Bayesian Nash equilibria and monetary policy
The MPs’ payoff function is as follows:
Π 𝑀𝑃 = − ( 𝜃 − 𝜋 ) 2
(5)
Where: •
Π 𝑀𝑃 = payoff for MPs.
• θ = inflation expectations updated based on CB's transparency. • π = Actual inflation rate.
This payoff function represents the MPs’ primary concern of maxim izing the accuracy of their expectations relative to actual inflation, as this allows them to optimize their economic decisions such as spending or requesting higher wages. They do not incur costs directly from the CB’s transparency level, but their expectations, θ, do. To reduce inflation expectations traps, the MPs’ expectations must be in line with those of the current rate of inflation, otherwise the CB will be forced to accommodate those expectations.
As a Bayesian game, the beliefs of market participants about future inflation are adjusted based on the degree of transparency of the central bank about current and future inflation. I define the Bayesian updating principle for this game as follows:
μpost = τ
priorτ×priorμprior+τ+likeτlike×πt
(6)
Where: • μ prior : prior mean represents the market's expectation of inflation before new information is disclosed. • π t : the true inflation rate reported by the CB for the period t. This serves as the new data that will be used to update the MPs’ expectations. • τ prior : prior precision is the degree of confidence in the prior inflation expectations. A higher value indicates a higher reliability of prior beliefs. • τ like : the likelihood precision is determined by the transparency level, this precision measures how much weight the actual inflation data carries in the updating process. It is computed as . This relationship ensures that as transparency increases, the precision of the likelihood increases. This reflects the increased trust in the actual inflation data relative to prior expectations due to more communication. τ like = The outcome of this game will be the variance between expectations and actual inflation, which is a testament to the effectiveness of the CB’s communication strategy. The Bayesian Nash equilibrium is where variance is at its lowest, and neither player has an incentive to deviate. Equilibrium will only be achieved when transparency is at its optimum.
Game simulation
To simulate this game at numerous different transparency levels, I have developed the following python code (figure 6) which processes BoE and Fed data to find the optimal level of transparency for the Fed and BoE that ensures inflation expectations are in line with their target of inflation. The code
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