utility from consumption, which is subject to their inflation expectations and the actual inflation rate, influenced by the CB policies. Let 𝐶 t denote the consumption level of EAs at time 𝑡 , and represent their inflation expectations for the same period. The utility function for EAs at time 𝑡 is defined as
20 Extends the concept of BCE to dynamic games, where players interact over multiple time periods.
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