Bayes correlated equilibrium and inflationary bias
where 𝜋 is described as the actual inflation rate and 𝜆 is a parameter indicating the sensitivity of the EAs’ utility to the discrepancy between expected and actual inflation.
EAs update their inflation expectations based on the CB’s signals and the observed inflation rate, following an adaptive expectation mechanism
where 𝜃 measures the speed of adjustment to observed inflation and 𝜉 captures the influence of CB’s signals 𝜎 t on adjusting expectations, with 𝑠 t denoting the state of the economy.
Building on the utility function and the dynamic adjustment of expectations, we create an optimization problem that reflects EAs’ aim to maxim ize their cumulative utility over a horizon 𝑇 , considering their consumption choices and the impact of inflation expectations:
subject to their budget constraint and the evolution of inflation expectations, where 𝛿 is the discount factor reflecting future utility’s present value.
Consider a repeated interaction model Γ where in each period 𝑡 , the CB issues policy signals 𝜎 t based on the state 𝑠 t and its objectives. EAs, in turn, adjust their actions 𝑎 EA based on these signals and their expectations. This iterative process reflects the ongoing adaptation of strategies in response to evolving economic conditions and policy landscapes. The CB’s policy signal 𝜎 T at time 𝑡 is a function of its intended policy action 𝑎 CB and the current state 𝑠 t , formalized as 𝜎 t = 𝑓 CB ( 𝑎 CB , 𝑠 t ). The signals are designed to influence EAs’ expectations and actions. A BCE in a dynamic setting is achieved when, at each period 𝑡 , for all 𝑎 CB ∈ 𝐴 CB and 𝑎 EA ∈ 𝐴 EA , the condition holds. This is the same for EAs, indicating that neither player has an incentive to deviate from the strategy suggested by the CE. By continuously adapting policy signals ( 𝜎 t ) and responses ( 𝑎 EA ), both the CB and EAs participate in a cooperative dynamic that stabilizes inflation expectation, and consequently limiting actual inflation deviations from the target without resorting to suboptimal discretionary policies.
Transition from theory to quantitative simulation
In the previous section, we established a framework where the CB and EAs interact within a dynamic game, adjusting their strategies based on the economic conditions and policy signals over time. To test our hypothesis regarding the effectiveness of BCE and adopted strategies, and the impact of CB’s policy signals on inflation dynamics (and ultimately the inflationary bias), we adopt simulation tools, as an empirical test is not obvious. Specifically, we simulate for 100 time periods the evolution of the main economic variables that affect the utility functions of CBs and EAs, i.e. inflation and unemployment from an initial state, and employ BCE and randomly generated strategies to measure the cumulative utility difference between them as an indicator of effectiveness.
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