Problem 58449. Compute rational expectations in a static, linear NKM model

Consider a static, linear approximation of the baseline New Keynesian macroeconomic model. This can be described by an IS equation of the form
,
a PC equation of the form
,
a (nominal) interest rate rule of Taylor type of the form
,
and a Fisher equation of the form
.
In these equations, x, π, i, r and denote Okun's output gap, the inflation rate, the nominal interest rate, the real interest rate and the natural real interest rate respectively; a superscript indicates expectations formed by agents, so is the expected output gap and the expected inflation rate. The terms , and are white-noise shock terms; b, β, κ, and are positive parameters (with ), and additionally, the Taylor principle holds so that . is the central bank's (exogenously chosen) target inflation rate, and is the implied target nominal interest rate.
We want to compute the rationally expected (model-consistent) inflation rate and output gap and that are implied by given values of the model's parameters and a given value of . To do this, we set for any variable z for which agents form expectations (e.g. x and π), where denotes the mathematical expectations operator and where the agents' information set I contains both the structure of the model equations and the values of all parameters (as well as and ).
Since for any such z by the law of iterated expectations, and since for the white-noise shocks (where z is x, π or i), this allows us to write down the IS equation as
,
the PC equation as
,
and so on. Please solve the model in expectations (you may find it helpful to write the nominal Taylor rule in terms of the real interest rate gap for this) and write a function that, for given parameter values and a given inflation rate target, computes the model-consistent expectations and for x and π.
Bonus question 1: what can you say about the relationship of and ? What happens if ?
Bonus question 2: why is there, in fact, always a unique solution for and ?
Bonus question 3: what role does the parameter b play?
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Last Solution submitted on Jun 27, 2023

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